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Heshan Al Fares, King Fahd U. of Petroleum and Minerals, Saudi ArabiaS.S. Appadoo, U. of Manitoba, CanadaMaria Albareda Sambola, Universitat Politècnica de Catalunya (UPC), SpainArvind Bhardwaj, NIT Jalandhar, IndiaSimon Hsiang, North Carolina State U., USAHarmesh Kansal, Sant Longowal Institute of Engineering and Technology (SLIET) - Longowal, IndiaArshinder Kaur, Indian Institute of Technology Madras, IndiaAnish Kumar Sachdeva, NIT Jalandhar, India

Gilbert Laporte, HEC Montreal, CanadaM.N. Qureshi, The Maharaja Sayajirao U. of Baroda, IndiaZ. Rahman, Indian Institute of Technology Roorkee, IndiaSantosh Rangnekar, Indian Institute of Technology Roorkee, IndiaAshutosh Sarkar, Indian Institute of Technology Kharagpur, IndiaNazrul I Shaikh, The Pennsylvania State U., USAVijay Wadhwa, American Airlines Dallas - Fort Worth, USAJoe Wilck, The U. of Tennessee Knoxville, USAKi Young, U. of Houston – Clear Lake, USA

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• Business and strategy • Case studies in industry and services • Decision analysis • Engineering economy and cost estimation • Enterprise resource planning and ERPII • Facility location, layout, design, and materials handling • Forecasting, production planning, and control • Human factors, ergonomics, and safety • Industrial engineering education • Information and communication technology and systems • Innovation, knowledge management, and organizational learning • Inventory, logistics, and transportation • Knowledge and technology transfers in a globalized network • Manufacturing, control, and automation • Operations management • Performance analysis • Product and process design and management • Project management • Purchasing and procurement • Reliability and maintenance engineering • Scheduling in industry and service • Service systems and service management • Supply chain management • Systems and service modeling and simulation • Technology transfer and management • Third party/fourth party logistics • Total quality management and quality engineering

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January-June 2012, Vol. 1, No. 1

ReseaRchaRticles 1 TheSelf-RegulatoryFocusasaDeterminantofPerceivedRichnessofa CommunicationMedium VicencFernandez,UniversitatPolitècnicadeCatalunya,Spain XavierArmengol,UniversitatPolitècnicadeCatalunya,Spain PepSimo,UniversitatPolitècnicadeCatalunya,Spain

10 ApplicationoftheTheoryofConstraints(TOC)toBatchSchedulinginProcessIndustry

Dong-QingYao,TowsonUniversity,USA

23 AssociationRuleMininginDevelopmentalPsychology D.A.Nembhard,PennsylvaniaStateUniversity,USA K.K.Yip,PennsylvaniaStateUniversity,USA C.A.Stifter,PennsylvaniaStateUniversity,USA

38 Supply andProduction/DistributionPlanning inSupplyChainwithGeneticAlgorithm

BabakSohrabi,UniversityofTehran,Iran MohammadRezaSadeghiMoghadam,UniversityofTehran,Iran

55 AProductionPlanningOptimizationModelforMaximizingBattery ManufacturingProfitability HeshamK.Alfares,KingFahdUniversityofPetroleum&Minerals,SaudiArabia

64 RetailerOrderingPolicyforDeterioratingItemswithInitialInspectionand AllowableShortageUndertheConditionofPermissibleDelayinPayments ChandraK.Jaggi,UniversityofDelhi,India MandeepMittal,AmitySchoolofEngineering&Technology,India

Table of Contents

international Journal of applied industrial engineering

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International Journal of Applied Industrial Engineering, 1(1), 1-9, January-June 2012 1

Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.

Keywords: Channel Expansion, Communication Medium, Media Richness, Organizational Communication, Self-Regulatory Focus

INTRODUCTION

The scientific research field on organizational communication reflects its importance in good organizational practices, as well as in the attain-ment of individual and organizational objectives both in the short and long term (Yukl, 1989). Within this field of investigation, the study of communication media in organizations has

formed its own area of work in which a large number of theories have proliferated, inves-tigations and publications focused mainly on the reasons why managers and workers in an organization use certain communication media to the detriment of others, as well as the effi-ciency of these choices.

The main theories that explain the choice process for a communication medium in organi-zations stem from literature in the psychological area and can be represented on a continuum, placing those theories based on a rational per-

The Self-Regulatory Focus as a Determinant of Perceived Richness of a Communication Medium

Vicenc Fernandez, Universitat Politècnica de Catalunya, Spain

Xavier Armengol, Universitat Politècnica de Catalunya, Spain

Pep Simo, Universitat Politècnica de Catalunya, Spain

ABSTRACTAt present, a large number of theories exist which explain the process for choosing communication media in organizations. Channel expansion theory combines a large part of the theoretical foundation for these theories, suggesting that the perceived richness of a communication medium varies according to experience based on the knowledge of the organization’s members. Equally, Regulatory Focus Theory also suggests that individuals behave in a different way when their self regulation states are different. This investigation intends to present a set of proposals based on the existing literature about how strategy type /focus (promotion and prevention) affects the perception of the richness of a communication medium, increasing the explanatory capacity of channel expansion theory.

DOI: 10.4018/ijaie.2012010101

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2 International Journal of Applied Industrial Engineering, 1(1), 1-9, January-June 2012


spective at one end and theories that emphasize social aspects at the other. The main and most relevant theories on the choice of communica-tion media in organizations are media Richness Theory (Daft & Lengel, 1984, 1986), (Treviño, Lengel, & Daft, 1987), Social Influence Theory (Fulk, Schmitz, & Steinfield, 1990; Schmitz & Fulk, 1991), Critical Mass Theory (Oliver, Marwell, & Teixeira, 1985; Markus, 1990b), Social Presence Theory (Short, Williams, & Christie, 1976), and Channel Expansion Theory (Carlson & Zmud, 1999).

This last of these, Channel Expansion Theory, has focused a large number of inves-tigations in recent years (for example: Tim-merman & Madhavapeddi, 2008; D’ Urso & Rains, 2008) relating the proposals of theories focused on rational decision making to those of social influence in the organizations. More specifically, Carlson and Zmud (1999) suggest that decisions on the choice of communication media are taken with the perceived richness of the media in mind, which depends partly on experiences based on knowledge. On the other hand, Regulatory Focus Theory (Higgins, 1997) also suggests that individuals behave differently when their self regulation states are different (with a focus on promotion and prevention). Individuals with a high promotion focus show an attitude aimed at maximizing benefits and therefore with a tendency to advance. On the contrary, individuals with a high prevention focus usually show a monitoring state aimed at maximizing safety and therefore with a tendency to avoid losses.

The objective of this investigation is to suggest a set of proposals from the existing literature on how the type of strategy/focus (promotion and prevention) affects the percep-tion of richness of a communication media. More specifically, the intention is to propose relations between the four experiences based on knowledge of Channel Expansion Theory and Social Influence with the focuses of Regulatory Focus Theory.

Is it Still Necessary to Investigate the Choice of Communication Media in Organizations?

Presently, communication technologies play a fundamental role as communication media between members in an organization. Some communication media such as electronic mail and videoconferencing have arrived to a mature stage becoming relatively common elements and “omnipresent in contemporary organiza-tions” (Minsky & Marin, 1999). For this rea-son, it is practically impossible to consider an organization without communication by means of computer-based technologies (Curls & Gat-tiker, 2001). Some could argue that this maturity makes the study on the choice of communication through this type of technology as obsolete. However, on the other hand, it is possible to argue that the fact that these technologies and media are now so generalized makes this type of study still more relevant (van den Hooff, Groot, & de Jonge, 2005). Since communica-tion media are an integral part of organizational communication, explaining how they are used and their effects on organizational efficiency give sufficient reasons to support investigation.

Roots on Theories on Communication Media Choice

Most investigations on organizational commu-nication are based on the theoretical foundations of communication developed in the fields of psychology and sociology. Nevertheless, some authors claim that organizational communica-tion has evolved and matured into an indepen-dent discipline to those of social psychology, sociology, administration or anthropology, as demonstrated by its solid trajectory in empirical and practical investigation. The foundations of these areas have served to develop mid-range theoretical proposals such as Functional Theory (Gouran & Hirokawa, 1983, 1986, 1996), Structuration Theory (Poole, Seibold, & McPhee, 1996, 1986; Poole & Doelger,

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1986; Poole, 1999) and Symbolic Convergence Theory (Bormann, 1996) that have allowed for the development of studies with new conceptu-alizations, methodologies and their own norms of investigation.

The Evolution and Revision of Theories on Communication Media Choice

The literature on the choice of communication media identifies three theoretical approaches on the use of communication media in orga-nizations (Katz & Rice, 2002; Minsky et al., 1999): the theories of contingency, subjectivistic theories and situational theories.

Contingency theories consider the com-munication medium and the task which the broadcaster wishes to undertake the communi-cation as the main determinants in the choice of communication media (Van den Hooff et al., 2005). According to these theories, the members of an organization look to reach an optimal adjustment between the characteristics of the task which they communicate and the characteristics of communication media in order to maximize communication efficiency. Media richness theory (Daft & Lengel, 1984, 1986), included within contingency theories, proposes that the intrinsic characteristics of communication media define their appropriate-ness in satisfying the information requirements of different tasks. The results of studies by Byrne and Lemay (2006) highlight the average task appropriateness in quality measures and communication satisfaction.

The second approach proposed by Van den Hooff et al. (2005) makes reference to subjec-tivistic theories, emphasizing the importance of social context in the process of communication media choice. Subjectivistic theories assume that the tasks are, to a large extent, subjectivistic and determined by the user’s social context; that not all characteristics are equally relevant for all individuals; and that the choices do not follow a cognitive evaluation process, but rather people rationalize their decisions after they have been carried out. These suppositions are

reflected in several subjectivistic theories such as Social Influence Theory (Fulk, Steinfield, Schmitz, & Power, 1987). Among these theories the concept of critical mass is also highlighted (Markus, 1987, 1990a) which defines the level of usefulness of a communication medium on the basis of the number of useful contacts that communication medium permits. As Oliver et al. suggest (1985) it is necessary that a suf-ficient number of workers in an organization take part in a communication process for it to become a valuable collective action, and this is conditioned by the communication medium. Finally, subjectivistic theories argue that the social environments perform two functions (Kraut, Rice, Cool, & Fish, 1998): a normative environment and a social environment.

Finally, the third approach is that of situ-ational theories which emphasize the impor-tance of a series of specific characteristics of communication media, user experience and the experience in the perception of these character-istics. Although investigations exist that support the explanatory capacity of contingency and subjectivistic theories, their results show that they are still not able to fully explain the choice of communication media (van den Hooff et al., 2005). For that reason and in a complementary way, different theories appear which reflect the capacity that communication media have in overcoming certain situational limitations. These theories recognize that experience with a communication medium significantly influ-ences the way in which the characteristics of that medium are defined. In this way, Otondo, Van Scotte, Allen, and Palvia (2008) suggest that users who adhere to the beliefs of “richness” as a justification in the choice of communication media could take into account the affective motivations instead of the information process-ing capacity of the medium. For this reason, situational theories believe that users decide to use a communication medium taking into account their individual communication style, their demands, the situational task demands with which they communicate, the organiza-tional environment, and their communication experiences.

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The importance of user experience in determining the perception of richness of the media is supported by Carlson and Zmud (1999) through their theory on channel expan-sion, which argues that the perceived richness of a communication medium depends on five elements: (1) the intrinsic characteristics of the communication medium, (2) the experience of the message broadcaster with the communica-tion medium, (3) the experience of the message broadcaster with the receiver of the message (or as usually defined, communication partner), (4) the experience of the message broadcaster with the topic dealt with in the message com-municated, and (5) the experience of the message broadcaster with the organizational context where the communication takes place. Through these experiences, individuals develop a knowledge base that can be used in a more effective way to codify and de-codify richer messages through a channel. In this way, an objectively poor channel can become the most appropriate and useful for rich communication. Also, it is also proposed that social influence is an important determinant in the perception of richness in a communication medium. The results of investigation by Carlson and Zmud (1999), as well as other later works such as Timmerman and Madhavapeddi (2008) suggest that management and workers decide to use one communication medium instead of another on the basis of these five factors. Complementary theories of behavior also exist which state that individuals behave and perceive their environ-ment differently based on their attitudes or aptitudes, like regulatory focus theory.

Regulatory Focus Theory

The essence of Regulatory Focus Theory (Hig-gins, 1997, 1998) is in the notion that people are motivated to minimize differences between their present states and their desired end-states (experiment pleasure); and to maximize the differences between their present states and undesired end-states (avoid pain). Nevertheless, the theory proposed by Higgins (1997, 1998), goes beyond this basic hedonistic principle,

pointing out that the end-states can be defined in terms of (a) ideal self-guides (e.g., what does one wants to be) and (b) ought self-guides (e.g., what do others think that one should do). Those individuals who try to diminish the differences between their ideal self-guides are those that have a clear promotion focus, however those that try to minimize the differences with their self-obligations have a preventive focus. Nev-ertheless, there can also be individuals whose ideals and obligations are significantly related.

So, Regulatory Focus Theory has direct implications on individual’s self-motivation. The focus of promotion and prevention reflects different motivational states. Individuals with a promotion focus will see themselves working towards the attainment of their ideals, whereas those with a preventive focus are aiming to comply with their duties and obligations (Meyer, Becker, & Vandenberghe, 2004). The motiva-tional states associated with a promotion focus or a prevention focus can act as strong forces affecting behavior. Nevertheless, the obligations associated with a preventive focus should be perceived in a clearer and more limited way than those associated with a promotion focus. When somebody has to do something, the specifica-tions and requirements are important. The ideals tend to be less specific and can broaden when it is necessary to satisfy a continuous challenge. In fact, Higgins (1998) argues that individuals with a strong prevention focus try to satisfy or comply with the minimum of requirements, whereas those with a promotion focus try to obtain the maximum level of compliance. In addition, recent investigations show that a high promotion focus is associated with higher creativity levels (Friedman & Foerster, 2001; Lam & Chiu, 2002). However, regulatory forces can be temporarily induced based on demands (Freitas & Higgins, 2002; Liberman, Molden, Idson, & Higgins, 2001; Shah & Higgins, 2001), in spite of the existence of fixed differences between preventive and promotion focuses when conveying the objectives and activities (Higgins, Friedman, Harlow, Idson, Ayduk, & Taylor, 2001; Shah & Higgins, 1997). It is, therefore, worth considering the promotion

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focus and prevention focus as two orientations with unequal particulars in the choice and use of communication media in organizations.

Conceptual Model

According to channel expansion theory (Carlson & Zmud, 1999), the members of an organization develop knowledge bases related to the com-munication process through their own experi-ences and improving their message coding and decoding mechanisms, consequently increasing the efficiency of the communicative process. The continued use of communication media combined with their interest for the communica-tion medium generates learning on how to use that medium more efficiently. Therefore, it is hoped that individuals perceive an increase in the perception of a medium’s richness, due to the increase in the skills in the use of that com-munication medium in several, different con-texts. Also, individuals will be able to interpret messages received faster and in a more efficient way when they know all the possibilities that the communication medium is able to offer. On the other hand, those that do not develop this knowledge base, independently of the number of messages sent, will not be able to codify richer messages through these communication media and could even produce a reduction in the perception of richness in that communication medium. In this case, it is hoped that a person with a prevention or promotion focus does not affect the perception of richness for communi-cation media. For example, a person can learn to use electronic mail for many reasons: some could develop their abilities for reasons related to their principles (promotion focus), whereas others do it for defensive or obligation reasons (prevention focus).

• Proposal 1: The perception of richness of a communication medium by an individual due to their experience in that communi-cation channel will not be affected by the type of focus (promotion/prevention) that the individual has.

Continual communication with an indi-vidual can develop a common knowledge base which allows for coding and decoding messages in a more efficient way. Repeated or similar situations at work, common contexts, past an-ecdotes and comments with a double meaning permit an improvement in the communication process between members of an organization. A simple phrase can lead to confusion or mis-understandings due to different perceptions of two individuals, but a continued relationship between them diminishes the probability of this situation. Also, a simple reference to a common past can explain as much as a long and detailed narration. For this reason, it is hoped that the experience acquired in a person’s communi-cation, through one or more communication media, increases the perception of richness of that communication media. For example, two people who know each other and usually speak face-to-face will have fewer problems when communicating through another communica-tion medium (such as the telephone) than two people who do not know each other and begin a telephone conversation. Also, an individual with a promotion focus will be more motivated to acquire psycho-social knowledge about others, especially about those with whom they have a stronger relationship, which will allow them, consequently, to develop still even closer relationships with the receivers of the messages (Walther, 1992). Therefore,

• Proposal 2: The perception of the richness of communication media by an individual due to their experience with the commu-nication partner will be increased, if the broadcaster has a promotion focus rather than a prevention focus in their job.

Individuals develop experiences with those topics they communicate about on a regular basis. This fact creates a base of knowledge and terminology for the individual on the topic which later allows them to communicate in a faster and more efficient way. In cases where two individuals with a low degree of knowledge on a topic wish to communicate, they will need

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a greater amount of primary and secondary information (nonverbal information, graphical, etc.) to transmit the message. For this reason, these individuals will consider that there is a greater difference between the richness of different communication media. On the other hand, two individuals with a high degree of knowledge on a topic will need very little sec-ondary information to transmit the message, meaning that the perceived richness of several communication media will be similar. For ex-ample, two physicists will have few problems in discussing the trajectory of a stone thrown through an existing relation of kinetic energy and potential energy, they will even be able to do it with a few words; however, the same discussion between a physicist and a student might need graphs and a greater amount of detail to reach the same conclusions. Topic knowledge should be based on the student’s needs and independent of his/her attitude. For this reason it is hoped that a person with a promotion or prevention focus does not affect the perception of richness of this communication medium.

• Proposal 3: The perception of richness of a communication medium by an individual due to their experience in the messaging topic will not be affected by the focus type (promotion/prevention) that the individual has.

As individuals develop communication ex-periences within a specific organizational con-text, they also develop a knowledge base about the operation of the organization which allows them to codify messages with greater richness, so they are able to transmit complementary information in a simple, straightforward way. References, symbols and shared norms within an organization allow individuals to communicate in a more efficient manner through a commu-nication medium. For example, a standardized form to communicate incidences between the purchasing department and production depart-ment can simplify the code of the message, as well as its understanding by the receiver of the message. Another situation which reflects

this is the reference to internal abbreviations in an organization which enables a message to be reduced and a detailed explanation of its parts. Because all members of an organization are faced with the same references, symbols and norms, prevention or promotion strategies should not affect the richness of the media through an organizational context where the communication takes place. The focus type of a member of an organization does not affect the organizational context of an organization, since it depends on other determinants. For this reason, it is suggested:

• Proposal 4: The perception of richness of communication media by an individual due to their experience with the organiza-tional context will not be affected by the focus type (promotion/prevention) that the individual has.

Finally, investigation by Carlson and Zmud (1999) in relation to channel expansion theory suggests that social influence is also an impor-tant determinant in the perception of richness of a communication medium. According to models of social influence on the use of technol-ogy, individuals’ beliefs about the appropriate use of a channel and about the richness of that channel are, partly, socially constructed and therefore are subject to the influence exerted by the individuals’ environment (Fulk, Schmitz, & Steinfield, 1990). For example, a particular way of working by workers in an organization, although not written down anywhere, can affect the way in which a recently contracted person works. If everybody asks for information about their payroll through electronic mail, most probably new workers will follow the socially established rules. However, there are several investigations which have had opposite results to those expected - for example: Rice (1993). This divergence in the results of previous investiga-tions could be explained through Regulatory Focus Theory, which suggests that a promo-tion focus is related to advancement, growth and achievement, where objectives are hopes and aspirations, and that a preventive focus is

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related to safety and responsibility. Therefore, it is hoped that a relation exists between social influence and the perception of richness of a communication medium.

• Proposal 5: The perception of the richness of communication media by an individual in relation to the social influence of an organization will be affected by the focus type (promotion/prevention) that the in-dividual has.

CONCLUSION

Investigation in organizational communication, and more specifically on the choice of commu-nication media in organizations, has been quite intense in recent decades. However, the results of these investigations are, on some occasions, contradictory. This paper proposes a conceptual model stemming from the existing literature that might explain these contradictions. Chan-nel expansion theory has been complemented with the behavior and attitude of human conduct

which experience different focuses or self-regulation strategies. Figure 1 summarizes these relationships graphically. Empirical testing of the conceptual model’s proposals would allow future studies on organizational management from the self regulatory states of the members of an organization with the aim of achieving more efficient communication.

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Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.

Keywords: Case Study, Computer Software, Critical Chain, Process Industry, Scheduling, Theory of Constraint (TOC)

INTRODUCTION

Scheduling is fundamental to most organiza-tions. During the past several decades, an extensive amount of research has been con-ducted and a variety of scheduling techniques or algorithms has been done (e.g., Gang et al., 2007; Chen et al., 2008a). Many studies have focused on discrete manufacturing rather than on process industries. Since there are major differences between discrete and process in-dustries, scheduling techniques developed for discrete industries can rarely be applied directly to process industries. As a result, the process

industries continue to fall behind the discrete industries in the identification and implementa-tion of effective scheduling technique (Daina & Meredith, 2000).

Processes industries may use either con-tinuous or batch processes. Economies of scale often require large-scale equipment with a high capital investment. Batch operations usually have the following characteristics (Fransoo & Rutten, 1994): 1) long lead time, 2) much work in process (WIP), 3) less impact of changeover times, 4) a large number of production/process steps and 5) a large number of products. Ex-amples in this category are the oil, drug, and food industries. There are two reasons that make the batch scheduling much more chal-

Application of the Theory of Constraints (TOC) to Batch

Scheduling in Process IndustryDong-Qing Yao, Towson University, USA

ABSTRACTThis paper presents a practical daily batch scheduling problem at one leading global food company for multi-stage, multi-batch scheduling with no-wait. After investigated different approaches of both traditional optimization and simulation technique, the concept of the Theory of Constraint (TOC) was adopted to identify the bottleneck activity first, then the problem was converted from a multi-product batch scheduling to a multi-project critical chain based scheduling, and the scheduling technique of Drum-buffer-rope (DBR) in TOC was specifically adopted to solve the company’s routine scheduling problem. With the help of professional computer software and customized output, it is very efficient and effective for the daily scheduling personnel to operate due to no complicated algorithm or programming involved. This new TOC approach has been implemented in its several plants in headquarter and is expected to expand to other plants.

DOI: 10.4018/ijaie.2012010102

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lenging than discrete scheduling. First, batch production usually involves a large number of production protocols, or process sequences for a given product. Each production protocol contains numerous steps utilizing a variety of materials and equipment. While processing multiple batches simultaneously, the sched-ule dictates the use of particular equipment at a specific time. Equipment conflicts arise frequently. Second, some process industries, such as the food industry studied in this paper, have a no-wait constraint on batch scheduling. A no-wait constraint means no interruption or waiting is allowed during the processing of a batch between successive machines/equipment (Lin & Cheng, 2001). So as long as a batch gets started at the first stage, resources must be guaranteed to be available for all the following stages whenever they are needed. Otherwise the process has to be interrupted and serious consequences would occur that adversely affect the quality and/or productivity of the product. This constraint makes the batch scheduling very challenging for process industries.

In this paper we conducted a case study on one global manufacturer with operations in more than 30 counties. We developed a batch scheduling technique to solve the multi-stage and multi-product no-wait batch-scheduling problem, and then implemented this technique in one of its main plants. This company is the world’s largest supplier of fragrances, flavors, and colors to the food beverage, pharmaceuti-cal, cosmetic, home and personal care products, specialty printing and digital imaging industries.

LITERATURE REVIEW

In this project, we aim to minimize the total time needed to complete a group of batches from the beginning of processing on the first batch until the completion of processing on the last batch. This objective is often referred to as minimizing the “makespan” in scheduling theory. Tradition-ally mathematical programming (e.g., mixed integer programming) can be used to find optimal solutions to such scheduling problem.

For example, Sadykov and Wolsey (2006) combined integer programming and global constraint to find minimum cost assignment of jobs. Magatão et al. (2004) also developed an optimization structure to schedule activities in the pipeline industry based on mixed integer programming. For large-scale batch scheduling involving multiple batches, too many variables and constraints would have to be considered. As such, it is very difficult, if possible, to find the optimal solution to the mathematical pro-gramming. Therefore many heuristic algorithms have been widely investigated in the scheduling research to approximate the optimal solution to the mathematical programming. For example, Chen et al. (2008b) presented a hybrid approach of genetic algorithm and extremal optimization to solve a class of manufacturing scheduling problems. Guo et al. (2008) adopted bi-level genetic algorithm to solve a flexible assembly scheduling problem. Hansen and Mladenovic (2001) also introduced the application of Variable neighborhood search on scheduling problem. However, it was extremely difficult to set up constraints for no wait conditions by mathematical programming. In addition, the implementation of these kinds of algorithms would require expertise at the plant level that was not generally available.

Another approach to the batch schedul-ing problem is to use simulation (e.g., Senior, 1995; Yang et al., 2007). Simulation software (e.g., ProModel) is used to simulate the process and to obtain various scheduling sequences. The best schedule is picked from among the feasible ones. However, there are several disadvantages using the simulation approach. First, simulation is extremely time-consuming. It takes weeks or even months to get familiar with the simulation program. Afterwards, it takes another several weeks to program and prepare the batch protocols in the program for simulation runs. In order to prepare a production sequences, it usually takes hours to simulate different scheduling sequences and to come up feasible schedules. Second, simulation only provides information of resource conflicts for different simulated sequences. Thus trial and

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error method has to be taken in order to find an acceptable “good-enough” solution. In addi-tion to the fact that simulation approach is very time-consuming, it can only be used by experts familiar with the simulation software and the production protocols, not by typical operating personnel. For a plant that cannot afford to hire or train an employee to become a simulation expert this becomes a major stumbling block for implementing a simulation-based program at plant level, particularly at smaller plants.

Actually the company we studied used both mathematical programming and simula-tion techniques previously to obtain production schedules with some success. However, due to complicated algorithms and time-consuming simulations involved, the implementation has no sustainability. Therefore, the company is interested to find a relatively easy-to-use ap-proach to the batch scheduling problem that can offer “good enough” solution.

NEW APPROACH USING THE THEORY OF CONSTRAINTS (TOC)

For simple cases, a trial and error method can be used to find an optimal solution. However in real cases like what we studied in this project, a typical batch could have as many as 20 to 30 steps utilizing a variety of resources/equip-ment. Many times it is very hard to even get a non-trivial feasible scheduling solution for multi-stage, multi-product no-wait batches, let alone optimal solution. Most practitioners use a simple and quick technique to find a heuristic decision that is not likely optimal, but is “good enough.” A heuristic decision is feasible, satisfy-ing the scheduling constraints, relatively easy to develop and implement, and approaches the prescribed scheduling goals such as minimizing time and cost. Successful scheduling heuristics must be simple enough to implement in the plant for routine scheduling problems. “Good enough” is an important concept in developing large scale scheduling plan for process industries

when no proven effective algorithm may exist to ensure an optimized schedule (Leach, 2000).

The theory of constraints (TOC) developed by Goldratt (1999) does away with much of the complexity often associated with scheduling problems by simply focusing on bottleneck operations (i.e., those for which there was insufficient capacity).

The scheduling technique used in TOC is Drum-buffer-rope (DBR). The drum is the processing capability of the constraint. The buffer is placement of work-in-process (WIP) to account for process fluctuations. The rope represents the synchronization of the sequence of operations to insure effective use of the bottleneck operations (Lawrence, 2000).

One extension of Drum-buffer-rope tech-nique is “Critical Chain” for project planning and performance (Goldratt, 1997). For a project, the critical chain is the constraint. The buffers are time buffers instead of material buffers. Ropes are release of activities for work, criti-cal chain resource buffers, and decisions made in buffer management (Lawrence, 2000). The central contribution made by this paper was to convert the multi-product batch scheduling to multi-project critical chain based scheduling. Multi-project critical chain relies on the com-mon five TOC steps:

1. Identify the constraint(s).2. Exploit the constraint(s).3. Subordinate everything else to the decisions

made in step 24. Elevate the constraint(s)5. Go back to step 1.

In step 1, for batch scheduling, the system’s constraint is identified from the resource pool shared by all the batches. In step 2, we do not need to prepare a critical chain schedule for each batch since there are no resource contentions within each individual batch protocol. However we need to determine the project priority for the constraint resource, and then create the constraint resource multi-project schedule: the drum schedule.

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The approach will be illustrated next with a simplified real scheduling case from this leading global food company.

AN ILLUSTRATIVE CASE STUDY

For simplicity, we study one plant that is sched-uling 4 batches for the next production period, starting from 12:00am, July 25. All batches’ names and corresponding tasks are disguised. Table 1 summarizes the total tasks of each batch and its processing time respectively.

For each batch, the task time are certain allowing its protocol to be stored in a database. No buffer is needed between tasks since no-wait is allowed between subsequent tasks. A batch buffer might be added at the end of each batch for needed changeover time or clean-ups. The batch buffer was eliminated in the example problem for the purpose of simplicity.

Detailed protocols and resources for each batch are listed in the Appendix (Figure 4 and Tables 7 through 10). For example, the process-

ing time of batch 1 is 15 hours with 7 tasks. The duration of task T1 is 6 hours and resource R1 is needed for the operation. For each individual batch, no resource contention exists. However since all the batches share same resource pool, which is all of the available plant capacity resources, there could be tremendous resource contentions if we schedule multi-products inappropriately.

All the resources are listed in Table 2. For example, Resource R1 has 1 unit available; R2 has 2 units available, etc. All the resources are available 24 hours a day, 7 days a week.

The overall objective is to schedule these batches to minimize the makespan. Since for each batch, there is a no-wait constraint, re-scheduling within each batch is not allowed. Any feasible solution to this scheduling problem requires these batches. The only possible solu-tion requires the synchronization of these batches.

An ideal solution is to schedule all batches to start on 7/25 12:00 am. Figure 1 gives this ideal solution, and the makespan for the project is the duration of the longest batch. However

Table 1. Summary of the batches

# Of Tasks Processing Time (hours)

Batch 1 7 15

Batch 2 11 41

Batch 3 11 53

Batch 4 20 45

Table 2. Available resources

Resources Units Resources Units

R1 1 R8 1

R2 2 R9 1

R3 5 R10 1

R4 2 R11 1

R5 1 R12 1

R6 1 R13 1

R7 1 R14 1

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for most of the time, this simplistic solution is not feasible since there are many resource contentions. In Figure 1, for example, we can see resource R1 conflicts severely.

Next we present a systematic approach to solve the multi-stage, multi-product no-wait batch-scheduling problem based on TOC.

ANALYSIS

The company overcame this batch-scheduling problem by adopting a new approach using the Theory of Constraints (TOC) production at some of its plants. This technique, with the help of some professional software, has proven to be very effective to the multi-product, multi-batch no-waiting scheduling situations.

There are two general schedule directions: backward scheduling and forward scheduling (Stevenson, 2008). Backward scheduling means scheduling backward from a due date; forward scheduling starts with either today’s date or the first open time, and computes the schedule date to determine the completion date. Since the objective of our scheduling problem is to finish all the batches as early as possible, we will be taking the forward scheduling approach.

A means of finding a heuristic solution for the example is presented in the last section, and TOC has proven to be effective in practice. We have also developed a critical chain scheduling based procedure to solve the case. The steps used to scheduling the batches are summarized as follows:

Step 1: Retrieve corresponding batch protocols from the database and assign priorities to these batches.

Step 2: Initially schedule all these batches concurrently.

Step 3: Identify the drum resource and synchro-nize the batches.

Step 4: Check the results, if there is no bottle-neck, go to step 5. Otherwise go to step 3

Step 5: Print the scheduling result, and imple-ment the schedule.

In step 3, critical chain method is applied to synchronize the batches. Here we will emphasize two issues on batch priority and selection of drum resource. The first issue is batch priority. Before we begin to schedule, we must prioritize all ongoing batches. This is to set the priorities for using drum resource.

Figure 1. Ideal scheduling

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For simplicity, here we assume these batches have the same priority. Thus the batches will be synchronized according to the order of their input. The second issue is how to select drum resource. For multi-stage, multi-product batch scheduling, resource conflicts are of primary concern. The drum resource can be chosen according to the severity of the conflicts or simply by experience since most practitioners have a prior knowledge with which resource should be leveled first, which one should be leveled next. This is a very important point. The scheduling program must be used by people who are familiar with the plant operation, especially with the capacity and technical limitations of equipment.

Software can be employed to automate critical chain scheduling. In this case study we used Scitor Corporation’s Project Sched-ule 8 (PS8). Other products available on the market with critical chain scheduling function are ProChain, an add-on to Microsoft Project, and Concerto.

One challenging and distinctive issue in this project is that no gap can be allowed within any individual batch due to the no-wait constraint during the critical chain scheduling. Thus, the individual batch does not need to be scheduled via critical chain method. Synchronization is only possible among batches, not within batches.

The previous case can be used to illustrate how to apply this procedure:

Step 1: First retrieve the 4 batches’ protocol from the database and assign them the same priority.

Step 2: Schedule the 4 batches concurrently which, is the ideal solution revealed in Figure 2.

Step 3: In this step, we will check the resource contentions.

Table 3 lists all the resource conflicts for this solution. There are 7 out of 14 resource conflicts. Resource R1 is the most contentious

Table 3. Resource conflicts

Resource Conflict (in hours)

R1 17

R2 3.5

R5 3

R9 4

R10 4

R12 4

R13 2.5

Table 4. Revised scheduling solution

Scheduling Duration Schedule Start Schedule Finish

Batch1 15h 7/25 12:00AM 7/25 3:00PM

Batch2 41h 7/25 6:00AM 7/26 11:00PM

Batch3 53h 7/25 1:00PM 7/27 6:00PM

Batch4 45h 7/25 10:00PM 7/27 7:00PM

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in this step, so we set it as the drum resource to synchronize the projects. Figure 2 is the dialog box popped up in PS8, and Table 4 is the revised solution after synchronization.

Step 4: In this step, we need to check the bottle-neck iteratively.

After 1 round, now the number of conflict-ing resources reduced to 4 (R1, R9, R10, R12), namely R1 for 1 hour, R9 for 4.5 hours, R10 for 4.5 hours, and R12 for 4.5 hours. Since there is a tie here, we will choose one resource arbitrarily. We set R9 as the drum resource this time and reschedule, now we get a feasible solution in

Table 5 with batch 1 starting at 12:00am, July 25, batch 2 starting at 6:00am, July 25, batch 3 starting from 1:00pm, July 25, and Batch 4 starting from 8:00am, July 26.

We check the solution again, and find no resource conflicts exist this time. Thus this is a feasible “good enough” solution. The makes-pan of the schedule is 77 hours. All detailed scheduling time for each task also can be ob-tained accordingly. For example, Table 6 gives the detailed schedules for batch 2, and Figure 3 provides part of the whole detailed task schedules of each batch from the PS8.

In the second round of this step, there are ties among resource R9, R10, and R12. We

Figure 2. Use of the critical chain method to synchronize batches

Table 5. “Good Enough” scheduling solution

Scheduling Duration Schedule Start Schedule Finish

Batch1 15h 7/25 12:00am 7/25 3:00PM

Batch2 41h 7/25 6:00AM 7/26 11:00PM

Batch3 53h 7/25 1:00PM 7/27 6:00PM

Batch4 45h 7/26 8:00AM 7/28 5:00AM

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Table 6. Detailed scheduling times for batch 2

Tasks Duration Schedule Start Schedule Finish

Batch2 41h 7/25 6:00AM 7/26 11:00PM

T1 7h 7/25 6:00AM 7/25 1:00PM

T2 3h 7/25 10:00AM 7/25 1:00PM

T3 3h 7/25 10:00AM 7/25 1:00PM

T4 1h 7/25 1:00PM 7/25 2:00PM

T5 25h 7/25 10:00AM 7/26 11:00AM

T6 2h 7/26 9:00AM 7/26 11:00AM

T7 1h 7/26 11:00AM 7/26 12:00PM

T8 5h 7/26 11:00AM 7/26 4:00PM

T9 7h 7/26 2:00PM 7/26 9:00PM

T10 7h 7/26 3:00PM 7/26 10:00PM

T11 7h 7/26 4:00PM 7/26 11:00PM

Figure 3. Output of the “good enough” scheduling solution

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arbitrarily choose R9 to proceed. However if we select R10 or R12, the solution will be same in this example. In practice, it could be useful to choose different sequence and pick the best solution.

Step 5: Last step is to print the scheduling result, and implement the schedule.

Sometimes customization of the output is necessary before the implementation. For example, the company also developed a Visual Basic Application (VBA) in Excel to customize the output from PS8 in order to meet differ-ent plants’ requirement, which proves to be very important for successful implementation of new scheduling technique. For example, Thomas Hsiang (2002) developed a similar output interchanging the relationship between equipment and batches by showing equipment on the vertical axis, and representing batches as horizontal bars, in order to conform more closely to the existing plant conventions.

DISCUSSION AND CONCLUSION

Theory of Constraint (TOC) as a new manage-ment philosophy has been extensively used in different business areas (e.g., Singh et al., 2006). This project suggests that a systematic approach of the TOC to multi-stage, multi-product no-wait batch scheduling problem that can provide a “good enough” solution quickly and effectively. For this case study, final solu-tion is reached after two rounds. For larger scale batching problems, such as 15 batches that is a typical daily operation capacity for one plant in this global company, according to our experiments, after 2-4 rounds, feasible solution also can be quickly reached. In practice, if some small conflicts can be tolerated, then

the procedure can be stopped when only such little conflicts exist.

For multi-stage, multi-product no-wait batch scheduling problem, the TOC approach illustrated in this paper has been proven very effective due to no complicated algorithms and programming involved, thus it is very convenient to plan daily scheduling in practice. The technique has been implemented in the company’s several plants. The scheduling ap-plication is currently being expanded to other plants. As a natural extension of the project, the next step is to build up a decision support system (DSS) to find the feasible good solu-tion by automatically running several rounds based on some criteria, and then coupled with customized output from VBA program, the daily (or even weekly) production scheduling can be generated and executed.

REFERENCES

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Chen, Y.-W., Lu, Y.-Z., & Yang, G.-K. (2008b). Hybrid evolutionary algorithm with marriage of genetic algorithm and extremal optimization for production scheduling. International Journal of Advanced Manufacturing Technology, 36(9-10), 959–968. doi:10.1007/s00170-006-0904-9

Daina, D., & Meredith, J. (2000). An analysis of process industry production and inventory manage-ment systems. Journal of Operations Management, 18, 683–699. doi:10.1016/S0272-6963(00)00039-5

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Gang, W., Gong, W., DeRenzi, B., & Kastner, R. (2007). Ant colony optimizations for resource- and timing-constrained operation scheduling. IEEE Transactions on Computer-Aided Design of Inte-grated Circuits and Systems, 26(6), 1010–1029. doi:10.1109/TCAD.2006.885829

Goldratt, E. M. (1997). Critical chain. Great Bar-rington, MA: North River Press.

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Guo, Z. X., Wong, W. K., Leung, S. Y. S., Fan, J. T., & Chan, S. F. (2008). A genetic-algorithm-based optimization model for scheduling flexible assembly lines. International Journal of Advanced Manufac-turing Technology, 36(1-2), 156–168. doi:10.1007/s00170-006-0818-6

Hansen, P., & Mladenovic, N. (2001). Variable neighbourhood search: Principles and applications. European Journal of Operational Research, 130, 449–467. doi:10.1016/S0377-2217(00)00100-4

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APPENDIX

The Appendix contains the detailed protocols for each batch used in the example and a Gantt chart for batch 1.

Table 7. Batch 1’s protocol

TaskName Duration ScheduleStart ScheduleFinish Resources

Batch1 Start 0h 7/25 12:00AM 7/25 12:00AM -

T1 6h 7/25 12:00AM 7/25 6:00AM R1

T2 6h 7/25 12:00M 7/25 6:00AM R2

T3 1.5h 7/25 6:00AM 7/25 7:30AM R2, R13

T4 9h 7/25 12:00AM 7/25 9:00AM R3

T5 3h 7/25 6:00AM 7/25 9:00AM R4

T6 1h 7/25 9:00AM 7/25 10:00AM R13

T7 8h 7/25 7:00AM 7/25 3:00PM R9

Figure 4. Gantt chart for batch 1

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T1 7h 7/25 12:00AM 7/25 7:00AM R1

T2 3h 7/25 4:00AM 7/25 7:00AM R5

T3 3h 7/25 4:00AM 7/25 7:00AM R2

T4 1h 7/25 7:00AM 7/25 8:00AM R13, R5,R2

T5 25h 7/25 4:00AM 7/26 5:00AM R3

T6 2h 7/26 3:00AM 7/26 5:00AM R4

T7 1h 7/26 5:00AM 7/26 6:00AM R13,R4

T8 5h 7/26 5:00AM 7/26 10:00AM R11

T9 7h 7/26 8:00AM 7/26 3:00PM R9

T10 7h 7/26 9:00AM 7/26 4:00PM R12

T11 7h 7/26 10:00AM 7/26 5:00PM R10




T1 9h 7/25 12:00AM 7/25 9:00AM R1

T2 3h 7/25 6:00AM 7/25 9:00AM R5

T3 3h 7/25 6:00AM 7/25 9:00AM R2

T4 1h 7/25 9:00AM 7/25 10:00AM R13,R5,R2

T5 33h 7/25/ 6:00AM 7/26 3:00PM R3

T6 2h 7/26 1:00PM 7/26 3:00PM R4

T7 1h 7/26 3:00PM 7/26 4:00PM R4,R13

T8 5h 7/26 3:00PM 7/26 8:00PM R11

T9 9h 7/26 6:00PM 7/27 3:00AM R9

T10 9h 7/26 7:00PM 7/27 4:00AM R12

T11 9h 7/26 8:00PM 7/27 5:00AM R10

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T1 4h 7/25 12:00AM 7/25 4:00AM R5,R6,R1

T2 4h 7/25 12:00AM 7/25 4:00AM R2

T3 1h 7/25 4:00AM 7/25 5:00AM R2,R5,R6,R13

T4 9h 7/25 12:00AM 7/25 9:00AM R3

T5 2h 7/25 7:00AM 7/25 9:00AM R4

T6 2h 7/25 7:00AM 7/25 9:00AM R7

T7 2h 7/25 7:00AM 7/25 9:00AM R2

T8 1h 7/25 9:00AM 7/25 10:00AM R2,R4,R7,R13

T9 8h 7/25 7:00AM 7/25 3:00PM R3

T10 2h 7/25 1:00PM 7/25 3:00PM R4

T11 2h 7/25 1:00PM 7/25 3:00PM R6

T12 2h 7/25 1:00PM 7/25 3:00PM R2

T13 1h 7/25 3:00PM 7/25 4:00PM R2,R4,R6,R13

T14 22h 7/25 1:00PM 7/26 11:00AM R3

T15 4h 7/26 7:00AM 7/26 11:00AM R4

T16 4h 7/26 7:00AM 7/26 11:00AM R8

T17 1h 7/26 11:00AM 7/26 12:00PM R4,R8,R13

T18 11h 7/26 8:00AM 7/26 7:00PM R9

T19 11h 7/26 9:00AM 7/26 8:00PM R12

T20 11h 7/26 10:00AM 7/26 9:00PM R10

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Keywords: Association Rules, Developmental Psychology, Human Development, Social Science, Time Series

1. INTRODUCTION

Development psychology is the scientific study of progressive psychological changes that occur in human beings as they age. This field examines change across a broad range of topics including motor skills and other psycho-physiological processes, problem solving abilities, conceptual understanding, acquisition of language, moral understanding, and identity formation.

Generally, multivariate time series data are collected to answer important questions

in developmental psychology research. These include questions related to emotional response patterns (Rovine, Molenaar, & Corneal, 1999), marital interactions (Laurenceau, Feldman-Bar-rett, & Rovine, 2005), parent-child interactions (Stifter & Braungart, 1995), cognitive process-ing (Gilmore & Johnson, 1998), therapeutic in-terventions (Molenaar, 1987), the development of motor coordination (Fitzpatrick, 1998), and postural models (Newell, 1998), among others. Often data are collected in the form of daily reports, diaries, relatively dense behavioral observations, electronic responses (beepers, online studies), or physiological responses (ECG, brainwaves). In these areas, common

Association Rule Mining in Developmental Psychology

D. A. Nembhard, Pennsylvania State University, USA

K. K. Yip, Pennsylvania State University, USA

C. A. Stifter, Pennsylvania State University, USA

ABSTRACTDevelopmental psychology is the scientific study of progressive psychological changes that occur in human beings as they age. Some of the current methodologies used in this field to study developmental processes include Yule’s Q, state space grids, time series analysis, and lag analysis. The data collected in this field are often time-series-type data. Applying association rule mining in developmental psychology is a new concept that may have a number of potential benefits. In this paper, two sets of infant-mother interaction data sets are examined using association rule mining. Previous analyses of these data used conventional statistical techniques. However, they failed to capture the dynamic interactions between the infant-mother pair as well as other issues relating to the temporal characteristic of the data. Three approaches are proposed in this paper as candidate means of addressing some of the questions that remain from previous studies. The approaches used can be applied to association rule mining to extend its application to data sets in related fields.

DOI: 10.4018/ijaie.2012010103

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methodologies include statistical analysis such as ANOVA, MANOVA, regression model, state space grids, Yule’s Q, and time series analyses (Bukowski, Adam, & Santo, 2006). See Bishop, Feinberg, and Holland (1975), Everitt (1996), Kiess (1996), Welkowitz, Ewen, and Cohen (1986), and Wilcox (2002) for implementations of these methods in developmental psychology and related fields.

ANOVA and MANOVA are statistical models based on the analysis of variance, which test for significant differences between means of different groups of observations. Regression models are used to predict one dependent vari-able from one or more independent variables. Both of these methods require the analyst to clearly define the dependent and independent variables. Yule’s Q is based on the odds ratio and a symmetric measure taking on values between -1 and +1, where the extremes imply perfect negative or positive association, respec-tively, with 0 representing no association. We remark that these methodologies are useful and straightforward when the researchers have a well defined set of questions to address. However, there may often exist underlying information and relations among the variables for which no questions have been formulated. In these cases, potentially valuable knowledge may go unde-tected. Additionally, temporal characteristics of these data while potentially of interest are not addressed by the existing set of statistical tools commonly used for these data.

To address some of these limitations, researchers have developed the state space grid, a methodology based on dynamic system principles (Hollenstein, 2005), as a graphical approach that utilizes observational data and quantifies these data according to two ordinal variables that define the state space for the system. Observations across time are plotted as a trajectory in the state space. Another method, time series analysis is a methodology that in-vestigates the underlying structures in a time series (i.e., sequence of data points measured at successive times), in order to make forecasts. Common models used in time series analysis include Autoregressive Moving Average models

(ARMA) and Autoregressive Integrated Mov-ing Average models (ARIMA). These methods analyze the dynamics of data across time. However, the resulting models are only applied to an individual. If for example, an experiment has collected times series data for 10 different subjects, using these two methodologies can efficiently describe the 10 subjects individu-ally, but the dynamic of the all the subjects as a group are not addressed.

As a means of overcoming some of these limitations, we will examine association rule mining. We remark that association rule min-ing has been shown to be a useful method in extracting information and uncovering hidden correlations in data. However, the approach is new to the field of developmental psychology. Further, conventional association rule mining does not consider the temporal nature that exists in some data. Each data point taken across time is usually treated as an individual record in the data set, where each record in the data set is assumed to be independent. A goal of associa-tion rule mining is to find associations between different attributes in the data set among these records. Hence, the correlation of the data with respect to time is collapsed. In order to deal with temporal data, researchers in the data mining field have developed advanced algorithms which adapted the ideas of temporal association rules and sequential pattern mining (e.g., Das, et al., 1998; Last et al., 2001). However, with additional variables, the temporal characteristics of the data may be captured with conventional association mining techniques as well, without the added complexity.

In this paper, two sets of infant-mother interaction data are examined (Jahromi et al., 2004). We develop three approaches to apply association rule mining to these data in order to address some of the questions that were not addressable with the statistical methodologies used in previous studies. Section 2 in the paper provides data descriptions and the proposed association rule mining approaches. Section 3 compares the results of the analysis using some of the statistical tools that are commonly used in the field of development psychology

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and along with association rule mining. In Section 4 we discuss the results from each of the methods used.

2. METHODOLOGY

In this section, the two data sets are described. The data collection procedures and coding, as well as the goals that guided these studies are discussed, followed by association rule min-ing and the descriptions of the three proposed approaches.

2.1. Mother-Infant Data Descriptions

The dynamic interactions between mother and infant is thought to have important develop-mental consequences for infants’ social and emotional development. Mothers’ soothing of their distressed infants is not only important for the immediate regulation of the infant but also is thought to be critical to children’s develop-ment of self-regulation of emotion (Stifter & Braungart, 1995).

The dataset used in this study was origi-nally collected in the research conducted by Jahromi et al. (2004), wherein maternal soothing of infant distress to inoculations at 2 months and at 6 months was investigated. 150 pairs of mother-infant samples were recruited from a community hospital in central Pennsylvania. The purpose of the study was to have a better understanding of the developmental trends in infants’ reactions to highly distressful situations, changes in maternal soothing behaviors used in these situations, and the association between maternal soothing and infant reactivity.

Researchers observed infants and mothers during the infant’s routine clinical visit at the age of 2-months and at 6-months. Infants and their mothers were videotaped for at least 1 minute prior to the administration of the shot and continued until the subject was calm for a period of 20 consecutive seconds following the inoculation. Other covariate information, such as gender, infant’s state of general irritability rating, doctor’s office, number of injections

received prior to the taping, family background (race, age and education level of mother, number of siblings, and etc.) were also noted.

Videotapes of infant reactivity to the inoculation and maternal soothing behaviors were coded in 5-second intervals for up to 48 intervals. Infant reactivity was coded accord-ing to a 4-point scale: 0 (audible vocalization), 1 (fussing, whining, or whimpering but not crying), 2 (low-intensity crying) and 3 (high-intensity crying). The presence or absence of the following maternal soothing behaviors were coded: affection, touching, holding, rocking, vocalizing, caretaking, distracting, presenting face, feeding and pacifying.

Previous analyses done by Jahromi et al. (2004) had used Pearson’s coefficient of correlation, repeated measures ANOVA and contingency analysis and had addressed the following four goals:

1. The effect of possible covariates on the study variables including infant’s general irritability, number of injections received, and doctor’s office.

2. Changes in the duration and intensity of infant reactivity across age (2-months to 6-months)

3. Changes in maternal behaviors used across age (2-months to 6-months)

4. Associations between infant reactivity and maternal behaviors such as the- relationship between overall infant reactivity and spe-cific maternal behaviors, the contingency between maternal behavior and infant reactivity level change, the contingency between maternal behavior and decrease in crying, and the contingency between maternal behavior and decrease in varying levels of crying.

Although researchers have made progress in their conceptualization of emotion regulation in infancy and its development, the statistical approaches to the data have not been satisfac-tory in capturing the dynamics of the process. Infants who turn away from the source of their distress (re-orienting) may decrease their

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arousal temporarily, but often they are observed to return to high levels of distress. Likewise, a mother may talk soothingly to her distressed infant but it is not clear how long she needs to maintain this behavior to get an effect. While the attempts to quantify the effectiveness with which infants self-regulate or mothers soothe their infants have informed the field about the emotion regulation process, the use of statistical techniques (chi-square and Yule’s Q) based on collapsing the behaviors over time may have obscured the important regulatory processes. Thus, what is needed is a technique that cap-tures the dynamics of the regulatory process both within the individual infant and between the infant and his/her parent. Not only may such techniques model the changes that take place when an infant is emotionally aroused but they may also identify intra-individual (intra-interactional) characteristics that predict developmental outcomes.

2.2. Association Rule Mining

The concept of association rule mining was motivated by market basket analysis. An as-sociation rule, of the form J J1 2→ , is relation-ship between premise (J1 ) and consequence (J 2 ), where J1 and J 2 are itemsets and J J NULL1 2∩ = . Given a dataset D, an item-set is a subset of items consisted in D . A rule has support = s if s % of the instances (or transactions) in the in the data set contain J J1 2∪ . A rule has confidence = c if c% of the instances containing J1 also contain J J1 2∪ , that is, c J J= ( )Pr |2 1 .

2.2.1. Association Rule Mining Approaches

Based on previous analyses (Jahromi et al., 2004), the dynamic interaction between the infant, mother and the temporal characteristics of the data were not captured. For example: the sequence or patterns of maternal behaviors, mothers’ responses to infant reactions, cor-relations between certain maternal behavior durations and infant’s reactivity, the circum-

stances when a maternal behavior is effective in soothing. The issues related with temporal characteristics are: durations of behavior, times of the event and also the patterns of behavior across time. We present three approaches for the discovery of useful association rules from these data, which follow.Approach 1: Time WindowsSuppose we are particularly interested in know-ing the correlation between changes in infants’ reactivity and maternal behaviors. In addition, we also want to know the responses of moth-ers when there is a change in infant’s reactiv-ity (i.e., whether she is going to maintain the same behavior or switch to another behavior). To answer these questions, the data set for rule mining may be augmented to include the fol-lowing attributes:

• The change in infant’s reactivity between two consecutive time stamps ( t −1 and t ) - This attribute can take a positive or negative value. If the value is negative, it implies a decrease in the infant’s reactivity, and hence successful soothing is associated with a mother’s behavior.

• To observe the maternal behaviors that trigger changes and the mothers’ response to changes, we use a “time window” tech-nique. For example, we can introduce a 4-time-stamp-window, which is to include the maternal behaviors at t −1 , t , t +1 and t + 2 . There are 11 possible maternal behaviors at each time-stamp: 0. idle, 1. affection, 2. touching, 3. holding, 4. rock-ing, 5. vocalizing, 6. caretaking, 7. distract-ing, 8. presenting face, 9. feeding, and 10. pacifying. The width of the time window can vary. Thus, some common patterns or sequence of maternal behaviors across time can be observed.

With this approach, we can observe the maternal behaviors that trigger positive/nega-tive changes in activity and correlate them with successfulness in soothing. In addition, we can observe the mother’s strategy when there is a change. For example, the mother may maintain

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the same behavior if soothing is successful (infant reactivity decreases) and switch to another behavior if the behaviors irritate the infant (infant reactivity increases). Within this data set, the infants usually remain at the same level of reactivity over several 5-second intervals. Since we are interested in knowing what mother behaviors trigger changes in in-fant reactivity, the instances with zero change in infant’s reactivity can be removed from the data set before rule mining.Approach 2: DurationsIf we wish to capture the correlations of the durations of maternal behaviors and the changes in infant reactivity, the data set for rule mining can be augmented to include the following attributes:

• The type of maternal behavior involved- [0, 10], where 0=idle, 1=affection, 2=touch-ing, 3=holding, 4=rocking, 5=vocalizing, 6=caretaking, 7=distracting, 8=presenting face, 9=feeding, and 10=pacifying.

• The duration of the maternal behavior involved- while the duration of the behavior is an integer (number of 5-second intervals), this attribute might be categorized into a nominal variable (e.g., Short: duration=1-3 intervals, Normal: duration=4-6 intervals, Long: duration= 7-15 intervals, Very Long: duration ≥ 16 intervals), in order to later simplify the interpretation of the resulting rules.

• Infant reactivity when a specific maternal behavior starts- this variable is an integer ranges [0, 3], where 0=audible vocaliza-tion, 1 =fussing, whining, or whimpering but not crying, 2=low-intensity crying, and 3 =high-intensity crying.

• Infant reactivity when a specific maternal behavior ends- this variable is an integer ranges [0, 3], where 0=audible vocaliza-tion, 1 =fussing, whining, or whimpering but not crying, 2=low-intensity crying, and 3 =high-intensity crying.

With this approach, we can correlate ma-ternal behavior durations with effectiveness

in soothing. For example, a long duration of affection may result in decrease in infant reactiv-ity (infant reactivity when a specific maternal behavior ends < infant reactivity when a specific maternal behavior ends). In addition, common durations of various behaviors, and the correla-tion between infant reactivity level, choice of maternal behavior, as well as the corresponding soothing effectiveness, have the potential to be detected. For example, the durations of feeding may be long; a mother may employ rocking if the infant reactivity is at high-intensity; a short duration of distraction may sooth the infant when the infant reactivity is at low-intensity.Approach 3: Time ZonesWe recall that the original dataset consists of data taken across 48 time intervals. If we are interested in knowing whether mothers chose different strategies at different points during the videotaping, we can augment the data set to include the following attributes for rule mining:

• Time zone- we divide the forty-eight 5-seconds intervals from the video into discrete time zones (e.g., zone1: intervals 1-5, zone2: period 6-10, zone3: intervals 11-20, zone4: intervals 21-30, and zone5: 31-48).

• Average of infant reactivity – infant re-activities over a time zone are averaged.

• Durations of each maternal behavior within a time zone- while the duration of the behavior is an integer (number of 5-second intervals), this attribute can be categorized into a nominal variable (e.g., Not Used: duration=0 intervals. Short: duration<20% of the time zone length, Moderate: duration≤ 50% of the time zone length, Long: duration>50% of the time zone length), to facilitate interpretations of rules discovered.

With this approach, we may be able to ob-serve the correlation of the use of each behavior within the time zone, the correlation of each behavior, time zone, infant reactivity and the corresponding duration, and also the potential correlation among different behaviors. For

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example, a mother may not feed in time zone 1; infant cries at high-intensity in time zone 1; a mother may feed for long duration in time zone 4; a mother may employ both short duration of rocking and short duration of vocalization in time zone 2; a mother may use short duration of affection and normal duration of rocking in the same time zone. We note that for a number of mother-infant pairs, mothers became idle and the infant had no reactivity toward the end of the videotaping. Hence, the records which have no infant reactivity and no maternal behavior (except idling) might be removed from the augmented data for this approach.

2.2.2. Apriori Algorithm

To examine the three candidate approaches, we employ the well known Apriori algorithm, which can be summarized as follows (Agrawal & Srikant, 1994).

Let Jl be a set of l ≥ 1 items where the items are selected from the set K . For a given Jt define Jt

h− to be the set obtained from Jt by omitting element h, where h is an element of Jt . The key idea of the Apriori algorithm is that the set Jt cannot possibly have m occur-rences unless, for each h in Jt , the set Jt

h− has m occurrences. Since the algorithm considers possible sets in order of their size, it has already gathered the information about all the sets of size l −1 before considering sets of size l. The algorithm is as follows:

1. FOR l from 1 to size of(K)2. FOR each set Jt such that for each h Jt∈

the set Jth− occurs k times DO

3. Examine the data to determine whether set Jt occurs in at lease k times. Those sets occur with at least k times are the frequent item sets.

4. END of FOR-DO (2)5. END of FOR-DO (1)6. Generate all rules from the frequent item

sets with the confidence ≥ the preset confidence threshold

3. RESULTS

The results presented in this section begin with a summary of results of several common develop-mental psychology research tools, correlation, ANOVA, Yule’s Q and contingency analysis, followed by summary of results from the three association rule mining approaches. We note that the same data set is used in each analysis.

3.1. Pearson’s Coefficient of Correlation

Using the Pearson’s Coefficient of Correlation analysis, it was found the infant’s general ir-ritability is related with the mean intensity of crying (Jahromi et al., 2004) (for 2-months, r=0.34, p<0.01; for 6-months, r=0.57, p<0.01), and the duration of crying (for 2 months, r=0.30, p<0.01; for 6 months, r=0.45, p<0.01). It was also found that the number of injections received is weakly correlated with cry intensity at 6-months (r=0.19, p<0.05). However, the number of injections received did not correlate with duration of crying, doctor’s office did not correlate with the intensity or the duration of cry.

Interrelations among the maternal be-haviors at 2-months and 6-months were also evaluated by Pearson’s correlations. Results showed that at 2 months, affection behaviors were positively associated with touching, vocalizing, holding and rocking (p<0.01), and caretaking behaviors were negatively associated with affection, touching, holding and rocking (p<0.01); at 6-months, affection behaviors were positively associated with vocalizing, holding and rocking (values of r ranged from 0.22 to 0.27, p<0.05), and caretaking were negatively associated with touching, presenting face, hold-ing and rocking (p<0.05).

3.2. ANOVA

To examine the differences in the mean propor-tion of time mothers engaged in each maternal behavior at 2-months and 6-months, repeated measures ANOVA was conducted by Jahromi et al. (2004). The results showed that there were significant differences in the mean proportions

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of time that mothers engaged in the 2-months and 6-months data. In addition, the changes in maternal behaviors with respect to gender differences and age (2-months and 6 months) were also examined using repeated measures ANOVA (with sex as the between-subject factor and age as the within-subject factor). Results concluded that there was no significant main effect or interactions associated with gender. It was also found that there were significant decreases in the proportion of time that mothers spent in affection and touching from 2-months to 6-months (affection: p<0.01; touching: p<0.05), and increases in time that mother used vocaliza-tion and distraction (p<0.01).

3.3. Contingency Analyses

Contingency analyses were conducted to inves-tigate the contingencies between the specific maternal behaviors and all changes in infant reactivity by Jahromi et al. (2004) in order to analyze the coexistence of specific maternal behaviors and the change in infant reactivity in the next 5-second interval, and between the specific maternal behaviors and decreases in infant reactivity. There are three possible changes in infant reactivity: decrease, increase, or no change. Results showed that most mater-nal behaviors were followed by no change in infant reactivity.

In the study of contingencies between the specific maternal behaviors and decreases in in-fant reactivity, there is a need to analyze whether specific maternal behavior yielded decreases in infant reactivity in the subsequent 5-second interval more often than expected by chance, where the expectation by chance is estimated by (total number of decreases displayed by the infant)*(total number of times the mother used the behavior)/(total number of 5-second inter-vals that the infant cried). The results showed that certain maternal behaviors (feeding/ pacify-ing, holding, rocking and vocalization) occurred more often with decreases in infant reactivity. Further investigation was also done to analyze the contingencies between maternal behaviors

and decreases in reactivity at varying levels of crying. That is, to analyze whether contingen-cies between maternal behaviors and decreases in infant reactivity were dependent on infants’ levels of distress. Results indicated significant differences in effectiveness of maternal behavior with respect to level of distress.

3.4. Yule’s Q

Yule’s Q was used to analyze the association be-tween any 2 maternal behaviors in the same time interval. It is found that feeding and pacifying are perfectly negatively associated (Q-value=1, which implies feeding → not pacifying, vice versa). Holding and rocking, holding and care-taking are also perfectly negatively associated. In the 2-months data, vocalizing, presenting face and holding are positively associated (Q-value for holding and presenting face =0.98, Q-value for vocalizing and presenting face =0.89, and Q-value for vocalizing and holding=0.51).

3.5. Association Rule Mining

The three proposed approaches are used to analyze the 2-month and 6-month inoculation data. Tables 1, 2, and 3 summarize the number of rules generated from using approaches 1, 2 and 3 respectively with various levels of support and confidence thresholds.

Note that for Approach 3 (Table 3), a maximum support threshold is set in order to limit the number of rules generated. The maximum support threshold removes items or item sets that have a higher support than the maximum support threshold from the “frequent item set” considered in the Apriori algorithm. Hence, the maximum support threshold will tend to filter out trivial rules. For example, if an item A is in 95% of all the records in the data set, this implies that there is a 95% chance that A happens regardless of the condition, and that A does not associate with other variables. A sample set of rules and the information can be drawn in each approach are summarized as follows.

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Approach1Examples:TimeWindows

• Data set: 6-months inoculation data• File size: 906 instances, 4 attributes• Attributes: change_in_cry (change in in-

fant reactivity), mother_at_t-1 (maternal behavior at time t-1), mother_at_t (ma-ternal behavior at time t), mother_at_t+1 (maternal behavior at time t+1)

• Minimum confidence thresholds: 40%• Minimum support thresholds: 10%• Number of rules generated: 52 (Table 1)• Sample rules:

• Information extracted from the sample rules in Table 4:i. The clauses in rules 4, 5 and 6 are a

subset of clause in rules 1, 2 and 3. Rules 1, 2, and 3 combined can be interpreted as a mother employs pre-senting face (maternal behavior #8) for 3 successive 5-second intervals.

ii. Rules 7, 8, 9 and 10 show that there are correlations between successive time intervals of presenting face (ma-ternal behavior #8), and decreases in infant reactivity (change_in_cry=-1). These four rules can be interpreted as a mother remains presenting face when infant reactivity reduces. Or similarly, 3 successive time intervals of present-ing face reduce infant reactivity.

Table 1. Number of rules generated by approach 1 (15-second time windows - three 5-second intervals)

Min Support (%) Min Confidence 9%) Number of Rules Generated

2-month data (1135 records)

6 month data (902 records)

10 50 48 43

10 40 52 52

10 30 63 65

5 50 74 142

5 40 84 170

5 30 105 202

Table 2. Number of rules generated by approach 2 (Durations)

Min Support (%) Min Confidence (%) Number of Rules Generated

2 - m o n t h d a t a (1298 records)

6 m o n t h d a t a (956 records)

10 50 17 14

10 40 21 21

10 30 36 34

5 50 26 28

5 40 35 36

5 30 63 78

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Approach2Examples:Durations

• Data set: 6-months inoculation data• File size: 956 instances, 4 attributes• Attributes: mother_act (the maternal be-

havior involved), actionduration (duration of the involved maternal behavior), before-cry (infant reactivity when the maternal

behavior starts), beforecry (infant reactivity when the maternal behavior ends)

• Minimum confidence thresholds: 40%• Minimum support thresholds: 10%• Number of rules generated: 21 (Table 2)• Sample rules:

Table 3. Number of rules generated by approach 3 (Time zones)

Max. Support (%) Min Support (%) Min Confidence (%) Number of Rules Generated

2-month data (518 records)

6 month data (403 records)

80 10 50 169 121

80 10 40 201 144

80 10 30 227 176

80 5 50 417 441

80 5 40 505 551

80 5 30 628 707

90 10 50 4864 3373

90 10 40 5839 3931

90 10 30 6722 4635

90 5 50 11693 6469

90 5 40 14981 7610

90 5 30 18485 9059

Table 4. Sample rules using the time windows approach (Approach 1)

Rules Confidence (%)

Support (%)

1. mother_at_t-1=8 mother_at_t+1=8 ➔ mother_at_t=8 85 23

2. mother_at_t=8 mother_at_t+1=8 ➔ mother_at_t-1=8 78 25

3. mother_at_t-1=8 mother_at_t=8 ➔ mother_at_t+1=8 76 25

4. mother_at_t=8 ➔ mother_at_t-1=8 75 34

5. mother_at_t=8 ➔ mother_at_t+1=8 74 34

6. mother_at_t+1=8 ➔ mother_at_t=8 71 35

7. change_in_cry=-1 mother_at_t-1=8 mother_at_t+1=8 ➔ mother_at_t=8 84 12

8. change_in_cry=-1 mother_at_t-1=8 mother_at_t=8 ➔ mother_at_t+1=8 77 14

9. change_in_cry=-1 mother_at_t=8 mother_at_t+1=8 ➔ mother_at_t-1=8 74 14

10. mother_at_t-1=8 mother_at_t=8 mother_at_t+1=8 ➔ change_in_cry=-1 54 19

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• Information extracted from the sample rules in Table 5:i. Rules 1 and 2 can be interpreted as

mother uses vocalization (maternal behavior #5) and presenting face (maternal behavior #8) when the infant cries at high-intensity (infant reactivity=3).

ii. Rules 3, 4, 5, 6 and 7 indicate there are correlations among “beforecry=3”, “aftercry=3” and “actionduration=1”. The five rules combined may be in-terpreted as a short duration of (any) maternal behavior does not reduce infant reactivity when the infant cries at high-intensity (infant reactivity =3).

Approach3Examples:TimeZones

• Data set: 6-months inoculation data• File size: 403 instances, 13 attributes• Attributes: timezone (the 48 5-second time

intervals in the data are grouped into five time zones), cry (the average infant reactiv-ity in a specific time zone), act0 (duration of maternal behavior #0-idle in a specific time zone), act1 (duration of maternal behavior #1-affection in a specific time zone), act2 (duration of maternal behavior #2-touching in a specific time zone), act3 (duration of maternal behavior #3-holding in a specific time zone), act4 (duration of maternal behavior #4-rocking in a specific time zone), act5 (duration of maternal

behavior #5-vocalizing in a specific time zone), act6(duration of maternal behavior #6-caretaking in a specific time zone), act7 (duration of maternal behavior #7-distract-ing in a specific time zone), act8 (duration of maternal behavior #8-presenting face in a specific time zone), act9 (duration of maternal behavior #9-feeding in a specific time zone), act10 (duration of maternal behavior #10-pacifying in a specific time zone),

• Minimum confidence thresholds: 40%• Minimum support thresholds: 10%• Maximum support thresholds: 90%• Number of rules generated: 3931 (Table 3)• Sample rules:

• Information extracted from the sample rules in Table 6 and discussions:i. Rule 1 implies that when an infant

cries at high-intensity, mother does not idle (maternal behavior #0). That is, mother always employs some strate-gies in attempt to sooth the infant.

ii. Rule 2 implies that, within a time zone, when a mother feeds (maternal behavior #9) the infant for a long du-ration, she does not pacify (maternal behavior #10). This rule reflects a negative correlation between long duration of feeding and pacifying.

Table 5. Sample rules using the durations approach (Approach 2)

Rules Confidence (%) Support (%)

1. mother_act=5 ➔ beforecry=3 96 32

2. mother_act=8 ➔ beforecry=3 93 32

3. aftercry=3 actionduration=1 ➔ beforecry=3 97 12

4. beforecry=3 actionduration=1 ➔ aftercry=3 54 22

5. actionduration=1 ➔ beforecry=3 aftercry=3 48 25

6. beforecry=3 aftercry=3 ➔ actionduration=1 48 25

7. aftercry=3 ➔ beforecry=3 actionduration=1 46 26

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iii. Rule 3 implies that mother does not pacify (maternal behavior #10) in time zone 1 (early stage of videotaping).

iv. Rule 4 implies that, in time zone 1 (early stage of videotaping) infant cries at low-intensity (infant reactivity =2), mother does not idle (maternal behavior #0).

v. Rule 5 implies that within a time zone, there are correlations among the items “act4=0”, “act6=0”, “act9=0”, “act10=0” and act5=0”, i.e., within a time zone, there are correlations for a mother does not use act4, act6, act9, act10 and act 5 (maternal behaviors #4-rocking, #6-caretaking, #9-feed-ing, #10-pacifying and #5-vocalizing respectively), or if a mother does not use act4, act6, act9 and act10 in a par-ticular time zone, then she does not use act5 in that same time zone. However, since there are 11 maternal behaviors, not using act5 does not imply the use or absence of a particular maternal behavior. What may be of interest to developmental researchers are the correlations of the usages of various maternal behaviors, not the absences of maternal behaviors. Rule 5 hence fails to provide information in this context.

vi. Rule 6 implies that in time zone 1, there are correlations among the items “act6=0”, “act7=0 and act5=0”, or if a mother does not use act6 and act7 in time zone 1, then she does not

use act5 in time zone 1. If there’s a correlation between time zone 1 and “act5=0”, or an implication that a mother does not use act5 in time zone 1, “timezone=1 → act5=0” should be one of the resulting rules in the gener-ated rule set. Hence, Rule 6 provides no additional information comparing to the rule “timezone=1 → act5=0”, and fails to give useful information in the context of associating usage of various maternal behavior.

In approach 3, a very large set of rules was generated. Specifically, with the maximum support, minimum support and minimum con-fidence thresholds set to 90%, 10% and 40% respectively, 3931 rules were generated (refer to Table 3). We remark that many of these rules will not be particularly informative (e.g., Rules 5 and 6 in Table 6). Hence, in order to be able to effectively extract useful information from the rule set, it is important to eliminate many trivial or non-informative rules. We describe two possible filters to screen many of the trivial or non-informative rules:

First, we need to define item set, R, which contains all items that we may want to remove from rules. In this context, we define R={“ac t0=0” , “ac t1=0” , “ac t2=0” , “act3=0”,“act4=0”, “act5=0”, “act6=0”, “act7=0”, “act8=0”, “act9=0”, “act10=0”}.

Filter 1: let Ik be the kth item in the premise (J1 ), Lm be the mth item in the consequence

Table 6. Sample rules using the time zones approach (Approach 3)

Rules Confidence (%) Support (%)

1. cry=3 ➔ act0=0 88 16

2. act9=4 ➔ act10=0 98 13

3. timezone=1 ➔ act10=0 95 37

4. timezone=1 cry=2 ➔ act0=0 87 12

5. act4=0 act6=0 act9=0 act10=0 ➔ act5=0 96 11

6. timezone=1 act6=0 act7=0 ➔ act5=0 95 11

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(J 2 ). A rule is removed from the rule set if I L Rk m, ⊂ , I J kk ⊂ ∀1, , L J mm ⊂ ∀2,and J J NULL1 2∩ = .

Filter 2: let Ik be the kth item in the premise (J1 ), Lm be the mth item in the consequence (J 2 ). A rule is removed if ∃ k , I Jk ⊂ 1,∃ m , L Jk ⊂ 2, I L Rk m, ⊂ , a n d J J NULL1 2∩ = .

Using filter 1, in the example, with the maximum support, minimum support and minimum confidence thresholds set to 90%, 10% and 40% respectively, the 3931 rules are reduced to 2805 rules. Using filter 2, the 3931 rules are reduced to 652 rules. The number of rules removed is dependent on the way that researchers define “non-informative” rules. To further help extracting information from the reduced rule set, the rules in the reduced rule can be sorted and grouped in the priority that interests the researchers. For example, the rules containing item “timezone=1” can be grouped together if the research in particularly interested in knowing the details in time zone 1.

The followings list summarizes several findings from the data using association rule mining:

1. Vocalizing and presenting face decrease in-fant reactivity when the behaviors continue for 3 or more time intervals (approach 1)

2. Durations of feeding are long in time zones 3, 4 and 5 (approach 3)

3. Vocalizing is used during time zones 1 and 2 (approach 3)

4. Rocking is used throughout all time zones, but the duration of rocking is short (ap-proach 3)

5. Affection is not used together with care-taking or feeding in the same time zone (approach 3)

6. When the infant reactivity is zero, mother gives short during of affection (approach 2)

4. DISCUSSION

Each of the commonly used methods in de-velopmental psychology has its advantages in providing useful insights into subject data. In this section, we will briefly describe how several of these provides information in this context, followed by descriptions of how the three association rule mining approaches have the potential to greatly augment and compliment these methods.

Pearson’s correlation coefficients give the sign and degree of association between two variables. This methodology is relatively straightforward to use and to interpret. However, it does not give information on the associations between more than 2 variables. For example, results showed that infant’s general irritabil-ity was positively associated with the mean intensity of crying, and it was also positively associated with the duration of cry. However, it did not give any information if the general irritability and mean intensity of cry combined together associated with duration of cry. Prior to applying Pearson’s coefficients, researchers have to predetermine the associations of vari-ables of interest. Hence, hidden correlations in the data may not be uncovered.

ANOVA (or MANOVA) is a practical method that indicates distributions or means differ between groups of treatments. That is, whether to support the hypothesis that the data in different groups have the same distribution. In addition, it can test the main effects and in-teractions of the factors. However, the method also requires researchers to predetermine a set of hypotheses to study. The dependent vari-ables and factors (both between-subject factors and within-subject factors) have to be clearly defined. Similar to the Pearson’s coefficients, hidden information in the data cannot be un-covered. While using age as the within-subject factor can investigate the effect on maternal behavior across time, the dynamic detail as-sociated with time series data is not captured.

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Contingency analysis checks the coexis-tence of different variables. Two-dimensional contingency analysis (coexistences of any two variables in the data) is straightforward to interpret. However, multi-dimensional analy-sis (coexistences of more than two variables) becomes somewhat more complex to interpret. While contingency analysis gives information about coexistence of variables, it does not give information about which variables are the causes or consequences. Similarly, Yule’s Q provides association of any two events, or coexistences of any two variables in the data, but it does not associate more than two variables at the same time.

There are several useful advantages in using association rules mining. First, the method is model free. Researchers do not need to prede-termine hypotheses or identify the dependent and independent variables. Second, similar to contingency analyses, association rule min-ing can identify relationships among multiple variables. Third, the output of association rule mining is in the format of “If A then B”, yielding results are relatively easy to interpret.

Using various approaches, association rule mining is also able to address the temporal characteristics and the dynamics of the data. For example, finding #1 states that vocalization decreases infant reactivity when the behavior continues for 3 time intervals. While contin-gency analysis had suggested coexistence of vocalization and decreases in infant reactivity, it failed to capture the necessity of the continu-ity of the behavior. In another example, finding #2 states that the durations of feeding are long in time zones 3, 4 and 5. While ANOVA had investigated the proportion of time that mothers engaged in feeding in general, and compared the quantity at 2-months and 6-months, it did not give information on when (during the taping) the mother did feed. In this case, the profiles of maternal behaviors with respect to time had been collapsed.

With the association rule mining approach-es introduced, the temporal characteristics of the data are included in the rule mining. The basic tactics are using “time windows” to capture

sequence of behaviors, “time zones” to capture at which point during the videotaping did the event occurred, and “durations” to capture the length of behaviors. We note that much of the data collected in the field of development psychology is time series data. By applying association rule mining with the basic tactics to these data, we can extend our knowledge about developmental processes. The advantages of using association rule mining are its power to uncover hidden patterns and association that are difficult to find using traditional statistical techniques (e.g., ANOVA, and Yule’s Q) and free of mathematical model between the variables.

In the past developmental research has focused on mean levels of behavior and whether that behavior changed with age. Developmental scientists, however, have been moving toward understanding the process that underlies these changes. The statistical approaches for exam-ining process that developmental scientists currently use, however, are limited. In other words, while researchers use procedures for ob-taining data that reflect process (microanalytic behavioral coding, physiological recordings, daily diaries), the methods for quantifying and analyzing these data are lacking.

The data presented here represent an at-tempt to understand how mothers regulate their infants’ distress. Previous methods of analysis (Pearson’s correlation, ANOVA, contingency analysis, and Yule’s Q) have moved us toward a better understanding of relationships between mothers’ behavior and infants’ crying but much more can be learned from the data. Observing the patterning of maternal behavior in light of the infants’ level of distress tells us more about process. For example, in a previous analysis of these data (Jahromi et al., 2004) we showed that vocalizations were effective in reducing infant distress. However, the contingency analysis we used collapsed these data across time thereby making it difficult to know how long the vocalizations need to persist for the effect to be seen. It is reasonable to assume that a brief vocalization by the mother may not be heard by the crying infant. So, our question regarding the duration of maternal vocalization

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and its effect on infant crying could not be ad-dressed. In the current analysis, association rule mining with three time-modeling approaches (time windows, durations, and time zones) was applied to these data. The results demonstrated that maternal vocalizations need to persist for at least 3 5-second epochs before infant distress is decreased. The same analysis could be applied to other maternal behaviors, as well. Another advantage of association rule mining is iden-tifying the when a change in infant distress occurs and what mothers do in response to that change. It may be that mothers who maintain a soothing strategy that is successful (infant distress decreases) reflects her sensitivity to her infant’s state, an indicator of later more positive outcomes for the infant. Likewise, a mother who switches her soothing strategy despite its success may indicate insensitivity and thus poorer outcome.

There is an important drawback in using as-sociation rule mining, in particularly associated with Apriori algorithm. One of the challenges is to set the most appropriate thresholds of mini-mum support and confidence. If the thresholds are set too high, interesting rules may not be uncovered. In contrast, if the thresholds are set too low, too many uninteresting rules may be found causing difficulty in interpreting results. Also, from the development psychologists’ point of view, associations between certain variables are more important than the others. Modifying the current method, such as by setting different weights among the variables, may address this issue. Further investigation on how to weight these could be useful in future studies.

ACKNOWLEDGMENT

The research was supported in part by the Na-tional Science Foundation Grant DMI-0527449. The authors thank Eric Loken, Peter Molenaar, Michael Rovine and Katerina Sinclair in provid-ing suggestions for this project.

REFERENCES

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Bishop, Y. M. M., Feinberg, S. E., & Holland, P. W. (1975). Discrete multivariate analysis: Theory and practice. Cambridge, MA: MIT Press.

Bukowski, W. M., Adam, R. E., & Santo, J. B. (2006). Recent advances in the study of development, social and personal experience, and psychopathology. Inter-national Journal of Behavioral Development, 20(1), 26–30. doi:10.1177/0165025406059970

Das, G., Lin, K., Mannila, H., Renganathan, G., & Smyth, P. (1998). Rule discovery from time series. In Proceedings of the 4th International Conference on Knowledge Discovery and Data Mining (pp. 16-22).

Everitt, B. S. (1996). Making sense of statistics in psychology. New York, NY: Oxford University Press.

Gilmore, R. O., & Johnson, M. H. (1998). Object-oriented attraction in 8-month-old infants. Develop-mental Science, 1(2), 221–225. doi:10.1111/1467-7687.00034

Hollenstein, T. (2005). Using state space grids to display describe, quantify and analyze synchronized time series or event sequences. Retrieved from http://psyc.queensu.ca/faculty/holl/StateSpaceGrids-Noldus2005.pdf

Jahromi, L. B., Putnam, S. P., & Stifter, C. A. (2004). Maternal regulation of infant reactivity from 2 to 6 months. Developmental Psychology, 40, 477–487. doi:10.1037/0012-1649.40.4.477

Kiess, H. O. (1996). Statistical concepts for be-havioral sciences. Needham Heights, MA: Allyn and Bacon.

Last, M., Klein, Y., & Kandel, A. (2001). Knowledge discovery in time series databases. IEEE Transac-tions on Systems, Man, and Cybernetics, Part B, 1-9.

Laurenceau, J. P., Barrett, L. F., & Rovine, M. J. (2005). The interpersonal process model of intimacy in marriage: A daily-diary and multilevel modeling approach. Journal of Family Psychology, 19(2), 314–323. doi:10.1037/0893-3200.19.2.314

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Rovine, M. J., Molenaar, P. C. M., & Corneal, S. (1999). Analysis of emotional response patterns for adolescent stepsons using p-technique factor analysis. International Studies on Childhood and Adolescence, 7, 261–286.

Stifter, C. A., & Braungart, J. M. (1995). The regula-tion of negative reactivity in infancy: function and development. Developmental Psychology, 31(3), 448–455. doi:10.1037/0012-1649.31.3.448

Welkowitz, J., Ewen, R. B., & Cohen, J. (1986). Introductory statistics for the behavioral sciences. Orlando, FL: Harcourt Brace Jovanovich.

Wilcox, R. R. (2002). Understanding the practical advantages of modern ANOVA methods. Journal of Clinical Child and Adolescent Psychology, 31(3), 399–412.

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Keywords: Buyer-Seller, Genetic Algorithm, Material Flow Modeling, Production/Distribution, Supply Chain Management

1. INTRODUCTION

Supply chain management (SCM) is the process of planning, implementing, and controlling the operations of the supply chain as efficiently as possible. Contrary to the uncoordinated and inefficient production management of previous years, a uniform supply chain is being preferred these days in order to manage properly the material flow, commodity, information and finance. However, for better performance, some old or new means like mathematical program-ming, simulation, meta-heuristic procedures

are applied. Since the design and management of supply chain flows (materials, information and finance) as well as their success are closely linked to each other; failures of electronic com-merce are often attributed to problems arising from poor design and management of supply chain flows (Chopra et al., 2004). By this way, the SCM is a collection of methods used for the effective coordination of suppliers, producers, storage and stores. It also makes the needful products available to customers, at a significant time and place. Further, this method minimizes the total cost of chain and satisfies customers’ needs with a high level service (Simchi-Levi & Kaminsky, 2000). Since, the failure or the

Supply and Production/Distribution Planning in Supply Chain with Genetic Algorithm

Babak Sohrabi, University of Tehran, Iran

MohammadReza Sadeghi Moghadam, University of Tehran, Iran

ABSTRACTThe present study, using genetic algorithm, tries to improve material flow management in supply chain. Consequently, in this paper, an integrated supply-production and distribution planning (SPDP) is considered despite the fact that in most of the Iranian industrial firms, SPDP is done independently. The effective use of integrated SPDP not only enhances the performance rather decreases inventory cost, holding cost, shortage cost and overall supply chain costs. A quantitative mathematical model is used to the problem articulation, and then it is solved by applying heuristic genetic algorithm (GA) method. The proposed model with genetic algorithm could provide the best satisfactory result with the minimum cost. The reliability test was carried by comparing the model results with that of the amount of variables.

DOI: 10.4018/ijaie.2012010104

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success of SCM depends on its inventory role; the level of coordination is much needed phe-nomenon. As a matter of fact, while the short-age of source adversely delays the production flow, extra inventories could also increase the needless costs. Additionally, there is an inven-tory shortage in this supply chain sections, but at another places, there unusually exists surplus inventory, which are unsatisfactory (Stevenson, 2004).

In the Middle Eastern region, most of the manufacturing units have the same traditional planning, with each supply chain units concerns optimizing its benefits alone. Hence, it often leads to raise the actual price level and con-siderably decreases the firms’ competitiveness. The rest of the paper is organized as follow: The subsequent sections two and three deal with literature review and experimental study. Section 4 covers the mathematical modeling of the supply chain. Section 5 explains the proposed GA, with brief description about the heuristic approaches used in the computational experiment. Section 6 gives model validation followed by conclusion in Section 7.

2. LITERATURE REVIEW

Previous researchers came up with different models on supply chains, to which we came across during the course of literature review. These models can be categorized as:

• Coordinated model of buyer-seller• Coordinated planning model of

production-distribution• Coordinated planning model of

production-invention• Model of location-allocation• Coordinated model of

purchase-production-distribution

Chandra and Fisher (1994) proposed a coordinated planning model of production and distribution, in which products from a single production unit are transported to retailers di-rectly from the plant. Significance of this model

is the demand of retailers for each product in the specific period, however; this also minimizes the total costs consisting manufacturing, transporta-tion and the inventories. Jayaraman and Pirkul (1998) presented yet another coordinated model of mixed zero-one programming. In this model, the input of the production plant are varied i.e., raw materials are supplied from different dealers and the output of finished goods are carried to different storages or distributed to markets based on customers’ demands. This model also aims to minimize the total costs including deploy-ment, operation and storage, production and distribution. In a separate research, Jayaraman and Pirkul (2001) also proposed the PLANWAR model, showing the location of plants and stor-ages with capacity constraints.

Cohen and Moon (1991) illustrated a mixed zero-one programming aimed at optimizing the flow of materials and products as well as combining the output in a supply chain network with stable structure. This model significantly emphasizes on sellers, manufacturing units, their capacities and distribution centers. It intends to minimize the overall material, pro-duction and transportation costs; and as such it concerns much about demand, supply and chain structure.

Considering different source constraints, Lee et al. (2002) determined the system structure based on multi-plant, multi-product and multi-periodic production and distribution. Kiahara and Toshiba (2003) highlighted the P.P.P model, which is about the problems of suppliers to the final assembling plants. It consists of a two-level BOM to allocate manufacturing and distribut-ing plans. In this model, the transportation time was considered later. Pyke and Cohen (1994) forwarded a coordinated production- distribu-tion model probably with the aim of minimizing the production and distribution costs. However, this model encounters with main constraints of demand and supply capacity. Sabri et al. pro-vided a coordinated multiple objective model for strategic and operational planning in the supply chain. Its main purpose was to minimize the chain costs hence; the model at operative level wanted to distinguish the purchase and

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distribution of materials through economical formulas (Sabri et al., 2000). Arntzen, et al. (1995) propounded the global supply chain model (GSCM), comprising products, tech-nologies, customers, suppliers, and production and distribution centers extending to other countries. The model combines the minimum costs of supply chain as well as the minimum time cycle to transfer the products and materi-als. The saturation in the forecasted demands and production capacities at different stages are the main constraints of this particular model. Zhou et al. (2000) came out with a goal-oriented model that aimed to optimize the complete sup-ply chain ranging from material purchasing to distribution of finished goods. As a whole, this particular model deals with economical, social, resource and environment aspects.

Jang et al. (2002) presented a combined model of network design and production/dis-tribution planning for a supply chain network. While they used a Lagrangean heuristic to design the supply chain network, a GA was proposed for the integrated production and distribution problem. Syarif, Yun, and Gen (2002) devel-oped a spanning tree-based GA approach for the multi-source, single-product, multi-stage supply chain network design problem based on Prufer numbers. To design the distribution net-work and manage the supply chain environment, Jayaraman and Ross (2003) proposed a heuristic approach based on simulated annealing. Few years later, Yeh (2006) developed a hybrid heuristic approach for the problem considered before by Syarif et al. (2002). This approach is a combination of greedy method, linear programming technique and three local search methods. Once again for the same problem, Yeh (2005) came up with a mimetic algorithm (MA) combining GA, greedy heuristic, and local search methods. The current author has extensively investigated the performance of the MA on the randomly generated problems. To design a single-source, multi-product, multi-stage supply chain, Altiparmak et al. (2007) have recently presented a solution procedure based on steady-state genetic algorithms with a new encoding structure.

Gumus et al. (2009) have structured an inventory management framework and deter-ministic/stochastic-neuro-fuzzy cost models for effective multi-echelon supply chains under stochastic and fuzzy environments.

3. EXPERIMENTAL STUDY

As a case study, the proposed research has taken into account the supply chain of one of the biggest sewing and embroidery machines manufacturer in the Middle East. This firm, which produces four different brands, has at-tracted both internal and external suppliers and has an extensive distribution network in the region. However, this firm too has encountered with many challenges recently due to intense competition and market saturation especially by the influx of low cost Chinese products. All of these essentially demonstrate for supply chain management and coordinated decisions.

Above machines are being manufactured at three singular production lines, each based on three different technologies from Germany, Japan and China. Studying all these units and interviewing with expert managers it was found that the production line based on the Japanese technology was unprofitable; hence this was selected for further study.

The components of sewing machine are supplied from abroad as well as internal purchas-ing and engineering products. With regard to the BOM of one kind of sewing machine, 266 parts of this product have been demonstrated. With A, B, C analysis, the recognition of vital components and agreement with the experts, some 57 pieces have essentially been recognized and selected for planning. Figure 1 shows the scope and piece flow of the supply chain.

4. MATHEMATICAL MODELING

This section presents a comprehensive math-ematical formulation, considering the real-world factors and constraints. The proposed supply chain modeling is a mixed integer pro-gramming, which has 1266 integer variables,

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99 binary variables, and 2100 constraints. In order to formulate the model, the following notations are used:

Model Indices

t: time period [t= 0,1,2,…,T]i: purchased piece [i=0,1,2,…,I]r: manufactured piece at workshop [r=

0,1,2,…,R]d: sale subsidiary [d=0,1,2, …,D]s: supplierj: manufactured product at assemble workshopp: storagee: working stations

Model Parameters

Djt: Demand of product j at period tNij: Numbers of piece units i used for manu-

facturing a product jNrj: Numbers of piece units r used for manu-

facturing a product jCjt: Cost of manufactured product j in period tCit: Cost of purchased piece i in period tCrt: Cost of manufactured piece r in period tHjtp: Holding cost of a unit of piece i in period

t in storage PHitp: Holding cost of a unit of product j in period

t in storage P

Hrtp: Holding cost of a unit of piece r in period t in storage P

π jt: Shortage cost of a product j in period tπ it: Shortage cost of a unit of piece i in period

tπ rt: Shortage cost of a unit of piece r in station

tMie: Time of a unit of piece i in station eMre: Time of a unit of piece r in station eMMj: Time of a unit of product j in final as-

semble stationMMMj: Time of a unit of product j in packag-

ing stationsCPet: Load time of production in station e in

period tCCPt: Load time of production in final assemble

station in period tCCCPt: Load time of production in packaging

stations in period tDMdjt: The demands of distributor d in period

t for product jRCjd: Transportation cost of a unit of product j

to the distributor dRCis: Transportation cost of a unit of piece i

from supplier sSSipt: Probable saving rate of piece i in period

t in storage PSSrtp: Probable saving rate of piece r in period

t in storage PSSjtp: Probable saving rate of product j in period

t in storage P

Figure 1. Illustration of a supply chain network

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APjtp: Storage capacity of product j in period t in storage P

APrtp: Storage capacity of piece r in period t in storage P

APitp: Storage capacity of product i in period t in storage P

Problem Variables

XMist: Rate of piece i purchased from supplier s in period t

XPjt: Rate of product j manufactured in period tXDjdt: Rate of product j consigned to the dis-

tribution center d in period tXXPret: Production rate of piece r in station c

in period tIpjt: Rate of product j stored in product storage

p at the end of period tIIpit: Rate of piece i stored in storage P at the

end of period tIIIrpt: Rate of piece r which stored in product

storage p at the end of period tOSt: If ordered the supplier s for purchasing

material s in period tODt: If we consign commodity to distributor d

in period tIQjdt: Rate of product j stored in distributive

storages d in period tLjt: Rate of shortage comes after not covering

product j demand at the end of period tLLit: Shortage rate of piece i at the end of period tLLLrt: Shortage rate of piece r at the end of

period t

Model Objectives

The objective functions of the model are as follows:

stp

P

s

S

i

I

t

T

it ist itp ipt it it is istO c XM h II LL RC====∑∑∑∑ + + +1111

( Π XXM

c XXP h III LLLrtp

P

rete

E

i

R

t

T

rpt rpt rt rt

)

( )

(

+

+ + +====∑∑∑∑1111

Π

jjtp

P

j

J

t

T

jt jpt jpt jt jt

dtd

D

j

J

t

T

c XP h I L

O===

===

∑∑∑

∑∑

+ + +111

111

Π )

∑∑ +( )jd jdt jpt dtRC XD h IQ

• To minimize the purchasing costs, the purchasing rate of each supplier can be distinguished.

• It can minimize all holding costs of pur-chased pieces and shortage cost.

• It can minimize all costs of manufacturing, holding of pieces and confront with the shortage of pieces.

• It can minimize all costs of manufactur-ing, holding of pieces and confront with its shortage.

• It can determine the rate of commodity transportation to the subsidiaries, consider-ing minimization of costs related to supply and demand of distribution centers.

• It can minimize the holding costs of the subsidiaries.

Model Constraints

(1)

prtp

P

rete

E

rjj

J

jt prtp

P

III XXP n XP III−= = = =∑ ∑ ∑ ∑+ − =

11 1 1 1

(2)

pjtp

P

jt pjt jdtd

D

p

P

I XP I XD−= ==∑ ∑∑+ − =

11 11

(3)

ist sti

I

XM O− ≤=∑ M1

0 (4)

M is a very high number.

ret re etr

R

XXP m CP× ≤=∑

1

(5)

iet ie eti

I

XXP m CP× ≤=∑

1

(6)

jtj

J

j tXP mm CCP=∑ × ≤

1

(7)

iptp

P

ists

S

ijj

J

jt iptp

P

II XM n XP II i t−

= = = =∑ ∑ ∑ ∑+ − = ∀

11 1 1 1

,

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jtj

J

j tXP mmm CCCP=∑ × ≤

1

∀t

jdtj

J

tXD ODM=∑ − ≤1

0 (8)

jdt dt dt djtXD IQ IQ DM+ − ≥−1

∀j d t, ,

iptp

P

iptp

P

II SS= =∑ ∑≥

1 1

(9)

rptp

P

rptp

P

III SS= =∑ ∑≥

1 1

∀r t,

rptp

P

jptp

P

I SS= =∑ ∑≥

1 1

∀j t,

jpt jptj

J

I AP≤=∑

1

(10)

ipt ipti

I

II AP≤=∑

1

∀p t,

rpt rptr

R

III AP≤=∑

1

∀p t,

ij jt ists

S

pitp

P

ij pjtp

P

itn D XM II n I LL− + +

=

= = =∑ ∑ ∑

1 1 1

(11)

rj jt rete

E

rptp

P

rj jtp

P

rtn D XXP III n IP LLL− + +

=

= = =∑ ∑ ∑

1 1 1

∀r t,

jt jdt jtd

D

D XD L− ==∑

1

∀j t,

ist pit itXM II LL, , ≥ 0 ∀i t s p, , , , integer (12)

jt jdt pjt jdtXP XD I L, , , ≥ 0 ∀j t d p, , , , integer

ret rpt rtXXP III LLL, , ≥ 0 ∀r t e p, , , , integer

dtIQ ≥ 0 ∀d t, , integer

stO = 0 or 1

dtO = 0 or 1

• The constraint (1) is related to the exist-ing confidence in the purchased pieces (piece existence balance): This constraint illustrates that the piece inventory i in the beginning and the rate of the pieces pur-chased from various suppliers in period t must be equal to the plus of the piece consumption rate i in the manufacturing of final product in month t and its inventory rate, which settle in the storage at the end of the month.

• The constraint (2) is also related to con-fidence of the purchased pieces (piece existence balance): This constraint distin-guishes that the piece inventory r in the beginning of period plus the rate of these pieces manufactured at different workshops in period t must be equal to the plus of the piece consumption rate in manufacturing the final product in the month t and its inventory rate, which were settled in the storage at the end of the month.

• The constraint (3) is related to the final product inventory balance: According to this constraint, the rate of the manufactured product j in the period t plus the product inventory rate j in the beginning of period in final product storage minus the product inventory at the end of period is equal to the product rate, consigned from this storage to the subsidiary.

• The constraint (4) is related to the piece ordered for purchase: This constraint il-lustrates that the purchase cost must be paid after the order is placed to the supplier s.

• The constraint (5) is related to the produc-tion capacity in manufacture units: Here, the piece manufacturing rate in the plant must be considered alone to the existing capacity of equipments there.

• The constraint (6) is the load process in assembling stations: This is similar to the constraint 5.

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• The constraint (7) is the final product process.

• The constraint (8) is about order supply of the distribution centers: According to the first group of constraints, after commodity is transported to the authorized agency d, the plant must pay the transportation costs. The constraints of the group two shows that the commodity rate transferred to the subsidiary each time plus their inventory at the beginning of the period minus the product inventory at the end of period in subsidiary must have been equal at least to the rate of the demand.

• The constraint (9) is about the safety stock in different storage: These constraints are necessary preservation of various pieces and products in all storages.

• The constraint (10) is related to the capac-ity of different storages: These constraints are about the load of different storages for preserving various products and pieces.

• The constraint (11) relates to the confront-ing of shortage of pieces and the final products.

• The constraint (12) is a model structural constraint.

5. PROPOSED GENETIC ALGORITHM

The Genetic Algorithm (GA) is one of a family of heuristic optimization techniques, including simulated annealing, taboo search, and the evolutionary strategies. Its application derives optimal solution to many diverse and difficult problems. For the specific engineering problems, however; the GA cannot guarantee optimality and sometimes suffer from the pre-mature convergence. Probably, this is due to lack of fundamental requirement of not using a priori knowledge and not exploiting the local search information. To overcome this problem, several hybridized techniques are introduced; the most popular is to combine the local search technique into the GA process. Here, a genetic search is used to perform global exploration among the population, whereas a local search

is used to perform local exploitation around chromosomes. Contrary to the local search techniques that are simple and computation-ally efficient, the global search techniques explore the global space without using local information about promising search directions. Consequently, they are less likely to be trapped in a local optimal solution, and have a higher computational cost as well. The distinction between the local and global search techniques is referred to as the exploitation /exploration trade-off (Lee et al., 2002). In the proposed current method, in order to carefully search the near optimal solution, the authors include their local search technique in the GA loop. However, the GA is used again for improving the local search. In this method, after the first generation, the population divides into N equal parts then for each part the crossover and the mutation operator are allocated differently. The GA procedure for this research is comprised of the steps shown in Figure 2.

5.1. Chromosome

While using the GA, one of the most important problems encountering with it is to choose an appropriate chromosome representation. In fact, this chromosome must translate the underlying real-world problem into a number of codes as accurately as possible. Consequently, each chromosome is represented by 1365 compo-nents where the 99 parts are binary and 1266 parts are integer numbers.

5.2. Fitness Function

This is the one where the variable rate of problem is placed and then the desirability of each reply is used as fitness function (Chan et al., 2005)

5.3. Population

A collection of chromosome is called popula-tion. Characteristically, the GA acts upon a population of chromosome rather concentrating on one point from searching space (Fogarty, 1989). In this problem, the model has been

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solved by using pop-size 500, 1000, 2000, 5000 and 10000 chromosomes.

5.4. Genetic Operators

Creating new chromosomes called child by using old chromosomes called parent is yet another important aspect of the GA. This process is accomplished through the genetic operator (Aarts, 1986).

5.4.1. Crossover Operator

This is a principle operation for generating new chromosomes in the GA. Alike itself, this operator mates in the nature and generates new individuals constituting from their parents’ components (gene) (Syarif et al., 2002). In the proposed research, the crossover operators are scattered, two-point, intermediate and heuristic.

5.4.2. Mutation Operator

In the natural developing cycle, mutation is an accidental process where to produce a new ge-netic structure, the content of agene is replaced with another gene. Often, the role of mutation is to assure that the search probability in chain had never been zero (Van, 1987). In this research, Gaussian (1, 1), Gaussian (2, 1) and uniform 0.1 are mutation operators. The Gaussian mutation operator adds an accidental number from the Gaussian distribution function with zero mean to each parent vector input. The variance of this distribution is regulated by the scale and the shrink parameters. With consecutive changes of variables, the current study has selected the scale 1 and 2 as well as the shrink 1.

• The standard deviations of the mutation is controlled by the scale at the first genera-tion. In other words, the scale multiplied by

Figure 2. The overall procedure of the proposed method

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a range of the initial population specified by the initial range option.

• The shrink controls the rate at which, the average amount of mutation decreases. Here, one finds linear reduction in the standard deviation and the final value equals 1-shrink times of its initial value at the first generation. For example, if the shrink has the value 1, the amount of the mutation decreases to 0 at the final step.

5.5. Generation

Repetition of each algorithm causing the new population production is called generation. In this study, a new generation production has been conducted for 100 times. In the due course, however, the strategy is encountering with the constraints like the penalty function. Here, procedures that prohibit from entering the unacceptable replies can exit. Consider-ing the unacceptable replies, one of the most prevalent techniques is the penalty strategy. This technique, at first doesn’t consider the problem constraint, rather a fine for each offense settled in fitness value (Gen et al., 2000). Table 1 show 10 cases of the best extracted replies found by running the model with various initial population, different operators and generation

production for 100 times. They indicate the run time of each model, too.

With the aim of showing model perfor-mance and proposing solution to the problems, the formulation was done in MATLAB software. It was executed over a PC with an Intel Pen-tium 4 with 2.6 GHz processor. Table 1 sum-marize process, the best solution is for the population rate 1000 chromosomes, mutation operator Gaussian (1,1), crossover operator scatter and generation production rate 100 which it has been replayed at 35 minutes and 17 sec-onds.

5.5.1. Hybrid Genetic Algorithm

Both the hybrid and the usual genetic algorithm have been used in the current research. While the model is solved by the hybrid GA, it divides the generation into sections and then utilizes various mutation and crossover operators at each generation. It means that the hybrid GA changes operators at each generation instead of changing PC and Pm. The last row of Table 1 clearly indicates the result of this algorithm.

Figure 3 shows the response curve from each generation, which is related to the best reply (Row 9 in Table 1). The horizontal and vertical

Table 1. Ten best replies by running the model

Generation POP-SIZE Crossover Mutation Objective Function Run Time (minute)

100 500 Scatter Gaussian(1,1) 3638564263,64 9,36

100 1000 Heuristic Gaussian(2,1) 3545127763,76 13,19

100 1000 Scatter uniform .01 3512478932,54 13,42

100 2000 two point Gaussian(1,1) 3175443648,32 16,2

100 2000 inter mediate Gaussian(1,1) 3262415130,59 16,4



100 10000 Heuristic Gaussian(1,1) 2301477410,28 36,24


100 10000 Hybrid Hybrid 2245502464,79 35,41

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axes of Figure 3 are related to the generation and fitness value, correspondingly.

6. MODEL VALIDATION

In the current study, all the existing procedures have been employed to validate the mathemati-cal modeling. At first, the model for the supply chain is designed, considering valid mathemati-

cal models given by eminent experts. To get the feedback and the valuable comments, the author could present the proposed model to at least 10 academic and industrial experts. Those opinions of the experts and researchers were utilized in this modeling process and finally the model results were compared with the present condition. Figure 4 shows the outcomes of the

Figure 3. The fitness value curve in 100 generation production

Figure 4. Comparison between the model result and present condition

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comparison and illustrates decreasing cost, which is about 8.6 percent.

7. CONCLUSION

In this paper, using genetic algorithm, a novel integrated model has been introduced for supply and production / distribution problem, which is an NP-hard problem. To demonstrate this subject, the author tried to formulate the prob-lem using the mixed-integer linear program-ming model (MIP) in LINDO software but no outcome could be found. Consequently, in the current study, a method has been propounded using genetic algorithm. The outcome of the GA methodology is used in a simulation model and developed to analyze the impacts of demand fluctuations. It seems that the algorithm can carefully search the best heuristic solutions to the problem. It is also believed that as an alternative method, this will give an advantage to solve the complicated problem. As such, a real case study was performed by using the proposed method and comparing the results with the traditional state. The comparison showed the superiority of the proposed algorithm. The model’s characteristics have been illustrated, following the triple parts i.e., the model first selects the suppliers considering their demand ratio, it plans about the production considering the results from supply section and finally it does the distribution planning and allocation of manufactured products among the subsidiaries. Appendix A highlights the results of supply based on the rate of each piece. Appendix B shoes the results of production section which covers the production rate of pieces at each working station, preservation rate of the piece inventory, and the authorized rates for confront-ing with their shortage. Also, the production rate of final product, the preservation rate of final product inventory at manufactured commodity storage and the authorized rate of confronting to its shortage have been determined in earlier section.

Appendix C illustrates the results of distri-bution section with transportation rates of sew-

ing machine to each of the internal subsidiary in the country, preservation rates of inventory for each agency and the authorized rates of confronting to the shortage for each of them.

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APPENDIX A

The rate of piece i purchased from supplier s in period t=1, 2, 3The rate of piece i stored at the end of period t=1, 2, 3 and the shortage rate of piece i at the end of period t =1, 2, 3

continued on following page

Table 2

I XMis1 S XMis2 S XMis3 S

1 906 1 1.171 1 1.752 2

2 2.598 8 1.561 8 659 8

3 5 1 6.275 2 6.754 1

4 1.377 8 1.177 8 109 8

5 99 9 775 9 1.500 9

6 570 7 380 7 228 7

7 1.088 9 865 9 82 9

8 3.350 3 3.047 3 241 3

9 1.153 10 238 10 437 10

10 118 8 475 8 201 8

11 1.008 8 267 8 459 8

12 369 6 323 6 351 6

13 18 11 11 11 11 11

14 1.177 9 737 9 1.987 9

15 1.609 8 207 8 1.137 8

16 274 7 142 7 282 7

17 535 9 224 9 88 9

18 4.413 8 363 8 61 8

19 359 8 268 8 995 8

20 105 7 54 7 52 7

21 83 8 379 8 193 8

22 1.108 8 94 8 1.036 8

23 758 9 1.099 9 337 9

24 1.686 8 1.702 8 2.535 8

25 4.795 9 233 9 1.225 9

26 464 7 1.523 7 1.262 7

27 220 9 608 9 329 9

28 1.675 8 3 8 1.934 8

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29 3.921 8 1.996 8 2.669 8

30 1.909 8 448 8 572 8

31 546 8 50 8 1.320 8

32 1.426 8 1.968 8 689 8

33 1.308 13 2.079 13 1.686 13

34 440 7 808 7 424 7

35 375 9 1.206 9 1.539 9

36 1.008 9 1.742 9 1.409 9

37 747 7 119 7 741 7

38 13 7 98 7 398 7

39 826 5 180 5 294 5

40 96 8 200 8 266 8

41 732 9 56 9 426 9

42 1.263 8 613 8 196 8

43 118 8 4 8 85 8

44 137 8 49 8 279 8

45 2.960 9 1.770 9 666 9

46 1.557 7 46 7 75 7

47 48 9 509 9 163 9

48 282 7 307 7 102 7

49 1.212 8 1.409 8 114 8

50 165 7 95 7 586 7

Table 2. continued

Table 3

i IIpip 11

4

=∑ LLi1 IIpip 21

4

=∑ LLi2 IIpip 31

4

=∑ LLi3

1 147126 0 134415 0 136079 0

2 68560 0 64442 0 65065 0

3 17931 0 19050 0 25785 0

4 72828 0 73374 0 73479 0

5 0 0 144 0 1641 0

6 24692 0 24441 0 24665 0

7 1217 0 1451 0 1528 0

8 8479 0 10895 0 11132 0

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continued on following page

Table 3. continued

9 4272 0 3880 0 4312 0

10 2673 0 2517 0 2714 0

11 164 0 0 0 455 0

12 814 0 505 0 852 0

13 10890 0 10271 0 10278 0

14 0 0 106 0 2089 0

15 0 0 0 0 1133 0

16 12777 0 12288 0 12566 0

17 4500 0 4092 0 4176 0

18 28410 0 28143 0 28199 0

19 0 0 0 0 991 0

20 8988 0 8411 0 8459 0

21 0 0 0 0 189 0

22 35603 0 35067 0 36099 0

23 0 0 468 0 802 0

24 36942 0 38014 0 40545 0

25 0 0 0 0 1221 0

26 21038 0 21930 0 23187 0

27 24021 0 23998 0 24323 0

28 78427 0 77799 0 79730 0

29 14614 0 15979 0 18644 0

30 75052 0 74869 0 75437 0

31 32725 0 32144 0 33460 0

32 15626 0 16963 0 17648 0

33 0 0 1448 0 3130 0

34 21881 0 22058 0 22479 0

35 52298 0 52873 0 54409 0

36 0 0 1111 0 2516 0

37 0 0 0 0 737 0

38 11425 0 10892 0 11286 0

39 4397 0 3946 0 4235 0

40 10545 0 10115 0 10376 0

41 0 0 0 0 422 0

42 1663 0 1645 0 1837 0

43 9042 0 8415 0 8496 0

44 39779 0 39197 0 39471 0

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45 0 0 1139 0 1802 0

46 20332 0 19747 0 19819 0

47 0 0 0 0 159 0

48 51207 0 50884 0 50982 0

49 1437 0 2215 0 2325 0

50 1437 0 902 0 1484 0

Appendix B

The production rate of piece r in period t =1, 2, 3 and the rate of piece r which was stored at the end of period t=1, 2, 3

The rate of production j which was manufacture in period t and the rate of production j which was stored at the end of period t:

The rate of shortage which comes after not covering product j demand at the end of period t =1, 2, 3.

Table 3. continued

Table 4.

r XXPre1 IIIrpp 11

4

=∑ XXPre2 IIIrpp 21

4

=∑ XXPre3 IIIrpp 31

4

=∑1 2.643 13154 1.126 13649 4.858 18503

2 17 0 1.876 1245 617 1858

3 1.152 0 3.446 2815 1.764 4575

4 251 0 1.841 1210 608 1814

5 2.230 88 236 0 255 251

Table 5

j XPj1 Ipjp 11

4

=∑ XPj2 Ipjp 21

4

=∑ XPj3 Ipjp 31

4

=∑1 2149 573 631 1001 240 217

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Appendix C

The rate of production j which was consigned to the distribution center d in period t =1,2,.

Table 6.

J Lj1 Lj2 Lj3

1 0 0 0

Table 7.

D XDjd1 XDjd2 XDjd3

1 61 0 0

2 688 0 0

3 105 0 0

4 160 0 0

5 115 0 0

6 106 0 0

7 550 0 97

8 0 48 0

9 0 105 0

10 0 50 0

11 0 0 73

12 0 0 30

13 0 0 167

14 0 0 145

15 0 0 59

16 0 0 39

17 0 0 86

18 0 0 250

19 0 0 50

20 0 0 68

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Keywords: Battery Manufacturing, Integer Programming, Optimization, Production Planning, Profitability

INTRODUCTION

Battery production requires a special manufac-turing process that involves several distinctive steps such as chemical curing, acid filling, and electrical charging. Due to the unique characteristics of the battery manufacturing process, battery production planning has its own distinguishing features. In this paper, an integer programming model is presented for maximizing the profitability of battery produc-tion at the formation stage of manufacturing for

the Middle East Battery Company (MEBCO) in Saudi Arabia. The model is formulated and applied to real production data, showing sig-nificant advantages over the current manual production planning approach.

The Middle East Battery Company (ME-BCO) is a joint venture between a group of Saudi Industrialists and Johnson Control International (JCI). MEBCO produces AC Delco, JCI and Toyota branded maintenance-free batteries. As an equity partner in MEBCO, JCI has manage-ment responsibility for the plant operation. MEBCO plant, which is located in Dammam, Saudi Arabia, started commercial production in

A Production Planning Optimization Model

for Maximizing Battery Manufacturing Profitability

Hesham K. Alfares, King Fahd University of Petroleum & Minerals, Saudi Arabia

ABSTRACTThis paper presents an integer programming (IP) model for production planning, which is used to maximize the profitability of battery manufacturing in a mid-size company. Battery production is a complicated multi-stage process. The formation stage, during which the batteries are filled with acid and charged with electricity, is considered to be the bottleneck of this process. The IP model maximizes the total profit of batteries produced in the formation stage, subject to limited manufacturing resources as well as time limitations and demand restrictions. The IP model is able to accommodate a large variety of battery models and sizes, and also dif-ferent charging circuit capacities and speeds. The model is formulated and optimally solved using Microsoft Excel Solver. Compared to the current manual production planning approach, the optimum IP-generated production plans lead to an average increase of 12% in daily profits.

DOI: 10.4018/ijaie.2012010105

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1998. The production has been steadily grow-ing and it has exceeded 2.5 million batteries since 2006. The plant produces batteries for all vehicle types, as well as marine, commercial and industrial applications. MEBCO’s primary market is the Middle East, but it also has growing markets in Asia, Europe, and North America.

MEBCO’s battery manufacturing plant is divided into 4 main areas; X-Met, Green Group, Formation, and Final Assembly. The Formation area’s main function is the electrical charging of batteries for periods that range from 14.5 hours to 24 hours, depending on the type of battery. Because of the limited time per day, and also because of space limitations that restrict the number of charging circuits, the Formation area is considered to be the bottleneck of the battery production capacity.

According to theory of constraints, the throughput rate of any multi-component system is controlled by the slowest component (called the constraint or the bottleneck). In order to maximize the productivity (profitability) of the battery production line, the company needs to optimize the production planning output in the formation stage. As shown in the following section, a variety of optimization and heuristic techniques have been used to solve battery production planning problems. In this paper, an integer programming (IP) model is used to maximize the profitability of the formation area. This IP model has been practically applied to optimally schedule the daily production of 74 battery models. The model considers different battery sizes, charging times, and daily demands, as well as limitations on times, capacities, and numbers of different types of charging circuits. Compared to manually generated production plans, the IP-optimized production plans in-crease daily profits by 12% on average.

The remaining sections of this paper are arranged as follows. First, previous literature on battery production planning is reviewed. Next, the manufacturing process, especially in the formation step, is described in more detail. Subsequently, the integer programming model for battery production planning is presented, and then the model’s performance is compared

to the current manual approach. Finally, results are analyzed, and conclusions are drawn.

LITERATURE REVIEW

Previous approaches to battery production plan-ning include simulation, network flow models, EOQ-type techniques, mathematical program-ming, and various heuristics. Turnquist (1991) described a software system for battery produc-tion planning developed by Cornell University for the Battery Strategic Business Unit of Delco Remy. The PC-based system includes three main components: (1) a forecasting module, (2) a multi-plant product allocations module, and (3) a scheduling module. The production scheduling module employs a network flow model to dynamically balance inventory and overtime costs given limited capacities and fluctuating demands for each plant.

Yenradeea (1994) combines simulation, the optimized production technology (OPT), and simple scheduling rules to schedule a four-stage battery production line. The OPT production plans successfully minimize inventory while maximizing the throughput rate. Using simu-lation experiments, these plans are shown to outperform both the push and the pull policies. Khadem and Ali (2008) develop a simulation model to optimize the cost effectiveness of a car battery manufacturer. The model is used to represent and analyze the dynamics of the battery assembly line, and also to make several recommendations for improving the line’s cycle time, productivity, and quality.

Sharma (2003) integrated the principles of lean production with the practices of Six Sigma to improve daily work life in a battery company on a continuing basis. The advantages realized by the company included significant cost reduction, more efficient manufacturing process, and higher customer satisfaction. Al-Turki (2000) presented a heuristic multi-product batch scheduling algorithm to schedule battery batch production with the minimum set-up time. The effects of set-up time/frequency and batch size on the production performance were

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investigated in a real-life battery manufactur-ing company. César et al. (2007) described a case study implementation of intermediate stock management in the production process for automotive batteries, highlighting the main positive aspects and the deficiencies.

Huston (1996) used a three-stage process for production planning at the International Fuel Cell Corporation (IFC) in the US. First, a spread-sheet model determines daily produc-tion and workforce size. Next, an EOQ model determines optimal lot sizes. Finally, a process improvement model increases bottleneck pro-duction by reducing setup time and/or cycle time. Elimam and Udayabhanu (1999) presented two procedures for planning the production of lead-acid batteries: a simple MRP-based heuristic and a mixed integer programming optimization model. In a later paper, Elimam and Udayabhanu (2001) modeled battery pro-duction as a multi-product, multi-period, and multi-component supply chain process. The trade-offs in this process – including the costs of holding, ordering, and defects – were tested in a small battery manufacturing plant.

Dunstall and Mills (2002) presented a mixed-integer programming (MIP) model to optimally schedule cyclic production of bat-teries for Exide Technologies in Australia. A production schedule is developed for 42 periods in a cyclic workweek. The objective is to maxi-mize throughout a multi-product battery charg-ing and finishing facility while maintaining a given product mix. Dunstall and Mills (2007) slightly modified their earlier MIP model and then used simulation experiments to evaluate its performance compared to two simple schedul-ing rules: shortest processing time (SPT), and random sequencing (RND). The simulation results confirmed that the optimal MIP sched-ules consistently yield higher throughputs than simple on-line dispatching rules.

The formation-stage production planning problem at MEBCO, which is described in detail later, does not involve any stochastic or nonlinear aspects. Although the problem has in-teger variables, it is fairly small and thus easy to solve. Therefore, optimization by integer linear

programming is the most suitable approach for this problem. Compared to Dunstall and Mills (2002) MIP model, this paper presents a pure-integer programming model with substantially different decision variables, objective function, and constraints. Before introducing the model, the battery manufacturing process is described in the following section.

BATTERY MANUFACTURING PROCESS

The manufacturing space in the plant is divided into 4 main areas; X-Met, Green Group, Forma-tion, and Final Assembly. The main processes in the X-Met area are lead strip manufacturing, oxide manufacturing, paste mixing, X-Met line, steaming, and curing. The Green Group main sections are encapsulation/collation, case preparation, battery assembly, cast-on-strap, side terminal weld, extrusion fusion weld, heat seal, and top terminal weld. The main processes in the Formation stage, which is described in more detail later in this section, are initial acid fill and formation. The Final Assembly’s main processes are formation acid dump, final acid fill, final heat seal, labeling, and packing. Fig-ure 1 illustrates the flow of the manufacturing process.

The first operation in the formation area is filling the batteries with a lower-specific-gravity acid called the forming acid. This weak acid contains more water than acid to facilitate and promote the chemical conversion of the plates. This forming acid also contains a spe-cific ingredient called formax. Controlling the amount of formax, acid gravity, and acid volume in the acid mixing process is very important in attaining proper battery formation and perfor-mance.

Once the Green Group batteries have been filled with forming acid, they are transferred by conveyors to the charging area, whose layout is shown in Figure 2. The charging area has 10 tables, 4 of which have 20-Ampere char-gers while the remaining 6 have 30-Ampere chargers. On the charging tables, batteries are

IGI GLOBAL PROOF



arranged in groups (known as circuits) to match the circuit output points of the chargers. Each charger has 36 output points (circuits), with up to 18 small-size batteries per circuit. However, due to space limitations, only 15 medium-size or 12 large-size batteries can be connected to each circuit. Within each circuit, the individual batteries are hooked together (positive of one battery attached to negative of next battery) by the use of charging straps.

Once the batteries are loaded on to tables in different circuits, they are subjected to a one-hour rest or pickling time. During this one-hour time frame, strap and circuit connec-tions are completed and the batteries are spaced apart to attain proper air flow cooling. Subse-quently, they are given two steps of electrical charges for a specific duration and ampere rates. Between two steps, a rest time of 3 hours is provided. The electrical charges are applied in

Figure 1. Flow chart of the manufacturing process

Figure 2. Layout of the charging tables in the formation area

IGI GLOBAL PROOF



order to turn the green sulfated plate material into formed plates capable of storing and de-livering electrical charge.

The electrical charging cycle of the batter-ies varies with the size of the battery. Smaller automotive batteries require less Ampere-hour (Amps-charger output multiplied by time on charge) charging than larger automotive bat-teries. Extra-large heavy duty batteries tend to have the longest charging cycles. Once the formation is completed, the batteries are un-strapped from each other and from the charging circuits. Finally, the batteries are off loaded from the charging rows onto conveyors that transfer the units to the finishing line. The steps of the formation process are depicted in Figure 3.

Sometimes during the peak season, the formation stage becomes a bottleneck that limits the battery production capacity because it involves many constraints and obstacles. These include long formation times, capacity constraints, lack of scheduling system, and inexact demand forecasting. In the following section, an integer programming optimization model for production planning is formulated for maximizing the productivity (profitability) of the formation stage.

FORMATION CAPACITY OPTIMIZATION MODEL

In this section, we present the assumptions, decision variables, objective function, and con-straints of an integer programming production planning model that is formulated to maximize

the profitability of batteries produced from the formation area.

Assumptions of the Model

1. There are 10 tables for charging batteries. The number of tables is currently limited due to scarce space availability in the formation area.

2. There are six 30-Amps and four 20-Amps charging tables. This is probably a sub-optimum allocation of the 10 tables among the two types, because more battery models can be charged on 20-Amps circuits than on 30-Amps circuits.

3. Each table has 36 charging circuits. This number is fixed by the table size and the space requirements of the different battery models and charging circuits.

4. Each circuit consists of 18, 15 or 12 bat-teries, depending on battery size.

5. There are three battery sizes: ◦ Size 1 (small): 18 batteries per circuit. ◦ Size 2 (medium): 15 batteries per

circuit. ◦ Size 3 (large): 12 batteries per circuit.

6. In any charging circuit, we cannot have any mix among the three battery sizes.

7. Some batteries can be charged only on 20-Amps circuits, some others can be charged on 30-Amps, while the rest can be charged on both.

8. If a battery can be charged on either 20-Amps or 30-Amps circuits, then the charg-ing time is shorter on the 30-Amps.

Figure 3. Detailed flow chart of the formation stage

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9. The larger the battery size, the longer the charging time required.

10. The total number of battery models pro-duced by the company is 74. Out of these, 35 models can be charged on 20-Amps circuits, 31 models can be charged on 30-Amps circuits, while 8 models can be charged on both types of circuits.

11. The maximum demand for each battery model is given per day, based on the overall weekly demand.

In reality, there are several other considerations that are not explicitly included in the model because they do not affect the model’s output. For example, storage time constraints require batteries are not kept for more than 96 hours after formation. In addition, batteries taking longer times are not loaded when there is an urgent need for finishing certain quantities of other models. Finally, where demand is normal, batteries with longer charging duration will be put on charge first.

Given Parameters

di = demand of model i battery, i = 1, ..., 74pi = profit per unit of model i battery, i = 1, ..., 74tia = formation time of model i battery if charged

on a 20-Amps tabletib = formation time of model i battery if charged

on a 30-Amps tableSk = set of indices (model numbers) of batteries

of size k, k = 1, ..., 3

Decision Variables

Xi = number of model i batteries charged on 20-Amps tables per day, i = 1, ..., 74

Yi = number of model i batteries charged on 30-Amps tables per day, i = 1, ..., 74

Ak = number of 20-Amps circuits assigned to batteries of size k, k = 1, ..., 3

Bk = number of 30-Amps circuits assigned to batteries of size k, k = 1, ..., 3

Objective Function

The objective of the integer programming model is to maximize the total daily profit of batteries processed in the formation stage of the battery manufacturing process:

Maximize Z = p X Yi i ii

( )+=∑1

74

(1)

Constraints

The maximum daily demand for each battery model cannot be exceeded:

Xi + Yi ≤ di, i = 1, ..., 74 (2)

The number of batteries charged in the 20-Amps (30-Amps) tables is limited by the available respective circuit capacity:

A1+ A2+ A3 ≤ 4(36) (3)B1+ B2+ B3 ≤ 6(36) (4)

For each of the three sizes, the total number of hours used for charging 20-Amps batteries cannot exceed the total available circuit-hours per day:

t X Aia ii S∈∑ ≤

1

18 24 1( ) (5)


2

15 24 2( ) (6)


3

12 24 3( ) (7)

For each of the three sizes, the total number of hours used for charging 30-Amps batteries cannot exceed the total available circuit-hours per day:

t Y Bib ii S∈∑ ≤

1

18 24 1( ) (8)

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2

15 24 2( ) (9)


3

12 24 3( ) (10)

Xi, Yi, Ak, Bk ≥ 0 and integer, i = 1, ..., 74, k = 1, ..., 3 (11)

MODEL APPLICATION AT MEBCO

All of the data required for the model were collected from the MEBCO Product Standards Manual for all 74 battery models produced by MEBCO. Some of these models can be charged either on 20-Amps or on 30-Amps charging circuits. Table 1 shows standard data (forma-

tion time, charge table type, and battery size) for a sample of 15 battery models. In addition, current unit profits and typical daily demands are shown for these 15 battery models.

For each work day, the above integer pro-gramming model was solved for different daily demands di, i = 1, ..., 74, using the Solver optimization tool in Microsoft Excel. Compared to the current manual schedules, the optimum solutions found by the model for produced significantly higher daily profits. The increase in daily profit ranged from $3,500 to $6,200, and averaged around $4,600. In relative terms, the increase in daily profit ranged from 8% to 16%, and averaged around 12%. Table 2 compares the daily profits of the model’s optimum solution and the company’s manual solutions for 5 typical work days.

Table 1. Typical data for a sample of 15 battery models

Model Formationtime(Hr) Tabletype Batterysize Profit($) Dailydemand

1 17.5 20 Amps 1 3.52 748

421 20 Amps

1 4.59 21817 30 Amps

5 24 30 Amps 2 6.72 4

20 22.5 30 Amps 3 6.72 180

21 23.5 20 Amps 2 5.28 75

22 23.5 20 Amps 2 5.28 225

28 24 20 Amps 1 5.23 0

29 24 30 Amps 2 11.12 0

30 21 30 Amps 3 7.15 72

3116 20 Amps

1 3.52 19215.5 30 Amps

32 23 30 Amps 3 7.87 0

3315.5 20 Amps

1 3.52 25616 30 Amps

34 24 30 Amps 3 10.40 9

35 18 20 Amps 1 3.04 0

36 16.5 20 Amps 1 3.04 0

37 20.5 30 Amps 2 9.20 144IGI GLOBAL PROOF



CONCLUSION

An optimum production planning model has been formulated and used to maximize the profitability of battery production for a real-life battery manufacturer. The integer programming model is specifically designed to determine the production plan at the formation stage (i.e., quantities of each battery model allocated to each type of charging table). The constraints of the model include demand for each battery model, availability of charging tables, and charging time limitations.

The IP model has several advantages. First, the model produced significant improvements (averaging 12%) in daily profit. Second, the model is flexible to changes in input parameters such as daily demands, or to changes in decision variables (e.g., new battery models). Another advantage is the use of standard Microsoft Excel Solver, which makes the model a very conve-nient low-cost tool to find the optimal battery production plan under a variety of scenarios. Limitations of the model are imposed by the given assumptions and constraints, reflecting the company’s current practice. These include space and time constraints and the number of charging tables of each type.

Several opportunities for future research could be pursued in order to improve perfor-mance further. First, daily demands for dif-ferent battery types could be estimated more carefully in order to take full advantage of the model. Second, the optimum number of 20-Amps and 30-Amps tables could be determined under space and budget constraints, subject to varying daily demands for individual battery

types. Third, mixing different battery sizes on the same charging circuit could be allowed to increase the flexibility to meet demands. Finally, the IP model could be combined with facility layout techniques in order to maximize the space utilization and increase the number of charging tables.

REFERENCES

Al-Turki, Y. A. Y. (2000, April 1-4). Multi-product batch scheduling for lead-acid battery assembly line. In Proceedings of the 11th Annual Production and Operations Management Society Conference, San Antonio, TX.

César, P., Rodrigues, C., & Oliveira, O. J. (2007, May 4-7). Intermediate stock management in the production process for automotive batteries: a case study. In Proceedings of the 18th Annual Production and Operations Management Society Conference, Dallas, TX.

Dunstall, S., & Mills, G. (2002, February 11-15). Scheduling the charging of batteries. In Proceed-ings of the Mathematics in Industry Study Group, Adelaide, Australia (pp. 87-105).

Dunstall, S., & Mills, G. (2007). Robustness of cyclic schedules for the charging of batteries. ANZIAM Journal, 48(4), 475–492. doi:10.1017/S1446181100003163

Elimam, A. A., & Udayabhanu, V. (1999, May 23-24). Production planning for lead-acid batteries. In Proceedings of the IIE Industrial Engineering Research Conference, Phoenix, AZ.

Elimam, A. A., & Udayabhanu, V. (2001, August 6-8). Multi-level supply chain in lead-acid battery manufacturing. In Proceedings of the First Interna-tional Conference on Logistics and Supply Chain Management, Coimbatore, India (pp. 99-103).

Table 2. Optimal solution profit versus company schedules’ profits for five work days

Day Companyprofit($) OptimumProfit($) %ProfitIncrease

1 37,387 43,200 15.55

2 41,605 46,160 10.95

3 32,803 37,099 13.10

4 41,369 45,077 8.97

5 35,973 39,767 10.54

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Huston, T. W. (1996). Production system design and cycle time reduction in a fuel cell manufactur-ing operation (Unpublished master’s thesis). MIT, Cambridge, MA.

Khadem, M., & Ali, A. (2008, November 16-18). Modeling and simulation for car battery manufac-turing for cost effectiveness. In Proceeding of the Intelligent Systems and Control Symposia: Compu-tational Biology and Bioinformatics, Orlando, FL.

Middle East Battery Company. (2006). MEBCO product standards manual. Dammam, Saudi Arabia: Middle East Battery Company.

Sharma, U. (2003). Implementing lean principles with the Six Sigma advantage: How a battery company realized significant improvement. Journal of Orga-nizational Excellence, 22(3), 43–52. doi:10.1002/npr.10078

Turnquist, M. A. (1991, October 27-31). University/industry cooperation on an integrated production planning software system. In Proceedings of the Conference on Technology Management: The New International Language, Portland, OR (pp. 456-459).

Yenradeea, P. (1994). Application of optimized production technology in a capacity constrained flow shop: a case study in a battery factory. Com-puters & Industrial Engineering, 27(1-4), 217–220. doi:10.1016/0360-8352(94)90274-7

Hesham K. Alfares is professor in the Systems Engineering (SE) Department of King Fahd Uni-versity of Petroleum & Minerals (KFUPM). He has a BS in electrical and computer engineering from the University of California, Santa Barbara, and an MS and a PhD in industrial engineering from the University of Pittsburgh and Arizona State University, respectively. Dr. Alfares research areas include employee scheduling, petrochemical industry optimization, maintenance modeling, production and inventory systems, and multi-criteria decision making. He has published more than 70 journal and conference papers in these areas.

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Keywords: Deterioration, Inspection, Inventory, Permissible Delay, Retailer Ordering Policy

1. INTRODUCTION

Usually in all public and private sector organi-zation, the quantity received is always subject to inspection because by and large all concern are likely to produce some defective items, although numbers may be small. The inspected items may be classified into two categories viz., non-defective and defective. Now, the

non-defective items enter into the inventory system. However, when the nature of the item is deteriorating then inspection of the lot becomes more meaningful.

The deterioration is now a well-established fact, which may take place in the form of gradual decay, damage or perishability and the process is affected by extraneous and invisible factors. Moreover, the deterioration may also happen due to weather conditions, insufficient and unscientific storage structure etc. This situa-

Retailer Ordering Policy for Deteriorating Items with Initial

Inspection and Allowable Shortage Under the Condition of Permissible Delay in Payments

Chandra K. Jaggi, University of Delhi, India

Mandeep Mittal, Amity School of Engineering & Technology, India

ABSTRACTWhile developing the inventory model with shortages under permissible delay in payments, it has been observed in the literature, the researchers have not considered the fact that the retailer can earn interest on the revenue generated after fulfilling the outstanding demand as soon as he receives the new consignment at the start of the cycle. Owing to this fact, the present paper investigates the impact of interest earned from revenue generated after fulfilling the stock out at the start of the cycle on a single commodity inventory model with shortages for deteriorating item, in which the whole lot goes through an inspection on arrival before entering into inventory system, under the conditions of permissible delay in payments. The results have been demonstrated with the help of a numerical example using the tools of Matlab7.0.1.

DOI: 10.4018/ijaie.2012010106

IGI GLOBAL PROOF



tion occurs particularly in case of foodstuffs, which are damaged due to insects, spoilage, and rodents, where as in other commodities deterioration may occur during normal storage facilities. Ghare (1963) proposed an economic order quantity (EOQ) model for items with an exponentially decaying inventory. Covert (1973) introduced variable rate of deterioration in Ghare (1963) model. A further improvement was introduced by Shah (1977) considering a model allowing complete backlogging of the unsatisfied demand. A good amount of work has been done by different researchers to explore the effect of deterioration on EOQ model under different circumstances (Dave, 1981; Mark, 1982; Hollier, 1983; Heng, 1991; Raafat, 1991).

Moreover, while developing a mathemati-cal model in inventory control, it is assumed that the payment will be made to the suppli-ers for the goods immediately after receiving the consignment. However, in the day-to-day dealing, supplier does allow a certain fixed period to settle the account, during which the supplier charges no interest, but beyond this period interest is charged by the supplier under the terms and conditions agreed upon. Now, in case debt financing, it is often a short-term financing. Thus, interest paid here is nothing but the cost of capital or opportunity cost. Also, short-term loans can be thought of as having been taken from the suppliers on the expiry of the credit period. However, before the account has to be settled, the customer can sell the goods and continues to accumulate revenue and earn interest instead of paying the overdraft that is necessary if the supplier requires settlement of the account after replenishment. Therefore, it makes economic sense for the customer to delay the settlement of the replenishment account up to the last day of the credit period allowed by the supplier.

Goyal (1985) presented the model by introducing permissible delay in payments for fixed time period and Aggarwal and Jaggi (1995) extended his work for deteriorating items. Furthermore, Jamal (1997) allowed shortages in the Aggarwal and Jaggi (1995) model, since then, many articles have been appeared under

different situations (Raafat, 1991; Shah et al., 2000; Goyal & Giri, 2001). The primary ben-efit of taking trade credit is that one can have saving in purchase cost and opportunity cost, which becomes quite relevant for deteriorating items, because in such cases, one has to procure more units than required in the given cycle to account for the deteriorating effect. In particular, when the unit purchase cost is high and decay is continuous, the saving due to delayed payment appears to be more significant.

Lot of work has been published by many authors for finding the economic order quantity with or without shortages for deteriorating items. However, in the literature, it has been found the inventory models with shortages under permissible delay in payments, researchers have not incorporated in the interest earned part by the retailer, which he could have also earned on the revenue generated after fulfilling the back order at the start of the new cycle. In this paper an attempt has been made to formulate the retailer ordering policy for deteriorating items with initial inspection under the condition of permissible delay in payments. Shortages are allowed and are fully backlogged.

2. ASSUMPTIONS AND NOTATIONS

The following assumptions are used in develop-ing the model:

(1) The lead time is negligible(2) Shortages are allowed(3) Demand is deterministic at a constant rate

and known(4) Interest paid is greater than interest earned(5) Replenishment is instantaneous(6) A constant fraction (θ ) of the on-hand

inventory deteriorate per unit time.

The notations adopted in this paper are:

A Order cost of inventory (dollars per order)N Number of items received before inspectionR Demand rate (units per unit time)

IGI GLOBAL PROOF



C Cost price of one item (dollars/unit)C1 Holding cost per unit per unit timeC2 Shortage cost per unit per unit timeS Selling price of one itemI Inspection cost per itemθ Constant fraction of the on-hand inventory

deterioration per unitNp Number of non defective item (t=0)p fraction of non-defective items (0≤ p≤1)Ie Interest earnedIp Interest paid, Ip ≥ IeM Permissible delay in settling the accountsT Inventory cycle lengtht1 Length of the period with positive stock of

the itemS1 Quantity consumed during t ime

= − − − + −

− − − −CI

Se e R t M e ep

M t t M11

1 11

θ θ θθ θ θ θ( ) ( ) ( )

D (T) Deteriorating units per cycleTVC1 (S1, T) Total variable cost per cycle per

unit time for case 1, when 0 ≤ M ≤ = − − − + −

− − − −CI

Se e R t M e ep

M t t M11

1 11

θ θ θθ θ θ θ( ) ( ) ( )

≤ TTVC2 (S1, T) Total variable cost per cycle per

unit t ime for case 2, when 0≤ = − − − + −

− − − −CI

Se e R t M e ep

M t t M11

1 11

θ θ θθ θ θ θ( ) ( ) ( )

≤ M≤ T

3. MATHEMATICAL FORMULATION AND SOLUTION

Let N be the items received at the beginning of the period. Further, let after inspection Np be the number of non-defective items (t=0), which are retained to fulfill demand. The fraction of non-defective items p (0≤ p≤1) is known, as it can easily be estimated from the past data.

At the beginning of the cycle, a batch of Np units enters the inventory system after inspection from which (Np - S1) units are delivered towards back-order leaving a balance of S1 units as the initial inventory (Figures 1 and 2). With the passage of time the inventory level gradually decreases mainly due to demands and partly due to deterioration of items up to time

= − − − + −

− − − −CI

Se e R t M e ep

M t t M11

1 11

θ θ θθ θ θ θ( ) ( ) ( )

(0≤= − − − + −

− − − −CI

Se e R t M e ep

M t t M11

1 11

θ θ θθ θ θ θ( ) ( ) ( )

≤T).

Further demands during = − − − + −

− − − −CI

Se e R t M e ep

M t t M11

1 11

θ θ θθ θ θ θ( ) ( ) ( )

,

T) are backlogged.

Let I(t) be the inventory level of the system at anytime t (0 ≤ t ≤ T). The instantaneous states of I(t) over the period (0, T) are given by

dI tdt

I t R t t( )

( )+ = − ≤ ≤θ 0 1

(1)

dI tdt

R t t T( )=− ≤ ≤1

(2)

The solution of the differential equation (boundary condition at t =0, I (t) = S1 = initial inventory) is

I tS e R e t t

R t t t t T

t t

( )[ ]

( )=

− − ≤ ≤

− ≤ ≤

− −1 1

1 1

1 0θ θ

θ

and (4) (3)

Since, at t== − − − + −

− − − −CI

Se e R t M e ep

M t t M11

1 11

θ θ θθ θ θ θ( ) ( ) ( )

,I ( = − − − + −

− − − −CI

Se e R t M e ep

M t t M11

1 11

θ θ θθ θ θ θ( ) ( ) ( )

) = 0, from equation (1) we get

S R e t11 1= −( )

θθ (5)

IGI GLOBAL PROOF



i.e.

111

1tSR

= +

θθ

log (6)

Further, total number of non-defective units is

Np R T t S= − +( )1 1 (7)

or

NpR T t S= − +[ ]1

1 1( ) (8)

The total demand during t1 is

Rt R SR111= +

θθ

log (9)

Therefore, number of units that deteriorated D (T), during one cycle is

D T R e Rtt( )= −( ) −θ

θ 1 1 1 (10)

3.1. Inventory Scenarios

Since we are considering the permissible delay in payments in this model, where stock out

Figure 1. Inventory system for case 1, M<t1 with complete back ordering

Figure 2. Inventory system for case 1, M<t1 with complete back ordering

IGI GLOBAL PROOF



are allowed. Under this situation there will be two cases:

Case 1: 0≤ M≤ = − − − + −

− − − −CI

Se e R t M e ep

M t t M11

1 11

θ θ θθ θ θ θ( ) ( ) ( )

≤ T

Case 2: 0≤ = − − − + −

− − − −CI

Se e R t M e ep

M t t M11

1 11

θ θ θθ θ θ θ( ) ( ) ( )

≤ M≤ T

Case 1:

0≤ M≤ = − − − + −

− − − −CI

Se e R t M e ep

M t t M11

1 11

θ θ θθ θ θ θ( ) ( ) ( )

≤T

In this case, it is assumed that one can earn interest on revenue generated from the sales up to t1. Although, he has to settle the account at M, for that, he has to arrange money at some specified rate of interest in order to get his re-maining stocks financed for the period M to t1.

Now, the interest paid per cycle is

= CI I t dtp M

t( )

1

∫

= − − − + −

− − − −CI

Se e R t M e ep

M t t M11

1 11

θ θ θθ θ θ θ( ) ( ) ( )

WhereS R e t11 1= −( )

θθ

= − − −−CI Re

CI Rt Mp t M p

θ θθ

2 11 1( ) ( )( )

(11)

Interest earned per cycle has got two parts:

Part 1

In first part, one can earn interest till the time period (t1),

Interest earned =S I Rt dte

t

0

1

∫ (12)

Part 2

Second part includes the revenue generated after fulfilling the back orders (at the start of the cycle) for the time period (M).

Interest earned =SI R T t Me ( )− 1 (13)

∴ Total interest earned in this case

S I Rt dte

t

0

1

∫ + S I R T t Me ( )− 1 (14)

Whereas Jamal (1997) has considered the interest earned part as

C I Rt dte

t

0

1

∫

i.e., he has not taken into account the interest earned from revenue generated after fulfilling the back orders at the start of the cycle, which has got significant impact on his total cost reduction.

Inventory Carrying Cost =C I t dtt

1 0

1

( )∫

= − + −

−Ce S R Rtt1

1 11 1

θ θθ( )( )

= − −

C Re tt1

2 11 1

θθθ( ) (15)

Shortage Cost = C2 R t t dtt

T( )−∫ 1

1

= −C RT t2 12 2( ) / (16)

Inspection Cost = IN (17)

The total variable cost per cycle per unit time (TVC1) is the sum of the order cost, cost of deteriorated units, inspection cost, carrying cost, shortage cost and interest paid, but the interest earned is subtracted from this sum, i.e.

IGI GLOBAL PROOF



TVC S T AT

CD TT

INT

C RTe t

C RTT tt

1 112 1

21 1( , )( )

( ) (= + + + − −

+ −

θθθ

112

2 11

2

11

) /

( ) ( )( )( )+ − − − −−

−−CI RTe

CI RTt M

SI R T t MT

SIRtp t M p e

eθ θθ 11

2

2T (18)

Now, we discuss some special cases.

Special Case 1

Without inspection and excluding interest earned after fulfilling the back orders (at the start of the cycle) the equation (18) reduces to,

TVC S T AT

CD TT

C RTe t

C RTT tt

11 112 1

2121 1( , )

( )( ) ( )= + + − −

+ −

θθθ //

( ) ( )( )

2

122 112

1+ − − − −−CI RTe

CI RTt M SI

RtT

p t M peθ θ

θ

(19)

When S=C, the above expression reduce to Jamal (1997).

Special Case 2

Without inspection and including interest earned after fulfilling the back orders (at the start of the cycle), the equation (18) reduces to,

TVC S T AT

CD TT

C RTe t

C RTT tt

12 112 1

2121 1( , )

( )( ) ( )= + + − −

+ −

θθθ //

( ) ( )( )( )

2

122 112

11+ − − − − −−−CI R

Te

CI RTt M SI

RtT

SI R T t Mp t M pe

e

θ θθ

TT

(20)

Whereas Jamal (1997) has not taken into account the interest earned from revenue gener-ated after fulfilling the backorders at the start of the cycle on the selling price which will help retailer to reduce his total variable cost.

Special Case 3

With inspection and excluding interest earned after fulfilling the back orders (at the start of the cycle), the equation (18) reduces to,

TVC S T AT

CD TT

INT

C RTe t

C RTTt

13 112 1

21 1( , )( )

( ) (= + + + − −

+ −

θθθ tt

CI RTe

CI RTt M SI

RtT

p t M pe

12

2 112

2

12

1

) /

( ) ( )( )+ − − − −−

θ θθ

(21)

Case 2

0≤ t1≤ M≤T

In this case, one can earn interest on sales revenue up to the permissible period (M), and pay no interest for the units kept in the stock.

Now, interest earned per cycle has got three parts.

Part 1

In first part one can earn interest till the time period t1,

Interest earned = S I Rt dte

t

0

1

∫ (22)

Part 2

Second part is having interest earned for the time period (M-t1),

Interest earned = S I R t (M-t ) e 1 1 (23)

Part 3

Third part includes the revenue generated after fulfilling the back orders (at the start of the cycle) for the time period (M),

Interest earned = SI R(T-t ) Me 1 (24)

∴ Total interest earned in this case

S I Rt dte

t

0

1

∫ + SI R t (M-t ) e 1 1 +

SI R(T-t ) Me 1 (25)

Whereas Jamal (1997) has considered the interest earned part as

C I Rt dte

t

0

1

∫ +CRt I M te1 1( )−

Similar to case1, here also he has not taken into account the interest earned from revenue generated after fulfilling the back orders at the

IGI GLOBAL PROOF



start of the cycle, which may lead to inaccurate analysis.

Carrying Cost = C I t dtt

1 0

1

( )∫

= − −

C Re tt1

2 11 1

θθθ( ) (26)

Shortage Cost = C2 R t t dtt

T( )−∫ 1

1

= −C RT t2 12 2( ) / (27)

Inspection Cost = IN (28)

The total variable cost per cycle per unit time (TVC2) is the sum of the order cost, cost of deteriorated units, inspection cost, carrying cost and shortage cost but the interest earned is subtracted from this sum, i.e.

TVC S T AT

INT

CD TT

C RTe t

C RTT tt

2 112 1

21 1( , )( )

( ) (= + + + − −

+ −

θθθ

112

12

2

2

) /

− − −SIRtT Te

S I R t (M-t )T

S I R (T-t ) Me 1 1 e 1

(29)

Now, we discuss some special cases.

Special Case 1

Without inspection and excluding interest earned after fulfilling the back orders (at the start of the cycle), the equation (29) reduces to,

TVC S T AT

CD TT

C RTe t

C RTT tt

21 112 1

21

21 1( , )( )

( ) ( )= + + − −

+ −

θθθ //2

212

− −SIRtTe

S I R t (M-t )T

e 1 1

(30)

When S=C, the above expression reduce to Jamal (1997).

Special Case 2

Without inspection and including interest earned after fulfilling the back orders (at the start of the cycle), the equation (29) reduces to,

TVC S T AT

CD TT

C RTe t

C RTT tt

22 112 1

21

21 1( , )( )

( ) ( )= + + − −

+ −

θθθ //2

212

− − −SIRtT Te

S I R (T-t ) M S I R t (M-t )T

e 1 e 1 1

(31)

Whereas Jamal (1997) has not taken into account the interest earned from revenue gener-ated after fulfilling the backorders at the start of the cycle on the selling price which will help retailer to reduce his total variable cost.

Special Case 3

With inspection and excluding interest earned after fulfilling the back orders (at the start of the cycle), the equation (29) reduces to,

TVC S T AT

CD TT

INT

C RTe t

C RTTt

2 3 112 1

21 1. ( , )( )

( ) (= + + + − −

+θ

θθ −−

− −

t

SIRtTe

12

12

2

2

) /

S I R t (M-t )T

e 1 1

(32)

Total variable cost functions for both the cases are:

TVC S TTVC S T M t TTVC S T t M T

( , )( , )( , )11 1 1

2 1 1

00

=≤ ≤ ≤≤ ≤ ≤

Our objective is to find such values of S1 and T (for case1 and case 2) which minimize the total variable cost function (TVC1 and TVC2), since this function is differentiable, therefore the general necessary condition for minimiza-tion of the function is,

∂∂

=∂∂

=TVC S T

TTVC S TS

( , ) ( , )1 1

1

0

In Case 1,

∂∂

=TVCT

1 0

IGI GLOBAL PROOF



⇒− −+

+

+ − − − − −

AT

CRT

SCR S

t IT p

Rt SCT

S Rt2 21

11 2 1 1

12 1 1θ θ

( ) ( )SSI RtT

SI RMtT

CIT

e RR S

e e

p M

12

21

2

21

2

−

− −+

−θ

θ( )

+

− −( )

+

S R R t MC RTT1

2 122

2

2θ θ θ( −− =t1

2 0)

(33)

∂∂

=TVCS

1

1

0

⇒+

++

++

− −+

C S

T R SI S

Tp R SC S

T R SSI RMT R S

eθθ

θθ θ θ

1

1

1

1

1 1

1 1( ) ( ) ( ) ( )

++

+ +

− +−CI

Te RR S

C RR S

tT

SI RtT

pM

eθ

θ θ θ( ) ( )( )

1

2

1

1 11(( )R S+

=θ 1

0

(34)

and in Case 2,

∂∂

=TVCT

2 0

⇒− −+

+

+ − − − − −

AT

CRT

SCR S

t IT p

Rt SCT

S Rt2 21

11 2 1 1

12 1 1θ θ

( ) ( )SSI R M t t

TSI RMtT

TSI R

M tR

e e

e

( )

( )(

−

−

−−

1 12

12

1

2

1 2++

+ − =

θSC RTT t

1

22

212

20

)( )

(35)∂∂

=TVCS

2

1

0

⇒+

++

++

− −+

C S

T R SI S

Tp R SC S

T R SSI RMT R S

eθθ

θθ θ θ

1

1

1

1

1 1

1 1( ) ( ) ( ) ( )

−−+

−+

++

−SI R M t

T R SSI Rt

T R SC RR S

tT

e e( )( ) ( ) ( )

211

1

1

1

2

1

1

θ θ θ

=0

(36)

Solving the equations (33), (34), (35) and (36), we get the optimum values of T* of T and S1* of S 1 for TVC1 and TVC2.

For sufficient condition, determine the following for both the cases (TVC1 and TVC2)

∂∂

∂∂

∂∂ ∂

21

12

21

2

21

1

TVC S TS

TVC S TT

andTVC S TS T

( , ),

( , ) ( , )

The sufficient condition for minimization TVC (S1 , T) is∂∂

>∂∂

>

∂∂

21

12

21

2

21

12

0 0TVC S TS

TVC S TT

andTVC S TS

( , ),

( , )

( , )∂∂

−∂∂ ∂

>

21

2

21

1

2

0TVC S TT

TVC S TS T

( , ) ( , )

Since, these expressions (Appendix) are complicated and it is very difficult to prove these conditions mathematically. Alternatively we have shown graphically both the cost func-tions TVC1 and TVC2 are convex in nature using plotting tools of Matlab7.0.1 in Figures 3 and 4.

4. NUMERICAL EXAMPLE

Example 1

Given p = 0.85, θ = 0.1, Ip = 0.15 /year, Ie = 0.13 /year, R = 1000 units/year, A=200 dollar/order, i = 0.12 /year, C = 20 dollars/unit, I = 0.05, C1 = 2.4 dollars/unit/yr, C 2 =20 dollars/unit/yr, S = 24 dollars/unit and M = 15 days.

Matlab7.0.1 direct search tool is used to solve the equation (18), we get following results,

= − − − + −

− − − −CI

Se e R t M e ep

M t t M11

1 11

θ θ θθ θ θ θ( ) ( ) ( )

= 100 days T = 122 daysN = 398 units S1 = 278 units

TVC1 = Rs.1137

Special Case 1

Without inspection and excluding interest earned after fulfilling the back orders (at the start of the cycle), by solving equation (19), we get:

= − − − + −

− − − −CI

Se e R t M e ep

M t t M11

1 11

θ θ θθ θ θ θ( ) ( ) ( )

= 102 days, T = 122 days, N = 398 units, S1 = 284 units and TVC11 is Rs.1100.

Special Case 2

Without inspection and including interest earned after fulfilling the back orders (at the start of the cycle), by solving equation (20), we get = − − − + −

− − − −CI

Se e R t M e ep

M t t M11

1 11

θ θ θθ θ θ θ( ) ( ) ( )

=100 days, T = 122 days, N = 399 units, S1 = 279 units and TVC13 is Rs.1077.

W h i l e J a m a l ( 1 9 9 7 ) , = − − − + −

− − − −CI

Se e R t M e ep

M t t M11

1 11

θ θ θθ θ θ θ( ) ( ) ( )

IGI GLOBAL PROOF



= 96 days, T = 117 days, N = 323 units, S1 = 265 units and total cost function value is Rs.1157.

Special Case 3

With inspection and excluding interest earned after fulfilling the back orders (at the start of the cycle), by solving equation (21), we get = − − − + −

− − − −CI

Se e R t M e ep

M t t M11

1 11

θ θ θθ θ θ θ( ) ( ) ( )

=102 days, T = 122 days, N = 398 units, S1 = 283 units and TVC12 is Rs.1159.

Example 2

Given p=0.85, θ=0.1 Ip = 0.15/year, Ie = 0.13/year, R =1000 units/year, A = 200 dollar/order, i = 0.12/year, C = 120 dollars/unit, I = 0.90, C1 = 14.40 dollars/unit/yr, C 2 =10 dollars/unit/yr, S = 138 dollars/unit and M = 30 days.

Matlab7.0.1 direct search tool is used to solve the equation (29), we get following results,

= − − − + −

− − − −CI

Se e R t M e ep

M t t M11

1 11

θ θ θθ θ θ θ( ) ( ) ( )

= 15 days T = 81 days N = 260 units S1 = 41 units TVC2 = Rs.1391

Special Case 1

Without inspection and excluding interest earned after fulfilling the back orders (at the start of the cycle), by solving equation (30), we get: = − − − + −

− − − −CI

Se e R t M e ep

M t t M11

1 11

θ θ θθ θ θ θ( ) ( ) ( )

= 24 days, T = 77days, N = 247 units, S1 = 66 units and TVC21 is Rs.1443.

Special Case 2

Without inspection and including interest earned after fulfilling the back orders (at the start of the cycle), by solving equation (31), we get:= − − − + −

− − − −CI

Se e R t M e ep

M t t M11

1 11

θ θ θθ θ θ θ( ) ( ) ( )

= 15 days, T = 81 days, N = 261 units, S1 = 41 units and TVC23 is Rs.332. Because interest earned after fulfilling the back order (at the start of the cycle) is very high and reduces to Rs.1203.

While Jamal (1997),

= − − − + −

− − − −CI

Se e R t M e ep

M t t M11

1 11

θ θ θθ θ θ θ( ) ( ) ( )

= 24 days, T = 78 days, N = 212 units, S1

Figure 3. Optimal total variable cost with respect to t1 and T for case 1

IGI GLOBAL PROOF



= 66 units and total cost function value is Rs.1481.

Special Case 3

With inspection and excluding interest earned after fulfilling the back orders (at the start of the cycle), by solving equation (32), we get t1= 24days, T = 77days, N = 247units, S1 = 66 units and TVC22 is Rs.2503.

Findings clearly indicate that in both the cases i.e., M < t1 and M > t1, it is beneficial for retailer to introduce inspection and interest earned on the revenue generated after fulfill-ing the back orders (at the start of the cycle) as he is able to reduce his total cost to that of Jamal (1997).

5. SENSITIVITY ANALYSIS

In order to gain more in sight of the above model the sensitivity analysis has been per-formed on certain parameters viz. purchase cost (C), Deterioration rate (θ), Shortage cost (C 2

) and permissible payment period (M).Now for fixed C2, taking θ = {0.1, 0.2}, C

= {20, 40, 180, 200} dollars/unit, S = {24, 48, 216, 240} dollars/unit., C1 = {2.4, 4.8, 21.6,

24} dollars/unit/yr, I = {0.05, 0.1, 0.45, 0.50} and M = {15, 30, 45} days and assuming rest of the data of example 1. No feasible solution marked as ‘----‘in Table 1.

When purchase cost (C) is fixed and taking θ = {0.1, 0.2}, C 2 = {20, 50, 100, 1000} dollars /unit/yr, S = 48 dollars/unit., C1 = 4.8 dollars/unit/yr, I = 0.1 and M = {15, 30, 45} days and assuming rest of the data of example 1. No feasible solution marked as ‘----‘in Table 2.

Further for case 2, keeping fixed C2 and taking θ = {0.1, 0.2}, C = {120, 150, 180,200} dollars/unit, S = {138, 172.5, 207, 230} dollars/unit, I = {0.90, 1.13, 1.35, 1.50}, C1 = {14.40, 18, 21.6, 24} dollars/unit/yr and M = {15, 30, 45} days and assuming rest of the data of ex-ample 2. No feasible solution marked as ‘----‘in Table 3.

It is evident from Table 1 that as M in-creases, t1 increases which suggests that the retailer should order more and total average variable cost decreases due to the fact that the revenue generated after fulfilling the back order(at the start of the cycle). However, when the deterioration rate (θ) increases, cycle length (T) reduces but the average total variable cost increases.

Figure 4. Optimal total variable cost with respect to t1 and T for case 2

IGI GLOBAL PROOF



However, Table 2 indicates that asC 2 in-creases, t1 approaches to T that implies stock out tends to zero, which endorses the fact that if the inventory system is operated at all it is never optimum to have back order.

Further from Table 3 one can conclude when M > t1 , as M increases and average total variable cost decreases significantly as interest earned after fulfilling the back order increases significantly.

6. CONCLUSION

The present paper investigates the effect of initial inspection, interest earned on selling

price and interest earned after fulfilling the back orders (at the start of the cycle) on the retailer ordering policy. Findings have been very encouraging as retailer is not only able to increase his order quantity but also his total variable cost decreasing in comparison able to the situation where inspection cost and interest earned on revenue generated after fulfilling the back order has not been incorporated to that of Jamal (1997). Further three special cases viz. without inspection and excluding interest earned after fulfilling the back orders (at the start of the cycle), without inspection and including interest earned after fulfilling the back orders (at the start of the cycle) and with inspection and

Table 1. Optimal cycle length and total variable cost with respect to C, θ and M

Table 2. Optimal cycle length and total variable cost with respect toC 2 , θ and M

IGI GLOBAL PROOF



excluding interest earned after fulfilling the back orders (at the start of the cycle) and have been discussed. Results have been validated with the help of examples along with a comprehensive sensitive analysis.

A future study would be extended to the proposed model for linearly increasing and price/stock dependent demand.

ACKNOWLEDGMENT

The first author would like to acknowledge the financial support provided by University Grant Commission (Grant No. Dean(R) /R&D /2009/487).

REFERENCES

Aggarwal, S. P., & Jaggi, C. K. (1995). Ordering policies of deteriorating items under permissible delay in payments. The Journal of the Operational Research Society, 46(5), 658–662.

Covert, R. P., & Philip, G. C. (1973). An EOQ model for items with weibull distribution de-terioration. AIIE Transactions, 5, 323–336. doi:10.1080/05695557308974918

Dave, U., & Patel, L. K. (1981). (T, S) Policy inventory model for deteriorating items with time proportional demand. The Journal of the Operational Research Society, 32(2), 137–142.

Ghare, P. M., & Schrader, G. F. (1963). A model for exponential decaying inventory. Journal of Industrial Engineering, 14(3), 238–243.

Goyal, S. K. (1985). Economic order quantity un-der conditions of permissible delay in payments. The Journal of the Operational Research Society, 36(3), 335–338.

Goyal, S. K., & Giri, B. C. (2001). Recent trends in modeling of deteriorating inventory. European Journal of Operational Research, 134(1), 1–16. doi:10.1016/S0377-2217(00)00248-4

Heng, K. J., Labban, J., & Linn, R. J. (1991). An order-level lot-size inventory model for deteriorat-ing items with finite replenishment rate. Comput-ers & Industrial Engineering, 20(2), 187–197. doi:10.1016/0360-8352(91)90024-Z

Hollier, R. H., & Mak, K. L. (1983). Inven-tory replenishment policies for deteriorating items in a declining market. International Jour-nal of Production Research, 21(4), 813–826. doi:10.1080/00207548308942414

Jamal, A. M. M., Sarker, B. R., & Wang, S. (1997). An ordering policy for deteriorating items with al-lowable shortage and permissible delay in payments. The Journal of the Operational Research Society, 48(8), 826–833.

Mark, K. L. (1982). A production lot-size inventory model for deteriorating items. Computers & Indus-trial Engineering, 6(2), 309–317.

Raafat, F. (1991). Survey of literature on continuously deteriorating inventory model. Journal of Opera-tional Research, 42, 27–37.

Table 3. Optimal cycle length and total variable cost with respect to C, θ and M

IGI GLOBAL PROOF



Raafat, F., Wolfe, P. M., & Eddin, H. K. (1991). An inventory model for deteriorating items. Com-puters & Industrial Engineering, 20(1), 89–94. doi:10.1016/0360-8352(91)90043-6

Shah, N. H., & Shah, Y. K. (2000). Literature survey on inventory model for deteriorating items. Economic Annals (Yugoslavia), 44, 221–237.

Shah, Y. K. (1977). An order-level lot-size inventory model for deteriorating items. AIIE Transactions, 9(2), 108–112. doi:10.1080/05695557708975129

Chandra K. Jaggi is an associate professor in the department of operational research, faculty of mathematical sciences, University of Delhi, India. He earned his PhD, M.Phil. and Master’s degree from the Department of Operational Research, University of Delhi. His research interest lies in the field of analysis of inventory system. He has published more than 55 papers in vari-ous international/national journals including International Journal of Production Economics, Journal of Operational Research Society, International Journal of Systems Sciences, European Journal of Operational Research, Investigacion Operacional Journal, Advanced Modeling and Optimization, TOP, Opsearch etc.. He has guided M.Phil./ Ph.D. theses in Operations Research. He is Editor-in-Chief of International Journal of Inventory Control and Management. He is on the Editorial Board of the International Journal of Services Operations and Informatics, International Journal of System Assurance Engineering & Management, Indian Journal of Mathematics and Mathematical Sciences, Australian Journal of Basic and Applied Sciences Journal of Applied Sciences Research. He has traveled extensively in India and abroad and delivered invited talks.

Mandeep Mittal is a research scholar in the department of Operational Research, faculty of mathematical sciences, University of Delhi, India. He has completed his MS (Applied Math-ematics) in 2000 from IIT Roorkee. At present he is pursuing his PhD in operational research. His area of interest is “Inventory Control and Management”. He has published two research papers, one in International Journal of Mathematics and Mathematical Sciences (IJMMS) and other in International Journal of Industrial Engineering Computations.

IGI GLOBAL PROOF



APPENDIX

In case 1, equations (33) and (34) lead to,

∂2 12

TVCdT

= 23

AT

+2 11

1

3

CR e t

T

tθ θ

θ

− −( )+2 1 1

3

I R Rt SpT( )−

+2 11 1

2 3

1C R e t

T

tθ θ

θ

− −( )+C RtT2 1

2

3+

−− + − −( )−2 11

1

2 3

CI R e t M

Tp

t Mθ θ θ

θ

( )

- −

2 13

S I RM tTe -

S I R tTe 1

2

3

∂2 1

12

TVCdS

= CRR S T

θθ( )+ 12

+ I Rp R S T

θθ( )+ 12

+C R

R S T1

12( )+ θ

+

C R T R S RR S T

2 1

12

1( log( ) / )( )

+ − ++

θ θθ

+C I RR S T

p

( )+ 12θ

-S I RMR S T

e θθ( )+ 12

-

−− + +

+

S I R R S RR S T

e ( log(( ) / ))( )

1 1

12

θθ

In case 2, equations (35) and (36) lead to,

∂22

2

TVCdT

= 23

AT

+2 11

1

3

CR e t

T

tθ θ

θ

− −( )+2 1 1

3

I R Rt SpT( )−

+2 11 1

2 3

1C R e t

T

tθ θ

θ

− −( )+C RtT2 1

2

3+

−− +2 1 13

S I Rt M tT

e ( )- −

2 13

S I RM tTe -

S I R tTe 1

2

3

∂2 2

12

TVCdS

= CRR S T

θθ( )+ 12

+ I Rp R S T

θθ( )+ 12

+C R

R S T1

12( )+ θ

+

C R T R S RR S T

2 1

12

1( log( ) / )( )

+ − ++

θ θθ

+-S I RMR S T

e θθ( )+ 12

- −− + +

+

S I R R S RR S T

e ( log(( ) / ))( )

1 1

12

θθ

- −− + ++

S I R M R S RR S T

e ( log(( ) / ) )( )

θ θθ

2 21

12

IGI GLOBAL PROOF

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