Robust production planning and capacity control for...

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Dávid Gyulai et al. / IFAC-PapersOnLine 48-3 (2015) 2312–2317

© 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

Robust production planning and capacitycontrol for flexible assembly lines

David Gyulai ∗,∗∗ Botond Kadar ∗ Laszlo Monosotori ∗,∗∗

∗ Fraunhofer Project Center for Production Management andInformatics, Computer and Automation Research Institute (SZTAKI),

Budapest, Hungary (e-mail: [email protected]).∗∗ Budapest University of Technology and Economics, Department of

Manufacturing Science and Technology, Budapest, Hungary

Abstract: The frequently changing order stream and high product variety require robustplanning and control approaches, as well as a flexible system structure in order to fulfill thehighest possible customer service level and to keep the production costs on a reasonable level.In the paper, a combined production planning and capacity control method for assembly lines isproposed aiming at balancing the workload of the human operators and decreasing the overallproduction costs on a given time horizon.Instead of using the idealistic cycle times and simple manufacturing control rules, the proposedplanning and control methodology is based on adaptive calculations taken from continuouslyupdated historical production data. The manufacturing execution-level data is applied forbuilding regression models predicting the capacity requirements of the future productionscenarios. Besides, the historical data is also used as direct input of discrete-event simulations, todetermine the proper control policies of human operator allocations for the different scenariosmentioned above. In order to calculate reliable and feasible production plans, the regressionmodels and control policies are integrated in a mathematical programming model that minimizesa cost function representing the total production costs.

Keywords: assembly, planning, production control, mathematical programming, simulation

1. INTRODUCTION

High product variety, competitive market and the fluctu-ating order stream are important characteristics of today’smanufacturing industry. Taking the automotive sector intoconsideration, suppliers have to meet strict due datesand provide high quality products for their customers,while keeping their production costs on the lowest possiblelevel. Flexible manufacturing/assembly systems exist fordecades, however, they are still one of the most funda-mental solutions to react to the changes and disturbancesefficiently.

1.1 Flexible assembly lines

In the automotive industry, the cars themselves are usu-ally assembled on platform-based mixed-model assemblylines, while most of the main components are producedin flexible assembly lines. The general difference betweenthese solutions lie in the control flexibility of the lines:while the mixed model lines are capable of assembling anarbitrary changing sequence of product variants withoutany changeovers, flexible flow lines are designed for pro-ducing different product variants in batches. Therefore,in latter systems precise production planning is crucial tominimize the changeovers required to setup the line fromone product variant to another. Besides, most of these linesare not fully automated, which means human operators arerequired to perform certain assembly tasks.

Focusing on the manually operated flexible assembly lines—which is the main topic of the paper— the most generalplanning decisions include the identification of requiredcapacities and production lot sizes. In the paper we fo-cus on the mid-term planning of the assembly lines thatprovides cost-optimal plans with the calculated productionlot-sizes, release dates as well as the capacity requirements.In order to handle the changes and disturbances in a robustway, the proposed planning method is combined with alower level capacity control, which means the assignmentof operators to the different tasks along the productionplan. In our view capacity control generally decides aboutthe work hours and when and to which workstations hu-man resources are allocated (Rossi and Lodding, 2012).While the objective of the mid-term planning is to de-crease costs by eliminating the unnecessary changeoversand decreasing the stock levels, the capacity control isresponsible for balancing the workload of the operatorsand eliminating the idle times.

The paper introduces a method, which combines mid-termplanning and capacity control by applying mathematicalprogramming and discrete-event simulation. First, the ca-pacity control policies for each product variants are deter-mined to minimize the negative effects of shop-floor leveldisturbances like reject rates and machine breakdowns.Then the control policies are integrated in the mid-termplanning computation via regression models calculatingthe real capacity requirements.

Proceedigs of the 15th IFAC Symposium onInformation Control Problems in ManufacturingMay 11-13, 2015. Ottawa, Canada

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1.2 Robustness in production planning and control

In general, production planning methods rely on deter-ministic input data hence fail to cope with a dynamiceffects of the execution environment and the considerableuncertainty of the underlying planning information. Inorder to face the challenges, robust techniques are requiredwhich can provide feasible production plans. Robustnessin production planning involves refined approaches thataim at handling predictable or unpredictable changes anddisturbances. They respond to the occurrence of uncertainevents (reactive approaches) or protect the performance ofthe plan by anticipating to a certain degree the occurrenceof uncertain events (proactive approaches) (Tolio et al.,2011).

In our methodology robust planning is the logical layerof the robust production. A production plan is calledrobust if it results in an acceptable level of the selectedperformance indicators even if unpredictable disruptionsoccur during the execution of the plan. The robustness ofthe systems often works against other efficiency criteria,hence, it means a trade-off is required if the objective isto increase system’s robustness. Efficient ways of takinguncertainties into account, and to achieve more robustsolutions are either applying stochastic models (Csajiand Monostori, 2008) (e.g., by estimating the underlyingstochastic processes), or using adaptive and cooperativeapproaches which allows prompt responses to changes anddisturbances (Monostori et al., 2010).

1.3 Production planning for flexible assembly lines

Generally, the production and supply-chain planning tasksare represented with a three-stage hierarchy (strategic,tactical and operational levels), where the middle, tacticallevel is responsible for the aggregate production plan-ning. These planning problems are usually representedby mixed-integer programming (MIP) models that can besolved by systematic algorithms (e.g. branch and bound)or heuristics, even though the NP-complete nature ofthe problems. These planning problems usually apply adiscretized time horizon, and calculate the optimal planby assigning the production batch releases to the identicaltime slots (time buckets). Besides the time representation,an important reason for using integer variables in suchplanning models is the existence of important discreteparameters like setup times, setup costs and machine as-signment decisions (Pochet and Wolsey, 2006). Althoughefficient methods exist to solve those problems in an opti-mal or quasi-optimal way, large-scale problem instancesoccurring in industrial practice are still often hard tosolve in a reasonable time, which is always an importantrequirement in practice.

The planing models applied for the manually operatedflexible assembly lines typically include the constraintsof due-dates, human and machine capacities and setups.The objective functions of these models are usually theminimization of the production costs including the cost ofthe operation, setups, inventories and backlogs (if allowed)(Sillekens et al., 2011). Even though precise modelingand efficient solver methods are available, these modelsprovide plans which usually quickly become unfeasible

in the execution phase due to the variability of someimportant parameters (i.e. manual processing times areusually a stochastic). Similarly, machine breakdowns andreject/scrap rates causing rework are disregarded in mostof the planning models.

1.4 Simulation-based optimization with regression models

In order to consider these important factors already in themid-term plans, the paper introduces a novel simulation-based optimization technique production planning of flex-ible lines. Advanced simulation-based optimization meth-ods exist for decades now, and they are already appliedto solve production planning problems. In general, theyconsist of a mathematical optimization model in whichthe objective function or constraint(s) are represented byfunctions that are approximated by using the results ofsimulations (Azadivar, 1999). The reason for applying sim-ulation in these cases are usually the computational com-plexity or the lack of analytical expression of the objectivefunction and/or constraints. These challenges are oftenfaced when stochastic functions have to be represented inthe optimization models (e.g. lead time as a stochasticfunction of the in-process buffer levels).

In production planning, simulation-based optimization isusually applied in an iterative form, when some of the pa-rameters are modified after executing the calculated planin a simulation environment. The parameters can be iter-atively adjusted according to the results of the simulation,until the target values of the performance indicators arereached (Byrne and Hossain, 2005; Laroque et al., 2012).In contrast, a simulation-based optimization method isproposed in the paper that relies on linear regressionmodels instead of iterations, thus requires less computa-tion and always relies on up-to-date data. Although linearregression models may seem overly simplistic, they repre-sent properly practical problems and can outperform moresophisticated models which require higher computationaleffort.

2. PROBLEM FORMULATION

2.1 Production and capacity planning

In the paper a production planning problem is consid-ered, where the planner has to decide about both, theproduction lot-sizes of different product variants and therelease dates of the lots. The target production system isa manually operated, flexible flow assembly line built-upby sequentially coupled workstations. The line is operatedin an unpaced way, which means that there is no conveyorbelt for the material flow but the operators pass the prod-ucts from one station to another. The number of operatorsis less than the number of workstations, therefore, differentcapacity control policies can be applied to operate the line,depending on the number of operators and the operator-task assignments. The assembly line includes at least onetesting station that performs a quality check on everyproduct. The products that do not pass the test proceed toa manual rework station that is separated from the line.After performing the rework, the repaired products areretested again. A lot should totally be completed before anew one is started on the line.

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The production orders of the different variants are avail-able for a certain planning horizon that is split up into aset production shifts. Each order can be characterized byits volume and a specific due date. Make-to-stock optionis available in every shift, therefore in case of capacityshortage, orders can be fulfilled from stocks, however,holding inventory is associated with extra costs. Orderfulfillment after the due date is possible (backlogging) butalso penalized with extra costs. The objective of the mid-term planning is to provide an optimal, executable planthat is based on the minimization of the production costson a certain horizon, and to increase the utilization of thecapacities (machines and human operators).

2.2 Capacity control

Within the capacity control part of our methodology, theproper assignment of the operators to the assembly tasksis solved, in order to balance their workload and decreasethe idle times caused by the shifting bottleneck —based onthe assembled product variant— and the reject rates. Inthis case, the objective is to determine the best assignmentpolicies for each product variant and each possible numberof operators. It means that the number of operators canbe changed from shift to shift, according to the productionrates. However, more production lots are released in oneshift requiring different operator-task assignments whilethe number of the operators cannot be changed.

Usually, standard work instructions and corporate policiesdefine how to operate the lines with a given number ofoperators, however, they are all based on norm times andidealistic data. In order to define proper capacity controlthe underlying stochastic processes have to be considered,that is solved by considering historical production data.Therefore, the objective is to define how to assemble acertain product variant with a given number of operatorsto minimize the losses and balance the workload of theoperators instantly applying the preprocessed historicaldataset.

3. SOLUTION APPROACH

At most companies, production planning and schedulingprocesses are supported by an enterprise resource planning(ERP) system, which can handle the integrated infor-mation and material flow as well as the corporate levelplanning activities. However, they often fail to perform theline-level control tasks, as for example the calculation ofthe proper operation control and usually, these problemsare solved manually by the production planners or shiftleaders based on their experience. In order to define aproduction planning method that combines the capacitycontrol with the mid-term planning efficiently, the co-operation between the logical and physical layers of theproduction system must be ensured. This means that themathematical model has to rely on the production log datathat reflect the real work contents, instead that of the normtimes that are predefined for each product. The notationused in the prediction and planning models is summarizedin Table 3.1.

3.1 Define the capacity control policies

In order to determine the proper operator-task assign-ments for each product and all the possible number of op-erators, a discrete-event simulation (DES) tool is applied.Even though state-of-the-art assembly lines are usuallyequipped with advanced sensor network, the real workloadof the operators is hard to monitor. A detailed DES modelof the assembly line can provide reliable results aboutthe utilization and several various control policies can beevaluated. The main advantage of using simulation forsuch purposes is its capability to include the stochasticnature of important processes. For the validation of thesimulation model, sensor-level as well as MES data is usedthat reflect the real processing times, the production ratesand personnel.

Besides the simulation of the historical production, severalrandom generated but possible operator control scenariosare analyzed. The input parameters of the simulationare the number of the operators and the assignment ofthe operators to different tasks. In order to define thebest policy for each product variant, only one variantis produced in each scenario. The main output of thesimulation analysis are the utilization of the operators andthe performance of the line.

Table 1. The applied notation

Sets

N the set of production ordersP the set of productsT the set of working shifts

Variables

xit production of order i in shift t (binary)ypt production of product p in shift t (binary indicator

variable)qpt produced volume from product t in shift t (integer)wt number of operators working in shift t (integer)

Parameters

pi product of order ivi volume of order i [pcs.]di due date of order i [shift]hi inventory holding cost of order i [cost/order/shift]li late delivery cost of order i [cost/order/shift]cit deviation cost of order i in shift ts duration of a shiftr cost of a setupk cost of an operator per shiftwmax the max. number of operators working in the same shiftQ(qt) overall manual time requirement in shift t [minutes]

3.2 Prediction of real capacity requirements

In case of paced or highly automated assembly lines thecapacity requirements can be represented with determin-istic values, as their variability is rather low. As alreadymentioned in the introduction, in case the line is unpaced,moreover, the number of workers is less than the numberof workstations, the capacity requirements cannot be rep-resented reliably in a general way. Stochastic optimizationcan be applied in that cases, however, this require highcomputation efforts and special solver algorithms. Addi-tionally, diverse reject rates of the product variants andthe varying amount of rework also increase the complexityof the planning models.


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The production orders of the different variants are avail-able for a certain planning horizon that is split up into aset production shifts. Each order can be characterized byits volume and a specific due date. Make-to-stock optionis available in every shift, therefore in case of capacityshortage, orders can be fulfilled from stocks, however,holding inventory is associated with extra costs. Orderfulfillment after the due date is possible (backlogging) butalso penalized with extra costs. The objective of the mid-term planning is to provide an optimal, executable planthat is based on the minimization of the production costson a certain horizon, and to increase the utilization of thecapacities (machines and human operators).

2.2 Capacity control

Within the capacity control part of our methodology, theproper assignment of the operators to the assembly tasksis solved, in order to balance their workload and decreasethe idle times caused by the shifting bottleneck —based onthe assembled product variant— and the reject rates. Inthis case, the objective is to determine the best assignmentpolicies for each product variant and each possible numberof operators. It means that the number of operators canbe changed from shift to shift, according to the productionrates. However, more production lots are released in oneshift requiring different operator-task assignments whilethe number of the operators cannot be changed.

Usually, standard work instructions and corporate policiesdefine how to operate the lines with a given number ofoperators, however, they are all based on norm times andidealistic data. In order to define proper capacity controlthe underlying stochastic processes have to be considered,that is solved by considering historical production data.Therefore, the objective is to define how to assemble acertain product variant with a given number of operatorsto minimize the losses and balance the workload of theoperators instantly applying the preprocessed historicaldataset.

3. SOLUTION APPROACH

At most companies, production planning and schedulingprocesses are supported by an enterprise resource planning(ERP) system, which can handle the integrated infor-mation and material flow as well as the corporate levelplanning activities. However, they often fail to perform theline-level control tasks, as for example the calculation ofthe proper operation control and usually, these problemsare solved manually by the production planners or shiftleaders based on their experience. In order to define aproduction planning method that combines the capacitycontrol with the mid-term planning efficiently, the co-operation between the logical and physical layers of theproduction system must be ensured. This means that themathematical model has to rely on the production log datathat reflect the real work contents, instead that of the normtimes that are predefined for each product. The notationused in the prediction and planning models is summarizedin Table 3.1.

3.1 Define the capacity control policies

In order to determine the proper operator-task assign-ments for each product and all the possible number of op-erators, a discrete-event simulation (DES) tool is applied.Even though state-of-the-art assembly lines are usuallyequipped with advanced sensor network, the real workloadof the operators is hard to monitor. A detailed DES modelof the assembly line can provide reliable results aboutthe utilization and several various control policies can beevaluated. The main advantage of using simulation forsuch purposes is its capability to include the stochasticnature of important processes. For the validation of thesimulation model, sensor-level as well as MES data is usedthat reflect the real processing times, the production ratesand personnel.

Besides the simulation of the historical production, severalrandom generated but possible operator control scenariosare analyzed. The input parameters of the simulationare the number of the operators and the assignment ofthe operators to different tasks. In order to define thebest policy for each product variant, only one variantis produced in each scenario. The main output of thesimulation analysis are the utilization of the operators andthe performance of the line.

Table 1. The applied notation

Sets

N the set of production ordersP the set of productsT the set of working shifts

Variables

xit production of order i in shift t (binary)ypt production of product p in shift t (binary indicator

variable)qpt produced volume from product t in shift t (integer)wt number of operators working in shift t (integer)

Parameters

pi product of order ivi volume of order i [pcs.]di due date of order i [shift]hi inventory holding cost of order i [cost/order/shift]li late delivery cost of order i [cost/order/shift]cit deviation cost of order i in shift ts duration of a shiftr cost of a setupk cost of an operator per shiftwmax the max. number of operators working in the same shiftQ(qt) overall manual time requirement in shift t [minutes]

3.2 Prediction of real capacity requirements

In case of paced or highly automated assembly lines thecapacity requirements can be represented with determin-istic values, as their variability is rather low. As alreadymentioned in the introduction, in case the line is unpaced,moreover, the number of workers is less than the numberof workstations, the capacity requirements cannot be rep-resented reliably in a general way. Stochastic optimizationcan be applied in that cases, however, this require highcomputation efforts and special solver algorithms. Addi-tionally, diverse reject rates of the product variants andthe varying amount of rework also increase the complexityof the planning models.


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In order to tackle these problems, in our approach aproduction planning model is introduced that determinesthe near optimal production plan and the number ofhuman operators simultaneously, even besides the abovementioned factors. The essence of the method is theintroduction of the capacity requirements as a generalfunction of the products produced in the same shift. Thesefunctions can be approximated by regression models andcan be integrated directly in the production planningcalculation.

In order to approximate the real capacity requirement ofa given production lot mix assigned to the shift (Q(qt)), amultivariate linear regression model is proposed. The effi-ciency of applying regression models for capacity planningin an uncertain environment was shown by the authorsin Gyulai et al. (2014a,b). The input variables of theregression are the volumes of the products assembled in thesame shifts (qpt), and the output is the total manual timethat is required to assemble the products. The regressionmodels are defined by historical data gathered from theSCADA (supervisory control and data acquisition) systemof the assembly line. The regression function is defined asfollows:

Q(qt) = β0 +

P∑p=1

βpqpt. (1)

By this way, the real capacity requirements (includingrework rates, machine downtimes operator movementsand capacity control policy effects) of the set of ordersassembled in the same shift can be estimated. The functioncan be integrated in the aggregate production planningmodel, which is described in the following section.

3.3 Production planning model

The core of the production planning is formulated as amixed-integer programming problem including the capac-ity requirement function, as well as the other constraintslike the order due dates and inventory holding costs. Thedecision variables of the model give the number of allo-cated operators for each shift (wt), the number of setups(ypt), the assembled volumes of products per shift (qpt) andthe production of the orders (xit). The model minimizes anobjective function that is the sum of the deviation (earlydelivery and holding), setup and personnel costs (2).

minimizeN∑i=1

T∑t=1

citxit + r

P∑p=1

T∑t=1

ypt + k

T∑t=1

wt (2)

subject toT∑

t=1

xit = 1 ∀i (3)

xit ≤ ypt ∀t, p = pi (4)

qpt =

N∑i=1

xitvi ∀t, p = pi (5)

wts ≥ Q(qt) ∀t (6)

wt ≤ wmax ∀t (7)

xit ∈ {0, 1} ypt ∈ {0, 1} wt ∈ Z+ (8)

cit =

{hi(di − t) if t < dili(t− di) otherwise

(9)

The constraints include the fulfillment of all customerorders (3), the calculation of the setups (4) and volumes(5), the capacity restrictions (6)(7) as well as integrityconstraints (8). Equation 9 defines the extra cost of the latedelivery and inventory holding. The resulting productionplan specifies the required number of operators over thehorizon, gives the assignment of the customer orders tothe production shifts.

3.4 Implementation of the method

The method is evaluated within a software framework thatis able to couple various modules developed for variousplanning and control purposes. The framework is calledSimulation and Navigation Cockpit, and developed for in-creasing the robustness of the plants and supply networksby applying state-of-the-art solutions. The software has acentral database that provides the integrated data storageand acquisition among the planning module, the simula-tion and the graphical user interface (GUI). The proposedmethod is implemented in the framework by applying aloop-based work flow (Figure 1). The capacity control loopuse the sensor network data and the artificially generatedscenarios to determine the proper control policies and theutilization of the resources. In the production planningloop, regression models are built over the simulation re-sults, and they are integrated in the production plannermodule that implements the MIP model. The whole pro-cedure can be controlled via the web-based graphical userinterface of the framework.

4. EXPERIMENTAL RESULTS

The efficiency of the combined production planning andcapacity control method is demonstrated by an industrialuse-case.

Fig. 1. Implementation of the method in the Simulationand Navigation Cockpit


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Fig. 2. Schematic of the assembly line

4.1 The characteristics of the assembly line

The target system is a flexible assembly line of an automo-tive supplier. The line consists of manually operated work-stations, an automated testing machine with five slots,final assembly stations and a rework station (Figure 2). Inthe line, three product families (A,B,C) are produced, eachfamily has several variants. The total number of productvariants produced on the line is approximately 150 andthe diversity of the yearly volumes is rather high. The lineoperates three shifts per day, currently the average numberof setups is 6-8 per shift. The reject rates of each producttype are distinct, therefore, it is important to balance theireffects with adjusted production sequence and the corre-sponding capacity control. To ensure the reliability of theresulted plan, sensor-based process monitoring provides alarge amount of data about processing times, setup times,and throughput.

4.2 Simulation and capacity control

In order to determine the best capacity control policiesof the product variants, all combinations of the possibleinput parameters were analyzed in the simulation module.The result of this step is a p × wmax matrix, containingthe operator-task assignment that resulted the highestthroughput and least idle times for each p and wt.

4.3 Prediction of the capacity requirements

Fig. 3. Results of the multivariate regression

The multivariate regression for the approximation of thecapacity requirements was computed using the R statisti-cal environment, by applying its general linear regressionfunction, which took less than 1 second. The regression

model was built over a historical dataset with 1728 shifts,that was split up into a training (576 shifts) and test (1151shifts) set. As for the input variables, the regression isbased on nine product variants (with the highest yearlyvolumes) that are the most significant variables accordingto the significance test (each product family is representedby one candidate product). According to the results, themultivariate linear model provides precise prediction forthe real capacity requirements: R2 = 0.937, and for allp values, p < 2 · 10−16. It is also important to notethat the real capacity requirements are significantly higherthan the ones calculated according to the capacity normsand applied in the general production planning tools. Thereason for the difference are the stochastic nature of theunderlying processes like the machine downtimes, rejectrates and processing times (Figure 3).

4.4 Production planning and capacity control of the line

The above regression model can be applied directly inthe production planning model that was implementedin FICO Xpress and solved by its default branch andbound method. The optimization algorithm was run untilan optimality gap of at most 2% was achieved, whichrequired 280 seconds on average. In the experimental case,a fixed-horizon planning problem was investigated, andsolved by the current norm-time based planning as wellas with the improved, regression-based model. The inputof the production planning was 357 production ordersconcerning nine products with the highest yearly volumes.The horizon of the planning was 20 days, the due date ofthe orders were given in production shifts from 1 to 60.

In the first case, the planning was done based on thecurrently applied norm times, and the resulted plan wasexecuted in a simulation environment in which stochasticeffects were also included. The execution of the planresulted in a significant amount of backlogs —due to thecapacity shortages— that means 907 products were notassembled and delivered in time (Table 2).

In the second case, the capacity constraints in the modelwere represented by equation 6, the rest of the planningmodel left unchanged. According to the simulation results,the improved planning model resulted in a production planthat is more robust against the negative effects of rejectrates, as the amount of backlogs was decreased to 106,and the objective function value —including the deviation,operator and setup costs— also much less than in theprevious case.


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Fig. 2. Schematic of the assembly line

4.1 The characteristics of the assembly line

The target system is a flexible assembly line of an automo-tive supplier. The line consists of manually operated work-stations, an automated testing machine with five slots,final assembly stations and a rework station (Figure 2). Inthe line, three product families (A,B,C) are produced, eachfamily has several variants. The total number of productvariants produced on the line is approximately 150 andthe diversity of the yearly volumes is rather high. The lineoperates three shifts per day, currently the average numberof setups is 6-8 per shift. The reject rates of each producttype are distinct, therefore, it is important to balance theireffects with adjusted production sequence and the corre-sponding capacity control. To ensure the reliability of theresulted plan, sensor-based process monitoring provides alarge amount of data about processing times, setup times,and throughput.

4.2 Simulation and capacity control

In order to determine the best capacity control policiesof the product variants, all combinations of the possibleinput parameters were analyzed in the simulation module.The result of this step is a p × wmax matrix, containingthe operator-task assignment that resulted the highestthroughput and least idle times for each p and wt.

4.3 Prediction of the capacity requirements

Fig. 3. Results of the multivariate regression

The multivariate regression for the approximation of thecapacity requirements was computed using the R statisti-cal environment, by applying its general linear regressionfunction, which took less than 1 second. The regression

model was built over a historical dataset with 1728 shifts,that was split up into a training (576 shifts) and test (1151shifts) set. As for the input variables, the regression isbased on nine product variants (with the highest yearlyvolumes) that are the most significant variables accordingto the significance test (each product family is representedby one candidate product). According to the results, themultivariate linear model provides precise prediction forthe real capacity requirements: R2 = 0.937, and for allp values, p < 2 · 10−16. It is also important to notethat the real capacity requirements are significantly higherthan the ones calculated according to the capacity normsand applied in the general production planning tools. Thereason for the difference are the stochastic nature of theunderlying processes like the machine downtimes, rejectrates and processing times (Figure 3).

4.4 Production planning and capacity control of the line

The above regression model can be applied directly inthe production planning model that was implementedin FICO Xpress and solved by its default branch andbound method. The optimization algorithm was run untilan optimality gap of at most 2% was achieved, whichrequired 280 seconds on average. In the experimental case,a fixed-horizon planning problem was investigated, andsolved by the current norm-time based planning as wellas with the improved, regression-based model. The inputof the production planning was 357 production ordersconcerning nine products with the highest yearly volumes.The horizon of the planning was 20 days, the due date ofthe orders were given in production shifts from 1 to 60.

In the first case, the planning was done based on thecurrently applied norm times, and the resulted plan wasexecuted in a simulation environment in which stochasticeffects were also included. The execution of the planresulted in a significant amount of backlogs —due to thecapacity shortages— that means 907 products were notassembled and delivered in time (Table 2).

In the second case, the capacity constraints in the modelwere represented by equation 6, the rest of the planningmodel left unchanged. According to the simulation results,the improved planning model resulted in a production planthat is more robust against the negative effects of rejectrates, as the amount of backlogs was decreased to 106,and the objective function value —including the deviation,operator and setup costs— also much less than in theprevious case.


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An important fact that these results were achieved byproper sequencing that can decrease the negative effectsof the rejects rates, and not by simply increasing the normtimes and thus the allocated capacities as well. Moreover,

the number of working shifts (∑T

t=1 wt, s = 480 minutes)is decreased from 290 to 240 that is a positive ’side effect’of the regression-based planning (Table 2 and Figure 4.4).Comparing the total production costs (objective functionvalue), the regression-based method provides much betterresults.

Beside the production planning, another important goal isto identify the proper capacity control policies. This taskwas solved by running the simulation model on a historicaldataset with different operator-task assignments. By thisway, the best scenarios were selected —for each of the nineproducts with the highest yearly volumes— and includedin the planning problem. In the production plan executionmode of the simulation model, these controls were applied,in order to make the plan even more robust.

Table 2. Comparison of the planning methods

Method Obj. function Backlog [pcs.]∑T

t=1wt [shifts]

Current 14 904 907 290Improved 12 467 169 240

Fig. 4. Execution of the plans: the amount of realizedbaklogs along the planning horizon

5. CONCLUSION

In the paper, a robust, regression based aggregate plan-ning method was introduced that is aimed at providingfeasible production plans that face with changes and dis-turbances occur during the production. The method isbased on a multivariate-regression to estimate the capacityrequirements of the orders that are assigned to the sameproduction period. The method relies on historical datagathered from the SCADA system providing reliable ca-pacity estimation that include the stochastic parameterslike the downtimes, varying rework rates occurred by therejects and the stochastic processing times. The capacityrequirements were represented by a multivariate linearfunction that can be integrated directly in the mathe-matical model of the aggregate planning model. By thisway, the production (order-shift assignment) and shiftplanning is done simultaneously. By introducing addi-tional constraints in the model, special requirements likepattern-based shift planning can be solved, considering thecompany-specific planning requirements. The efficiency ofthe planning method is proven to be robust against the

reject rates by evaluating its feasibility with discrete-eventsimulation.

As for the future work, the primary aim is to generalizethe planning method to be able to apply it for differenttypes of assembly system. Another important goal is todefine a self-building modeling framework that appliesuniform data structure to build-up the simulation modelof the systems simultaneously with the correspondingmathematical models in order to ensure their co-evolutionand validation.

ACKNOWLEDGEMENTS

Research has been partially supported by the EuropeanUnion 7th Framework Programme Project No: NMP 2013-609087, Shock-robust Design of Plants and their SupplyChain Networks (RobustPlaNet) and by the NationalDevelopment Agency, Hungary, Grant No. ED 13-2-2013-0002.

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