Optimization for Hierarchical Production Planning of...

International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:14 No:02 90

I J E N S IJENSIJENS April 2014 -IJMME-5757-029914

Optimization for Hierarchical Production Planning of

Industrial Processes Asmaa A. Mahmoud

1, Islam H. Afefy

2, M. Abdel-Karim

3 1. Post Graduate student -Industrial Engineering Department, Faculty of Engineering, Fayoum University

2. Assistant Prof. Industrial Engineering Department, Faculty of Engineering, Fayoum University. 3. Professor, College of Engineering & Information Technology, University of Business & Technology, Jeddah, Saudi Arabia

(On leave absence, Dept. of Industrial Engineering. Fayoum University, Fayoum, Egypt).

* Corresponding Author

Islam Helaly Afefy: Industrial Engineering Department, Faculty of Engineering, Fayoum University: e-mail:

[email protected]

Abstract-- In this paper, a generalized mathematical model formulation for cellular manufacturing system (CMS) using

hierarchical production planning (HPP) approach is proposed.

The model is applied to two different real case studies and is

solved by using operation research optimization software (Lingo

(12.0) program). The model is divided into three main steps as

follows: data collection, mathematical model formula, and

results. The proposed mathematical model of the optimization

can solve the problems of the system under utilizing the limited

resources in a production plan. The results show that the

proposed mathematical model can be used to minimize

manufacturing total costs of products for similar cases. To prove

the work, two case studies are introduced; for the first case

(electric water heater with capacity 50 liter (EWH1)), the results

show that the total cost decreases by 8.46 % for the optimum

conditions. For the second case, (electric water heater with

capacity 80 liter (EWH2)), as global results, the total cost

decreases by 3.7% for the optimum conditions.

Index Term-- Hierarchical Production Planning (HPP), Cellular Manufacturing System (CMS), Group Technology (GT) and

General Manufacturing Company (GMC).

1. INTRODUCTION

The hierarchical production planning (HPP) approach is firstly introduced by Hax and Meal (1975). In this approach,

Hax and Meal structured the problem into three levels

hierarchy based on types, families and items. Firstly,

aggregation planning was done on types which include all

production families. Secondly, families are the groups of

item within types classification. These items require a

major equipment setup before they are produced. The HPP

divides the large planning optimization problem known as

monolithic problem into sub- problems, as described in

Figure 1.The components of different levels of aggregation

for products are:

1- Top level result from an aggregate planning model using

linear programming and linear decision rules for variable

according to its types. These types are sets of items that are

similar in term of seasonal demand pattern and production

rates.

2- Middle level considers a heuristic disaggregation of the

type into families. These families are sets of items having

similar setup costs.

3- Low level decision consists of disaggregation families that

sets of items based on equalizing run out times.

The HPP approach is necessary to demonstrate the CMS. The CMS is subset of group technology (GT). The GT is a

new approach for batch production (small lot- production) in

which similarities exist among component part or processes.

While applying the GT, it depends on formation of part

families on basis of design and/or manufacturing.

Furthermore, applying GT leads to a significant improvement

on product design, manufacturing engineering, and the CMS

(Akturk and Wilson, 1998). The group technology family can

have more than one feasible cell for its production. Inside such

feasible cell, a family can be processed entirely. In general,

the feasible cell consists of two sub- cells defined as:

Primary cell, which is capable of producing the family at the lowest possible cost.

Secondary cells, those are ones in which the manufacture of the family is possible at a higher cost

due to the increase of both setup and material handling

costs.

Now, cell loading is a decision activity that determines

the kind of items and quantities to be produced in each cell

within specific time period. The production process is

subject to production capacity and demand forecast. In batch

production (BP) environment, there are two different

approaches for cell loading problem which are monolithic

and hierarchal approaches.

The First approach known as Monolithic Approach, formulates cell loading problem as a large Mixed-

Integer Linear Programming (MILP) problem at an

individual item level. In this case, a heuristic

procedure can be applied to solve cell loading

problem.

The second approach known as Hierarchical Approach, divides the overall problem into a

hierarchy of smaller problems (Akturk and Wilson,

1998).

Hierarchical approach that is used directly toward

enhancements in several ways utilizing knowledge of group

technology based on manufacturing system. These

enhancements can be summarized as:

Formulation of production planning problem with adding capacity constraints of the systems. The



results of the production rates are determined

within capacity of the system.

At the HPP approach the decision making is top-down, being from highest level to lower level.

At cell loading level, there are three basic ways of responding

to change in demand:

1. Holding alternatively constant production rate and using inventory to satisfy demand peaks.

2. Using changes in level of production which include overtime and assigning some of items to their

secondary cells with an additional production cost.

3. Combining these two strategies to meet demand modified by Akturk and Wilson (1998).

This paper is divided into five sections. In section 1, the

HPP approach is described shortly. In section 2, the

background of our research (literature reviews) is discussed.

In section 3 a methodology for solving the CMS is given. In

section 4 a description of case study and application

methodology for HPP are introduced. Finally; conclusions and

recommendations in section 5.

2. LITERATURE REVIEWS

The main objective of many researchers was to obtain an optimization method to solve the problems of the system

under utilizing the limited resources in a production plan to

meet variability in products demand. So a classification for the

HPP approach at many cases can be summarized as following:

Goloven (1975); based on Hax and Meal was first

introduced the HPP approach for single-stage production

planning and the concept of hierarchal planning through the

difference between planning and operational. He also

modified multi-stage production planning as an extension to

the HPP approach for single stage models to systems with

multi-stages. A pioneer at the HPP approach at the case of

seasonal demand was introduced by Haas (1979), as proposed

a heuristic to perform the levels of hierarchical planning.

Gfrerer and Zipfel (1995) were proposed the HPP approach at

the case of uncertain demand for multi-period model .It

consist of an aggregate planning level and a detailed planning

level. Then, this approach modified by Wu and Ierapetritou

(2007). Carravilla and Sousa (1995) discussed a complex

analysis production planning problem in a make- to- order

company using hierarchical production planning approach.

Katayama (1996) proposed a relevant production planning

procedure for multi-item continuous production in term of

two- stage automated hierarchical planning procedure. The

HPP approach for complex manufacturing systems is studied

in details by Mehraz et al., (1996). Akturk and Wilson (1998)

discussed the CMS by proposing a hierarchical cell loading

approach to solve the production planning problem. Ozdamar

and Birbil (1999) described the HPP approach at case of

energy intensive production environment for energy intensive

industries (e.g. steel, tile and glass ware industries).This

search was to minimize the number of active kilns throughout

the year besides optimizing the process design in the curing

department. El-Houssaine et al., (2011) investigated a robust

HPP approach for a two-stage real world capacitated

production system operating in an uncertain environment.

Habibie et al (2012), developed multi objectives model using

Lingo program (software).

3. METHODOLOGY

In this research, the CMS problem is solved using Lingo

program, which is a comprehensive tool designed to make

building and solving mathematical optimization models easier

and more efficient. Lingo (12.0) provides a completely

integrated package that includes a powerful language for

expressing optimization models and a full-featured

environment for building and editing problems. In addition, it

has a set of fast built-in solvers capable of efficiently solving

most classes of optimization models. Lingos results of the

CMS are reducing costs in order to be competitive, saving

time, increasing achieving market demand and achieving goal

of the system under certain criteria and constraints design.

3.1 Mathematical Modeling

This section will develop three parts for solving the CMS

problem. The first part illustrates objective function; the

second part defines the mathematical formulation while the

third part is devoted to illustrate index and notation. These

parts are described briefly in the following subsections.

A. Objective Function

The objective function is to minimize variable production

cost. Such as (production cost, cells setup cost, inventory

holding cost and regular capacity cost). These variables can be

simplified as:

4

1

4

1

2

1

7

1 )(

.)/()(t t i

itit

j

jtjtijt

Pi

ijtij

jFSi

ijtijt IFhRrXQBPXCCostMinj

(1)

B. Mathematical Formulation

There are three types of constraints at hierarchical model

for cell loading problem in the CMS. These constraints can be

defined as:

Production Constraints for families and items described by Eqs.(2) and (3)

Inventory Constraints for families and items described by Eqs.(2), (3), and (4)

Production and resources capacity constraints over the planning horizon described by Eqs (5), (6), (7), (8),

and (9)), respectively.

7

1

1,

j

itittiijt dIFIFX (2) , for j=1, 2,7,

t =1,2, 4



)(

0iTIk

n

Iw

itktw IFI

(3)

jtPi

ijtijtij

jFSi

ijtij RXQbpXaj

)./()(

(4)

)lim(0 itupperR jt j (5)

)( )(

)(jFSi iTIk

LT

n

Iw

kjtwkl RRZPR LR(j),j,t (6)

)1).(lim(0 LtLt itupperRR L,t (7)

0)(

jt

jLRl

Lt RRR (8)

0,,,,,,, LtLtktwkjtwjtjtitijt RRORIZROIFX j,t (9)

C. Index and Notation

In this section, the notations in the formulation will be

described. It should be noted that, all cost parameters are

measured in Egyptian pound (L.E). The notations are

classified into parameters and decision variables as shown in

details in the following:

Parameters:

ija

: Average total time required to produce one unit of

family i at cell j, (h/unit).

ijBP

: Setup cost for family i in primary cell j, (L.E).

ijbp

: Setup time for family i in primary cell j, (h/unit).

ijtC

: Average unit cost for producing one unit of

family i by cell j in period t, (L.E).

itd

: demand for family i in period t, (Unit).

ktwd

: Demand for item k in sub period w of period t,

(Unit).

FS (j): A feasible set of families assignable to cell j.

ith

: Average holding cost for family j in period t,

(L.E).

J : number of cells

K : number of items

L : number of resources

jP

: Set of families which their primary cell is j, (Unit)

klPR

: Average total time required to produce one unit

of item k using resource l, (h).

ijtQ

: Initial estimation for the lot size of family i in cell

j in period t, (Unit)

jtr

: Average cost of one regular time unit for cell j

during period t, (L.E).

T : planning horizon.

TI (i): A set of items belonging to family i

t : n*w, where n is an integer multiple , and w is sub

period.

lt

: Average proportion of time resource l is down in

period t.

Decision variables:

kitwI

: Inventory of item k at the end sub period w of

period t, (Unit).

itIF

: Inventory of family i at the end of period t, (Unit).

jtR

: Regular time used by cell j in period t, (h/month).

ltRR

: Regular time used by total resources l in period t,

(h/month).

ijtX

: Number of units of family i produced by cell j in

period t, (Unit).

kitwZ

: Number of units of item k produced by cell j in sub

period w of period t, (Unit).

The most important part in this study is conceptual

framework and mathematical model which were proposed for

CMS. The study shows the basic principles of calculations of

the method for CMS model. Through; define Lagrange theory,

apply decomposition procedures, update the values of

Lagrange method using gradient method and use optimization

software program Lingo (12.0) to solve mathematical model.

Moreover, the flowchart of algorithms framework is carried

out. This chart concluded in details all the steps of solving the

problem, and the final part is devoted to illustrate the proposed

software methodology in solving the mathematical

optimization models in an easier and more efficient way. In

the previous study M. Akturk, and G. Wilson, were stopped at

using Lagrange theory but our solution complete the previous

study as it consists of formulating a Lagrange relaxation of the

initial model, solving this dual problem by efficient and

iterative solution procedures, applying decomposition

procedures, updating the values of Lagrange method using

gradient method and using optimization software program

Lingo (12.0) to solve mathematical model.

4. CASE STUDY This case study analyzes the impact of the changes in

parameters of several goals that must be achieved. First, lets

define a real case study applied at General Manufacturing

Company (GMC) for manufacturing engineering in Egypt.

The company manufactures three products: full automatic

washing machine, gas cooker and electrical water heaters

(EWH) with capacity of (30, 40, 50, 80,100 Liter). Its main

With respect to



interest is studying the last product i.e. EWH especially with

capacity of 50 L and 80 L in details. Hence, the proposed

mathematical model is applied through data collection as a

first step then through mathematical formula as a second step

and finally through obtaining results in the last step.

4.1. Problem Statement

In this section, we describe the problem of planning

production in this application. We first discuss the problem

statement, and then we introduce the solution method. In

general, systems play an important role in many industrial

processes. Particularly, the main objective of the CMS is to

obtain the optimum method to solve the problem of

minimizing the variable production costs (production cost,

cells setup cost, inventory holding cost and regular capacity

cost). The problem of our case study is to minimize the total

product cost, and increase profitability ; therefore, we use the

HPP approach for solving this problem .To achieve this target,

it is necessary to manage and control the relation between the

limited resources in the plan of the production and variability

in products demand.

4.2. Solution Method

A schematic flow chart for framework of algorithm used

to the CMS is illustrated in details in figure 2. Using this

algorithm provides optimal solution for the system. It should

be noted that, the algorithm consists of three main steps

named as: data collection, mathematical model and results.

These steps are discussed too. This schematic is introduced in

details at Appendix. A and B and its steps of optimal solution

are shown.

4.3 Data Collection In order to illustrate the capabilities of the proposed-model,

actual data from the GMC (with emphasis on the case of EWH

with only two capacities i.e. 50 Liter and 80 Liter) have been

collected. A comprehensive numerical example is presented to

illustrate the applicability of the proposed model in

environment. This example is solved with the Lingo (12.0)

software.

In this case, all different types of cost parameters that are

used for the computational study are carried out. The data

collections concerning of studying EWH which is presented as

following: 2 families of EWH with capacity of 50L (EWH1)

and EWH with capacity of 80L (EWH2). In addition to 7

manufacturing cells, 4 planning horizon period (4 weeks), 5

resources of production (machine, manpower, material ,

maintenance (spare parts)and energy), 65 (parts) items, no

secondary cells in production , no overall overtime, no

backorder, P1,P2,,P7 ={family 1.2}, where P1 is a set

of families at primary

cell(1),FS(1),.FS(7)={family1.2},where FS(1) is a

feasible set of families assignable to cell (1), TI (1) =TI (2)

=65 items, where TI (1) is a set of items to family(1),and

LR(1)=LR(2),LR(7)={resource1,2,3,4,5},where

LR(1) is a set of resources belonging to cell (1). Moreover,

variable parameters in the system are summarized in table (1),

where UN~ [a, b] represents a uniformly distributed random

variable in interval [a, b]. It should be noted that, most of the

value of these parameters are constant throughout the planning

horizon .In addition, the processing time of each item at each

feasible resource klPR , and the number of operations for each

item are fixed.

4.4 Results In this section, the results for different cases which are

under consideration, will be presented and discussed in details.

Two comprehensive examples with full input data and results

analysis are presented. Also, an explanatory scheme shows the

results of our system for the planning period. First, let us

consider the first product i.e., the EWH1 product. The

comparison between the initial and optimum total product cost

for EWH1 is presented in figure 3. A significant difference is

founded in the total product cost. It can be observed that the

optimum total product cost (Z) of EWH1 decreases with

respect to the initial values from 4.48*108 L.E in the initial

value conditions to 4.1*108 L.E in the optimum.

There are different types of variables which are shown in

table 2. We have 15 variables, but the most effective three

were discussed. These three effective variables cause

decreasing in the total product cost as plotted in figure 4 and

6. The optimization problem has been solved for the variable

parameter values given in table 2. The optimum value of each

variable was calculated, in which minimizing the total product

cost for EWH1 is represented by that variable. This table

shows the average cost of one regular time unit of cell

decreases from 21164.73 L.E in the initial conditions to

20251.47 L.E in the optimum conditions. Moreover,

approximately 8.43% of the total product cost for EWH1 was

saved according to the optimization results in the initial

conditions.

A Comparison between Initial and Optimal Variables

Values for EWH1 Product are plotted in figure 4. It is found

that there are improvements of these variables which reflect

the reducing total product cost of EWH1. For example,

inventory of the product at the end of planning period

decreases from 1295 unit/ month in the initial

conditions to 1275 unit/ month in the optimum conditions.

Also, the total product cost decreased with the decreasing of

the inventory cost.

As shown in figure 5, is presented the comparison between

initial and optimum total product cost of EWH2 as obtained

from Lingo program. A significant difference in the total

product cost of EWH2 can be observed. This means reduction

in the total product cost of EWH2 can be obtain by using a

suitable optimum technique. This figure shows approximately

5.4*106 L.E was saved according to the optimization results in

the case study.

To discuss these results in details, the comparison between

initial and optimum variable values is presented in table 3.

The critical securitizations for this table indicate that. There is

notable improvement at specific effective variables

parameters. It found that the optimum total product cost

decreased with existence of the initial values. This table shows

that the average cost of one regular time unit of cell decreases



from 36751.25 L.E in the initial conditions to 36001.29 L.E in

the optimum conditions. Approximately 4.04 % of the total

product cost was saved according to the optimization results in

the case study of (EWH2). These results agree with the results

of those previous investigations for Akturk and Wilson, 1998.

For more clear and discussed data in table 3, where there

are effective variable parameters. These effective variable

parameters are presented and analyzed in graph as shown in

figure 6. From figure 6, it is observed that there are

improvements at these variables that reflect the reducing of

total product cost. For example, initial estimation for the lot

size of the product in cell at planning period decreases from

2055 unit/ month in the initial conditions to 2047 unit/ month

in the optimum conditions. Also, the total product cost is

decreased with respect to the decrease of the production cost.

4.5 Sensitivity Analysis on the Optimization Model for CMS Results

In order to carry out an analysis to determine the sensitivity

of the problem developed for CMS using the HPP approach, a

different parameters utilized in the case study for (EWH1,

EWH2) with emphasis on the effective variable parameters:

ijtX , ijtQ , itIF , itd ,and kitwI . Those values, of all these

parameters, that resulted in minimum total product cost are

chosen to be used for sensitivity analysis. The sensitivity of

the approach against variations in the numbers of ijtX , ijtQ ,

itIF , itd , and kitwI is presented in figures 7, 8, 9and 10.We

observe the effective variable parameters for (EWH1, EWH2)

against the variance and the variance percentage (%)

respectively. This sensitivity analysis agrees with Adnan.

Tariq (2010) and Tarik. Aljuneidi (2013). Figure 7 is plotted that the most effective variable on the

variance is jtR = 913.26 h/ month, which indicate that it is

effective variable in minimization of total product cost for the

case study. Further, as far as analysis is presented in figures 8,

9and 10 respectively. The analysis on whole is showing the

most effective parameter variance, and variance (%) for

(EWH1, EWH2).

Figure 8 discusses the most effective variable on the

variance (%) which is ijtQ = 21.45 %, and ijtX = itd = 4.46

%, that indicats these effective variables in the same sound at

minimization of total product cost for the case study.

As shown in figure 9, which presents an analysis regarding

that there are two equal effective variables on the minimizing

of total product cost which are ijtX and itd .therefore; these

two variables play an important role in the case study. The

same as for figure 10, which introduces an analysis of the

effective variables which cause minimizing the total product

cost for the case study .These variables are ijtX and itd .

Also there is a relation between total product costs saved

for (EWH1, EWH2) against the variance (%) that is shown in

figure 11. It found that there are big variances (%) through

using the proposed mathematical model. For more clear data

figure 12 presents the total product costs saved for (EWH1,

EWH2).

It can be concluded from the analysis of the effective

variable parameters which is presented above that serves the

satisfactory performance of the proposed algorithm. The total

product costs are saved for two products and the aim achieved

by using the HPP approach for CMS, which saved 37.85*106

L.E (34.8%), 54.57*106 L.E (4.04 %) respectively. These

results are obtained from the improvement.

As a result of the HPP approach for the CMS, the

following recommendations (critical issues for future study)

could be outlined as: Development of the model under more

variables parameters such as costs,

processing routes and machine availability, Aggregating

proposed model with other assumptions like layout problem

considerations of CMS, as the layout design of CMS is more

important, Lingo (12.0) can be used rather than Lingo (8.0)

because it is the most effective optimization software program

for solving the problem of CMS . The variables sound great

effects as they are significant for current study, but they could

be variable for another. Finally, verification should be applied

to check the validity of the proposed model in the study

5. CONCLUSIONS AND RECOMMENDATIONS

In this paper, the proposed mathematical model is

developed and applied to two different real case studies. The

proposed mathematical model is solved using the Lingo

program and good results. By analyzing and comparing theses

results with the current value, many improvement at total cost

result for similar cases .consequently; applying such technique

can be used to minimum manufacturing total costs of products

that is required at any manufacturing system to:

Reduce costs in order to be competitive.

Achieve goal of the system under certain criteria and constraints design.

Save time and increase achieving market demand.

Acknowledgement

The authors wish to express their thanks to the staff

members of the (GMC) for manufacturing engineering

Company Egypt, for their support during carrying out this

work.

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1984.

[4] Adnan. Tariq, Operational Design of a Cellular manufacturing System, PH.D. Thesis, National University of Science and

Technology (NUST), Rawlpindi, Pakistan, 2010.



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A robust hierarchical production planning for a capacitated two-

stage production system, Computers and Industrial Engineering, 60, pp. 361372, 2011.

[7] Elizabeth Ann Haas,Hierarchical production planning, PH.D thesis, Massachusetts Institute of Technology, August 1979.

[8] Hiroshi Katayama,On a two-stage hierarchical production planning system for process industries, International Journal of

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Appendix A

Family /Cell Aggregation Sub-

problem

Item /Resources Disaggregation

Sub-problem

Solution procedures of mathematical model using:

Lagrange multipliers

Decomposition procedures

Update the values of Lagrange method using gradient method

1. Solution procedure for cell loading consists of formulating a Lagrange relaxation

jtjLRl

Lt

J

j

T

t

jtjt

jLRl

Lt

J

j

T

t

jt

iTIk

n

Iw

itktw

n

i

T

t

it

T

t

T

t

N

i

itit

J

j

jtjtjtjt

RRROOR

IFIIFhRrOoAMinL

)(1 1

2

)(1 1

1

)(1 11 1 11

21 .),,(

Part (2): According to Item / Resource Disaggregation sub-

problem for each t (IRDt):

)(

2

)(

1

)(

)(jLRl

Ltjt

jLRl

Ltjt

iTIk

n

Iw

ktwit RRORIZMin

Part (1): According to Family / Cell Aggregation Sub-

problem (FCA):

T

t

T

t

N

i

ititit

J

j

jtjtjtjtjtjt IFhRrOoAYMin1 1 11

21 ).()()()(

Outline for the Proposed Mathematical Model

1 1



3. Update the values of Lagrange method using gradient Method A gradient optimization method is used to update the values of Lagrange

multipliers .for example the it values at step c are updated by the following

formula:

itc 1 it

c + c

)i(TIk

n

Iw

itktw IFI

Where:

Optimal solution c =

2)(

21 )),,((

iTIk

n

Iw

itktw

c

IFI

LZ

Cell loading decision

(Detailed reports)

citjtjtjtjtijijtjijtijtLtLtktwkjtwjtjtitijt hRrOoBPQBSCQRRORIZROIFX ,,,,,,,,,,,,,,,,,,

To shop floor execution

End

Solving last equation to find the values of it , 1

jt , 2

jt

Solving last equation to find the values of it , 1

jt , 2

jt

Obtain the value of it

, 1

jt , 2

jt Obtain the value of it

, 1

jt , 2

jt

1 1

For Variable :

2

it

1

it

IT

0t0i

it

iZ

iY

0

it

it

For Variable 2

2212

00

22

2

2

2 0

iZ

iY

jtjt

jt

jt

jt

JT

tj

For Variable 1

2111

JT

0t0j

11

1

iZ

iY

0

jtjt

1

jt

jt

1

jt



Fig. 1. Components of levels of aggregation for products

Fi. 2. Schematic flow chart of algorithm used to the CMS

390

400

410

420

430

440

450

460

Z ,

(10

6L

.E/

Mon

th)

Eletrical Water Heater (50L)

Intial

Optimal

Fig. 3. Comparison between Initial and Optimum Total Product Cost for EWH1.

Equipment type

and amount

Production

allocation

among plants

Plant capacity

planning

Performance characteristics,

operating results

Constraints

Item production

schedules

Plant sizing and

location

Start

Data Collection

Mathematical Model

Results

End

See Appendix. A

GT families and items

GT cells and resources

Demand information on each item

Process plans (processing time routs, set of feasible cells)

Cost figures

See Appendix. A

Cell Loading (production

Planning) problem



Fig. 4. Comparison between Initial and Optimal Variables Values for EWH1.

1.27

1.28

1.29

1.3

1.31

1.32

1.33

1.34

1.35

1.36

Z ,

(10

9L

.E/

Mon

th)

Electrical Water Heater (80L)

Intial

Optimal

Fig. 5. Comparison between Initial and Optimum Total Product Cost for EWH2.



Fig. 6. Comparison between Initial and Optimal Variables Values for EWH2

Fig. 7. Effective Variable Parameters for EWH1on the Variance.

Fig. 8. Effective Variable Parameters for EWH1on the Variance (%)



Fig. 9. Effective Variable Parameters for EWH2on the Variance.

Fig. 10. Effective Variable Parameters for EWH2 on the Variance (%).

Fig. 11. The Total Product Costs Saved for (EWH1, EWH2) Against the Variance (%).



Fig. 12. The Total Product Costs Saved for (EWH1, EWH2).

Table I Different parameters used through data collection (Fixed parameters).

Parameters Set of Values

Total number of items , k 65

Total number of resources ,L 5

No of periods, T 1

No of sub periods, w 4

Cost of production, ijtC UN~[500,750]

Cost of regular time, jtr UN~[80,100]

Inventory Holding Cost, ith UN~[10,12]

Setup cost for the families, ijBP UN~[50,65]

Processing time, klPR UN~[0.07,0.075]

No of operation per part 130

Setup time, ijbp UN~[0.361,0.331]

Total time required to produced one unit of family i at cell j, ija UN~[3.37,4.5]

Table II

Comparison between Initial and Optimal Variables Values for EWH1 Product.

Variables IV OP V V %

Cijt ( L.E/Unit) 500 510 -10 -2

Xijt (Unit/ Month) 1570 1500 70 4.46

BPij ( L.E/ Month) 233 320 -87 -37.34

Qijt (Unit/ Month) 2000 1571 429 21.45

Rjt ( h/ Month) 21164.73 20251.47 913.26 4.32

hit ( L.E) 40 45 -5 -12.5

IFit (Unit/ Month) 1295 1275 20 1.54

Dit (Unit/ Month) 1569 1499 70 4.46

Aij (h/unit) 13.48 13.5 -0.02 -0.15

BOij (h/unit) 1.44 1.54 -0.1 -6.94

Dktw (Unit/ Month) 68595 68589 6 0.01

Zkitw (Unit/ Month) 68596 68590 6 0.01

Iktw (Unit/ Month) 323.75 318.75 5 1.54

RRlt (h/ Month) 21165.02 22634.78 -1469.76 -6.94

PRkl (h/ Month) 0.3085 0.33 -0.0215 -6.97

TotalProductCostZ(L.E/ Month) 4.49E+08 4.11E+08 37850400 8.43

V = Initial value - Optimum value

V % = ( V/ Initial value) *100

IV : Initial value

OP : Optimum value



Table III

Comparison between Initial and Optimal Variables Values for EWH2 Product.

Variables IV OP V V %

Cijt ( L.E/Unit) 750 780 -30 -4.00

Xijt (Unit/ Month) 2100 2000 100 4.76

BPij ( L.E/ Month) 210 260 -50 -23.81

Qijt (Unit/ Month) 2055 2047 8 0.39

R jt (h/ Month) 36751.25 36001.29 749.96 2.04

Hit ( L.E) 47 48 -1 -2.13

IFit (Unit/ Month) 1495 1480 15 1.00

Dit (Unit/ Month) 2099 1999 100 4.76

Aij (h/unit) 17 18 -1 -5.88

BOij (h/unit) 1.22 1.32 -0.1 -8.20

Dktw (Unit/ Month) 74425 74416 9 0.01

Zkitw (Unit/ Month) 104778.7 102710.3 2068.4 1.97

Iktw (Unit/ Month) 373.75 370 3.75 1.00

RRlt (h/ Month) 36751.5 36001.54 749.96 2.04

PRkl (h/ Month) 0.35 0.35 0 0.00

TotalProductCostZ (L.E/Month) 1.35E+09 1.30E+09 54576000 4.04

V = Initial value - Optimum value

V % = ( V/ Initial value) *100

IV : Initial value

OP : Optimum value

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