FACILITY LAYOUT DESIGN CONSIDERING RISK FOR SINGLE-PERIOD AND MULTI-PERIOD CASES. A Dissertation by. Id Jithavech

Transcription

1 FACILITY LAYOUT DESIGN CONSIDEING ISK FO SINGLE-PEIOD AND MULTI-PEIOD CASES A Dissertation by Id Jithavech Master o Science, Wichita State University, 2003 Bachelor o Science, Wichita State University, 2001 Submitted to the Department o Industrial and Manuacturing Engineering and the aculty o the Graduate School o Wichita State University in partial ulillment o the requirements or the degree o Doctor o Philosophy May 2008

2 Copyright 2008 by Id Jithavech All ights eserved ii

3 FACILITY LAYOUT DESIGN CONSIDEING ISK FO SINGLE-PEIOD AND MULTI-PEIOD CASES I have examined the inal copy o this dissertation or orm and content, and recommend that it be accepted in partial ulillment o the requirement or the degree o Doctor o Philosophy with a major in Industrial Engineering. Krishna K. Krishnan, Committee Chair Haitao Liao, Committee Member Don Malzahn, Committee Member Gamal Weheba, Committee Member Bayram Yildirim, Committee Member Hamid Lankarani, Committee Member Accepted or the College o Engineering Zulma Toro-amos, Dean Accepted or the Graduate School Susan K. Kovar, Dean iii

4 ACKNOWLEDGMENTS First, I thank Dr. Krishna K. Krishnan, my advisor, or his time, patience, and guidance throughout my research. I also thank Dr. Bayram Yildirim, Dr. Don Malzahn, Dr. Gamal Weheba, Dr. Haitao Liao, and Dr. Hamid Lankarani or serving on my dissertation committee. Second, I thank the entire aculty o the Department o Industrial and Manuacturing Engineering at Wichita state University or providing me with knowledge and valuable advice. Further, I would like to thank my amily and close riends or their endless support and encouragement. Finally, I would like to give a special appreciation to my parents, Jan Jithavech and Payongsiri Jithavech, my brother and his wie, Tawan Jithavech and oongtiwa Jithavech, and my good riend, Omika Tangvesakul or their endless love, support and encouragement, and or believing in me. iv

5 ABSTACT The most desirable characteristic o a acility layout is its ability to maintain its eiciency over time while coping with the uncertainty in product demand. A traditional acility layout design method is governed by the low intensity between departments, which is the product low quantity between two departments. Hence, an error in the product demand assessment can render the layout ineicient with respect to material handling costs. Most o this research integrates uncertainty in the orm o probability o occurrence o dierent rom-to charts. In an environment where the variability o each product demand is independent, the derivation o probabilistic rom-to chart based scenarios cannot be used to address uncertainty o individual demands. This dissertation presents a acility layout problem approach to deal with the uncertainty o each product demand in the design o acility layout. Two procedures are presented: the irst procedure is utilized to assess the risk associated with the layout, while the second procedure is used to develop the layout that minimizes risk. esults rom case studies have shown that the procedure results in reduction o risk by as much as 68 percent. v

6 TABLE OF CONTENTS Chapter Page 1. INTODUCTION Problem Background Facility Layout Problem (FLP) Static Facility Layout Problem Dynamic Facility Layout Problem Stochastic Facility Layout Problem esearch Purpose esearch Objective Chapter Outline LITEATUE EVIEW Introduction Static Facility Layout Problem SFLP Formulation SFLP Approach Additional Considerations and Applications in SFLP Dynamic Facility Layout Problem DFLP Formulation DFLP Approach Stochastic Facility Layout Problem Stochastic FLP Method (Static) Stochastic FLP Method (Dynamic) Fuzzy Approach in the FLP Genetic Algorithm Application o GA in FLP esearch Motivation FACILITY LAYOUT ISK ASSESSMENT METHOD Introduction Notations Facility Layout isk Assessment Procedure Identiication o Signiicant Flows Assessment o elative Increase in Flow Intensity (Mathematical Formulation Method) Assessment o elative Increase in Flow Intensity (Simulation Method) Facility Layout isk Calculation isk-anking vi

7 TABLE OF CONTENTS (continued) Chapter Page 3.4 Conclusion ISK-BASED FACILITY LAYOUT APPOACH FO SINGLE-PEIOD POBLEM Introduction Notations Layout Generation or Projected Demand GA Procedure Analysis o GA Procedure isk-based Facility Layout Design Procedure (Single-Period Case) Single-Period Case Studies Nine-Department Case Study (Zero isk) Nine-Department Case Study (With isk) Sixteen-Department Case Study Twenty-Five-Department Case Study Conclusion ISK-BASED FACILITY LAYOUT APPOACH FO MULTI-PEIOD POBLEM Introduction Notations isk-based Facility Layout Approach (Multi-Period Case) Negligible Transition Cost High Transition Cost Medium Transition Cost isk-based Facility Layout Design Procedure (Multi-Period Case) Nine-Department Case Study (Multi-Period) Nine-Department Multi-Period Case Study (t = 1) Nine-Department Multi-Period Case Study (t = 2) Nine-Department Multi-Period Case Study (t = 3) Sixteen-Department Case Study (Multi-Period) Sixteen-Department Multi-Period Case Study (t = 1) Sixteen-Department Multi-Period Case Study (t = 2) Sixteen-Department Multi-Period Case Study (t = 3) Twenty-Five-Department Case Study (Multi-Period) Twenty-Five-Department Multi-Period Case Study (t = 1) Twenty Five-Department Multi-Period Case Study (t = 2) Twenty Five-Department Multi-Period Case Study (t = 3) Conclusion vii

8 TABLE OF CONTENTS (continued) Chapter Page 6. ADDITIONAL CASE STUDIES Introduction Single-Period Case Study Multi-Period Case Study Conclusion CONCLUSIONS AND FUTUE ESEACH Conclusions Discussion Future esearch EFEENCES APPENDICES A Design Matrix B Normal Probability Plot, esidual Plots, and Outlier Plot C From-Between Chart or Twenty-Five-Department Single-Period Case Study D isk-flow Table or Twenty-Five-Department Single Period Case Study F From-Between Chart or Twenty-Five-Department Multi-Period Case Study (t = 2) G isk-flow Table or Twenty-Five-Department Multi-Period Case Study when t = H From-Between Chart or isk-based Layout or Twenty-Five-Department Multi-Period Case Study (t = 2) I From-Between Chart or Twenty-Five-Department Multi-Period Case Study (t = 3) viii

9 LIST OF TABLES Table Page 3.1 Summary o product demand and process sequence From-between chart Summary o the E[ ] values (mathematical ormulation method) esults o 3.5 Summary o the 13 E[ 18 ] values using simulation (50 iterations) E[ ] values (simulation method) Summary o the 95 percent conidence interval o E[ ] values using simulation method isk-low table isk-low table or nine-department case sutdy Analysis o varaince table or ull model Analysis o varaince table or reduced model Product demand and process sequence or nine-department single-period case study with zero risk From-between chart or nine-department single-period case study with zero risk Product demand and process sequence or nine-department single-period case study with risk From-between chart or nine-department single-period case study with risk Summary o E[ ] values or nine-department single-period case study with risk isk-low table or nine-department single-period case study with risk From-between chart or risk-based layout or nine-department single-period case study with risk ix

10 LIST OF TABLES (continued) Table Page 4.10 Summary o material handling cost comparison or nine-department case study with risk Product demand and process sequence or sixteen-department single-period case study From-between chart or sixteen-department single-period case study Summary o E[ ] values or sixteen-department single-period case study isk-low table or sixteen-department single-period case study From-between chart or risk-based layout or sixteen-department single-period case study Summary o risk ranking and risk reduction or sixteen-department single-period case study Summary o material handling cost comparison or sixteen-department single-period case study Product demand and process sequence or twenty-ive-department single-period case study Summary o E[ ] values or twenty-ive-department single-period case study Summary o risk ranking and risk reduction or twenty-ive-department single-period case study Summary o material handling cost comparison or twenty-ive-department single-period case study Product demand and process sequence or nine-department case study (multi-period) Summary o 1 E[ ] values or nine-department multi-period case study (t =1) isk-low table or nine-department multi-period case study (t = 1) From-between chart or risk-based layout or nine-department multi-period case study (t =1) x

11 LIST OF TABLES (continued) Table Page 5.5 Summary o risk ranking and risk reduction or nine-department multi-period case study (t =1) Summary o total loss comparison or nine-department multi-period case study (t = 1) Summary o 2 E[ ] values or nine-department multi-period case study (t = 2) isk-low table or nine-department multi-period case study (t = 2) From-between chart or risk-based layout or nine-department multi-period case study (t = 2) Summary o 3 E[ ] values or nine-department multi-period case study (t = 3) isk-low table or nine-department multi-period case study (t = 3) From-between chart or risk-based layout or nine-department multi-period case study (t = 3) Product demand and process sequence or sixteen-department multi-period case study (t = 2) From-between chart or sixteen-department multi-period case study (t = 2) Summary o 2 E[ ] values or sixteen-department multi-period case study (t = 2) isk-low table or sixteen-department multi-period case study (t = 2) From-between chart or risk-based layout or sixteen-department multi-period case study (t = 2) Product demand and process sequence or sixteen-department multi-period case study (t = 3) From-between chart or sixteen-department multi-period case study (t = 3) Summary o 3 E[ ] values or sixteen-department multi-period case study (t =3 ) isk-low table or sixteen-department multi-period case study (t = 3) xi

12 LIST OF TABLES (continued) Table Page 5.22 From-between chart or risk-based layout or sixteen-department multi-period case study (t = 3) Product demand and process sequence or twenty-ive-department multi-period case study (t = 2) Summary o 2 E[ ] values or twenty-ive-department multi-period case study (t = 2) Summary o risk ranking and risk reduction or twenty-ive-department multi-period case study (t = 2) Summary o total loss comparison or twenty-ive-department multi-period case study (t = 2) Product demand and process sequence or twenty-ive-department multi-period case study (t = 3) Product demand and process sequence or nine-department single-period case study From-between chart or nine-department single-period case study Summary o E[ ] values or nine-department single-period case study isk-low table or nine-department single-period case study From-between chart or risk-based layout or nine-department single-period case study Summary o material handling cost comparison or nine-department single-period case study Product demand and process sequence or nine-department multi-period case study (t = 1) Summary o 1 E[ ] values or nine-department multi-period case study (t =1) isk-low table or nine-department multi-period case study (t = 1) xii

13 LIST OF TABLES (continued) Table Page 6.10 From-between chart or risk-based layout or nine-department multi-period case study (t = 1) Summary o risk ranking and risk reduction or nine-department multi-period case study (t = 1) Summary o total loss comparison or nine-department multi-period case study (t = 1) Product demand and process sequence or nine-department multi-period case study (t = 2) Product demand and process sequence or nine-department multi-period case study (t = 3) Summary o 3 E[ ] values or nine-department multi-period case study (t = 3) isk-low table or nine-department multi-period case study (t = 3) From-between chart or risk-based layout or nine-department multi-period case study (t = 3) xiii

14 LIST OF FIGUES Figure Page 2.1. Typical scenario tree o probable utures (Montreuil and Laorge, 1992) Facility layout based on projected demand values (a) and the scenario where low intensity consisting o product 3 is equal to product 2 (b) Simulation process low chart Department placement scheme Interaction plot between problem size and percentage crossover operation Three-dimensional interaction plot between problem size and percentage crossover operation Layout based on projected demand value or nine-department case study (zero risk) Best layout based on projected demand or nine-department single-period case study isk-based layout or nine-department single-period case study with risk Best layout based on projected demand values or sixteen-department single-period case study isk-based layout or sixteen-department single-period case study Best layout based on projected demand values or twenty-ive-department single-period case study isk-based layout or twenty-ive-department single-period case study Layout based on projected demand or nine-department multi-period case study isk-based layout or nine-department multi-period case study (t = 1) Layout based on projected demand or nine-department multi-period case study isk-based layout or nine-department multi-period case study (t = 2) Layout based on projected demand or nine-department multi-period case study isk-based layout or nine-department multi-period case study (t = 3) xiv

15 LIST OF FIGUES (continued) Figure Page 5.7. Best layout based on projected demand values or sixteen-department multi-period case study (t =2) isk-based layout or sixteen-department multi-period case study (t = 2) Best layout based on projected demand values or sixteen-department multi-period case study (t =3) isk-based layout or sixteen-department multi-period case study (t = 3) Layout based on projected demand values or twenty-ive-department multi-period case study (t =2) isk-based layout or twenty-ive-department multi-period case study (t = 2) Layout based on projected demand values or twenty-ive-department multi-period case study (t =3) Best layout based on projected demand or nine-department single-period case study isk-based layout or nine-department single-period case study Layout based on projected demand or nine-department multi-period case study (t = 1) isk-based layout or nine-department multi-period case study (t = 1) Layout based on projected demand or nine-department multi-period case study (t = 2) Layout based on projected demand or nine-department multi-period case study (t = 3) isk-based layout or nine-department multi-period case study (t = 3) xv

16 CHAPTE 1 INTODUCTION 1.1 Problem Background Under current global manuacturing paradigms, the reduction in material handling cost is essential to maintaining a competitive edge in the market. Material handling cost is estimated to contribute up to 20 to 50 percent o the operating expenses in a manuacturing acility, and it is shown that this operating expense can be reduced by 10 to 30 percent with eicient acility layout design (Tompkins and White, 1996). In general, the acility layout problem (FLP) deals with the allocation o departments, machines, aisle space, etc., to available loor space with an objective to minimizing overall material handling costs. Traditionally, these allocations are determined based on product low intensities between machines or departments. However, due to the current market environment where product demand and level o product mix are continuously changing, material handling cost luctuates and is oten increased. In addition, introduction o new products and machines or discontinuation o existing products and machines can also alter the material handling requirements. Material handling cost is also commonly categorized as a non-value adding cost. Thereore, irms must ocus their eorts to reduce this cost to the minimum level. In order to maintain the eiciency o the acility layout, the layout design should be evaluated periodically to ensure its optimality under a new production setting. A new production setting may involve numerous changes in production requirements. In general, ive changes may cause a reduction in layout eiciency: (1) change in product demand and product mix, (2) change in process sequences, (3) change in scheduling and sequencing strategies, (4) change in resources, and (5) change in saety regulations. 1

17 Most o these actors are interrelated, and variations in any o these actors may increase the material handling cost. In most cases, when a change in product demand and product mix occurs, a change in process sequences and scheduling and sequencing strategies may be necessary. In addition, changes in product demand and product mix may consequently lead to changes in resource requirements, which in turn trigger the change in capacity level and number o material handling devices. Frequent additions and adjustment in saety constraints and regulations can also disrupt the existing acility. As a result, an eective method to design a acility that is lexible, responsive, and adaptive is a must in order to maintain the overall eiciency o the acility. Extensive research has been conducted in the area o acility layout design to address these issues, and they are summarized in the ollowing sections. 1.2 Facility Layout Problem (FLP) esearch in the acility layout area is commonly categorized as a static or dynamic FLP problem, depending on the nature o the input requirements and the time periods under consideration. The static acility layout problem (SFLP) approach generally assumes that low between machines, product demand, and level o product mix are constant and usually perormed or a single time period. On the contrary, a dynamic acility layout problem (DFLP) approach is perormed a periodic layout evaluation and modiication with respect to changes. In addition, in recent years, many researchers have addressed the SFLP and DFLP in a stochastic environment where many o the actors (i.e., product demand, level o product mixed, and process sequence) considered in the FLP are subjected to variability. The stochastic FLP approach aims to incorporate the true nature o many manuacturing environments, where the uncertainties are unavoidable. Much research in the area o SFLP, DFLP, and stochastic FLP can be ound in the literature. The ollowing sections provide a brie overview o these approaches. 2

18 1.2.1 Static Facility Layout Problem In the static acility layout problem approach, the low relationship between departments is used as input, and data is usually in the orm o a rom-to chart. SFLP assumes that there is no variability in product demand. Comprehensive reviews o the SFLP can be ound in Kusiak and Heragu (1987) and Meller and Gau (1996). Several optimization ormulations have been developed to solve SFLP: quadratic assignment problem (Koopmans and Beckman, 1957, and Armour and Bua, 1963), quadratic set-covering problem (Bazaraa, 1975), linear integer programming problem (Lawler, 1963, Love and Wong, 1976), mixed-integer programming problem (Kauman and Broeckx, 1978, Bazaraa and Sherali, 1980, Burkard and Bonninger, 1983, Frieze and Yadegar, 1983, and Montreuil, 1990), or graph theoretic problem (Foulds and obinson, 1976) Dynamic Facility Layout Problem In the situation where product demands vary rom one period to next, the DFLP is acilitated. Fluctuations in product demands could be the result o changes in product mix, introduction o new products, or discontinuation o existing products. Changes in product demands render the current acility layout ineicient and can increase material handling costs (Aentakis et al., 1990). In order to maintain good acility layout, it is necessary to continuously assess the changes in product demands, the low between departments/machines, and the existing layout to determine the need or redesigning the layout. There have been several attempts in the area o dynamic acility layout problems. osenblatt (1986) proposed a procedure that considered material handling costs as well as rearrangement costs to generate an optimal layout or multiple periods. The approach develops an upper-bound solution to the minimization problem by identiying the best layout or each 3

19 period and adding material handling costs and transition cost rom one period to the next or rearranging the layout. A second upper-bound value is generated by identiying the minimal total costs when the same layout is used or all periods. A population o solutions is generated randomly or each period, and the best solution is identiied. esearch in the DFLP ollowing this attempt can be categorized into that which that develops either a single robust layout or multiple production periods or lexible or agile (Kochhar and Heragu, 1999) layouts, which can be easily customized to adapt to changes in production requirements. The robust layout approach assumes that the product demands are known in the initial acility layout design stages. This approach generates a single layout that minimizes cost over the entire period under consideration (Kouvelis and Kiran, 1991). This is typically used when the cost o transition or a layout rom one period to another is high. It also can be employed to determine the most robust layout or a period under consideration with respect to multiple scenarios (osenblatt and Lee, 1987, osenblatt and Kropp, 1992). The periodic redesign approach assumes that layouts can be changed rom time to time. When there are wide changes in product mix and demands and when the redesign cost is low, the periodic redesign approach is more eective in reducing material handling costs than the single robust layout design approach. As in the SFLP, heuristic approaches have been used to generate layout design or the DFLP. Urban (1993) acilitated a steepest-descent pair-wise exchange method to obtain the layout design, which minimizes material handling costs in dynamic acility layout problems. This procedure utilized a time-window approach to consider the multiple periods under consideration. Balakrishnan et al., (2000) extended Urban s procedure and used a backward process with a single time window to improve the solution. They also combined Urban s procedure with a dynamic programming approach or each o the periods under 4

20 consideration to generate multiple layouts. Conway and Venkataramanan, (1994) used a genetic algorithm (GA) to solve the DLP. Balakrishnan and Cheng (2000) proposed a nested-loop GA, which used an inner loop to replace unit solutions and an outer loop to replace solutions at random. Balkrishnan et al. (2003) developed a hybrid genetic layout algorithm, which used dynamic programming and pairwise exchange heuristics to ind solutions along with the GA. Baykasoglu and Gindy (2001) proposed a simulated annealing (SA) approach to solve the DFLP, which was shown to be computationally eicient compared to GA approaches. In most DFLP approaches, researchers assumed discrete data points at the end o the period as the production demand or each period. Krishnan et al. (2006) modeled product demands using a continuous unction rather than discrete data points. They developed the concept o the dynamic rombetween chart (DFBC), which tracks the changes in demand continuously and identiies the need or redesign based on demand luctuations. DFBC was ound to be eective in reducing material handling costs in the presence o time-varying production demand Stochastic Facility Layout Problem All SFLP and DFLP research assumes that a single deterministic product demand is orecasted or an individual period. However, product demand orecasts are usually inaccurate, and thus manuacturers develop several possible demand scenarios or the period under consideration. One approach to designing layouts in such situations is to plan or the most likely scenario. However, this may result in large losses i a dierent scenario occurs. Another approach is to develop a compromise layout that can be used or multiple scenarios. A robust approach to a single-period plant layout problem or multiple scenarios in the same period is given by osenblatt and Lee (1987). Their model develops layouts to minimize the total expected material handling costs and minimize the maximum losses using a total enumeration 5

21 strategy. However, as the problem size increases, the total enumeration strategy becomes computationally ineasible, and it becomes worse in terms o computational easibility when solving multi-period multi-scenario problems. osenblatt and Kropp (1992) employed the methodology o compressing low matrices, whereby a set o probable scenario-based deterministic low matrices are chosen to evaluate a stochastic lexible layout or a single period. Mehrabi et al. (2000) addressed the need or reconigurable manuacturing systems to deal with demand scenarios that are highly unpredictable. They showed that it is vital to generate layouts that are quicy adaptable and reliable. Irani and Huang (2000) proposed a module-based technique such that dierent material low patterns can be decomposed into a set o modules and these modules can be reconigured on a timely basis to deal with uncertainties. Bruccoleri et al. (2003) concluded that mass customization is a key actor or any industry s survival. When uncertainty associated with demand is high, reconigurable layouts are capable o addressing such environments. This paper detailed the complexities o the decision-making process in reconigurable layouts and proposed an agent-based system to handle reconigurable systems. Several researchers have also used uzzy models to cope with the uncertainty o production or demand in FLP. Dweiri and Meier (1996) employed the Analytical Hierarchy Process (AHP) to generate an activity relationship chart that yields better layouts. Enea et al. (2005) used a uzzy model to represent the uzzy low patterns between departments and a genetic search approach to deal with uncertain production demand scenarios. A robust layout that can perorm eectively or a wide variety o demand scenarios by maximizing the system eiciency over a long term was designed. The model adopted arithmetic operators to calculate the material handling cost based on possible lows o inite-capacity constraint between departments. However, the lows are assumed to be deterministic. 6

22 Benjaaar and Sheikhzadeh (2000) presented a demand scenario-based procedure, which assigns the layout strategically to achieve reduction in total low. Multiple inite-demand scenarios along with their probabilities o occurrence are used to generate multiple rom-to low matrices. A heuristic approach iterates the problem separately in two steps: a acility layout problem and a low allocation problem. Yang and Peters (1998) developed layouts to reduce the material handling cost using the expected low density (EFD) model or multiple production scenarios. The EFD approach may not be the best option in such situations where widely varying production scenarios exist, because the layout generated is biased to the scenario with the highest probability o occurrence rather than compromising between dierent scenarios. Krishnan et al. (2008) developed models or the multi-scenario-single-period problem and the multi-periodmulti-scenario problem. They used a GA to minimize the maximum losses or both problems. 1.3 esearch Purpose In a manuacturing environment, where product demands are highly volatile, assessing uture product demand is very critical. Forecasting is one o the most widely used techniques or predicting the uture product demand. However, such orecasts may not be accurate. Indeed, in many cases, the uncertainty o projected product demands should be considered during the design o the acility layout. Any acility layout design has an element o risk, and risk actors should be quantiied. Usually, the larger the variances o the product demands, the higher the risk associated with the layout. To reduce the risk, the layout design process should incorporate advanced techniques that reduce the impact o errors. 1.4 esearch Objective The objectives o this dissertation were to investigate the models required or quantiying risk in acility layout design and to develop the acility layout problem approach or a single- 7

23 period and multi-period case, which minimize risk o the acility layout design. Numerous tasks are required to complete, veriy, and validate the developed ramework: isk assessment method development o Mathematical ormulation method o Simulation method isk based acility layout design development or single-period case o Mathematical ormulation o Solution algorithm using GA o Case studies (9, 16, and 25 departments) isk based acility layout design development or multi-period case o Mathematical ormulation o Solution algorithm using GA o Case study (9, 16, and 25 departments) 1.5 Chapter Outline This dissertation is organized into seven chapters. Chapter 2 presents a detailed survey o the existing FLP approaches in the literature. This survey mainly covers the most relevant literature on our topics: (1) SFLP approach, (2) DFLP approach, (3) Stochastic FLP, and (4) GA application in the FLP. Chapter 3 presents the development o a acility layout risk assessment method in detail. This includes product demand probability o occurrence assessment and methods to quantiy its relative change, and the acility layout risk calculation. Chapter 4 discusses procedure to solve the single-period FLP, which minimizes risk o the layout and development o the GA algorithm. In addition, several numerical examples to demonstrate the usage o the ramework or a single period are also presented in this chapter. The risk-based 8

24 acility layout approach or solving the multi-period FLP is presented in Chapter 5. The mathematical ormulation along with the solution procedure and several case studies are demonstrated in this chapter. Chapter 6 provides additional case studies to demonstrate use o the risk-based acility layout approach or dierent types o projected demand scenarios. Finally, Chapter 7 provides the conclusions o the developed ramework. It also presents the importance or contribution o this research to the FLP research area, and provides possible extensions o this research. 9

25 CHAPTE 2 LITEATUE EVIEW 2.1 Introduction The purpose o this research was to address the acility layout problem (FLP) in a stochastic environment, both statically and dynamically. To oer a comprehensive understanding o the FLP, this chapter provides a thorough review o the existing literature in the FLP research area. It is categorized into our major sections, which address dierent aspects o the FLP: the static acility layout problem (SFLP), the dynamic acility layout problem (DFLP), the stochastic acility layout problem, and the heuristics or solving the FLP. First, an overview o the SFLP along with existing solution approaches is presented. Next, an in-depth review o the DFLP, actors that are commonly considered in the DFLP and existing approaches or solving the DFLP are discussed. Furthermore, the concept o stochastic acility layout problem is introduced, and a detailed survey on the existing solution procedures is oered. Lastly, a review on applications o the heuristic approach, namely genetic algorithms (GA) to solve the SFLP and DFLP are discussed. 2.2 Static Facility Layout Problem In the past, the SFLP approach has been widely used as a tool to generate acility layouts. To determine the eectiveness o a layout generated by such an approach, material handling cost is one o the most commonly used perormance measures. Material handling cost comprises the cost o handling, distance between two locations, and number o trips required to move products rom one location to another. The objective unction in SFLP approach aims to minimize the total material handling cost while maximizing the closeness ratio. This type o approach assumes that the product demand and cost o handling are certain and already known or a single period. 10

26 Comprehensive reviews o the SFLP were conducted by Kusiak and Heragu (1987) and Meller and Gau (1996). The ollowing three sections provide a brie overview o SFLP modeling techniques, solution approaches, and additional considerations and applications o the SFLP SFLP Formulation Conventionally, the SFLP is ormulated as one o the ollowing: quadratic assignment problem (Koopmans and Beckman, 1957; and Armour and Bua, 1963), quadratic set-covering problem (Bazaraa, 1975), linear integer programming problem (Lawler, 1963; Love and Wong, 1976), mixed integer programming problem (Kauman and Broeckx, 1978; Burkard and Bonninger, 1983; Frieze and Yadegar, 1983; and Montreuil, 1990) or graph theoretic problem (Foulds and obinson, 1976). In quadratic assignment problem (QAP), the objective unction is derived in the orm o a second-degree unction o the variable, while constraints are ormulated in the orm o a linear unction o the variable. The principle o the model is to assign every machine/department to one position and only one machine/department to each position. The cost o allocating a machine/department at a speciic position depends on the position o the interacting machine/department (Meller and Gau, 1996). However, the QAP is requently used to solve the FLP, but its application is limited to an equal-area FLP. This is due to the input requirement o the QAP, which mandates the location o the machine or department to be known and speciied ahead o time. A modiied version o the QAP was later proposed to enable the traditional QAP to solve an unequal-area FLP (Kusiak and Heragu, 1987; and Liao, 1993). In the modiied QAP model, the departments are broken into smaller grids o equal size, and a large artiicial low between those grids o the same department is assigned to guarantee that they are not split to solve an unequal-area problem. However, the problem size that this modiied QAP can handle is 11

27 very limited due to an increase in the numbers o departments that need to be considered (Meller and Gau, 1996). The quadratic set-covering problem (QSP) is an alternative ormulation method to solve the FLP. The ormulation o QSP is similar to those o QAP with an additional concept o dividing the total area taken by the acilities into smaller blocks. Due to the ormulation similarity and the input requirement, likewise, QSP can only be used to solve an equal area FLP. Linear integer programming (LIP) is another modeling technique used to represent the FLP. In 1963, Lawler (1963) proposed LIP or solving an FLP and also proved that the proposed LIP is an equivalent o the QAP. Due the number o variables and constraints required to model the FLP using LIP (Lawler, 1963), an alternate simpler LIP, which signiicantly reduces the numbers o required variables and constraints, was proposed (Love and Wong, 1976). FLP can also be solved using graph theoretic (GT) techniques. The objective o a GT is to maximize the weight on the adjacencies (arcs) between department pairs (nodes) (Meller and Gau, 1996). To solve the GT, the ollowing three steps are perormed: (1) develop the adjacent graph rom department relationships, (2) construct the dual graph o the adjacent graph, and (3) convert the dual graph into an adjacent block (Meller and Gau, 1996). However, as in the QAP, the GT can only solve a small-sized problem. A more detailed the GT approach to solve a FLP can be ound in the study o Foulds and obinson, (1976). Mixed integer programming (MIP) is also one o the most extensively used modeling techniques to ormulate a SFLP. One o these models was proposed by Montreuil in Unlike the traditional QAP, which is a discrete ramework, he uses the distance-based objective or a continuous layout representation (Montreuil, 1990). The objective unction was based on the low time rectilinear distance between department centroids. The ollowing constraints in the 12

28 ormulation: each department must be within the acility, maximum and minimum length o the department, and the departments should not overlap. Later on, a special case o Montreuil s MIP model, which allows the user to speciy width, length and orientation o the department priory, was developed to provide a more powerul and detail ormulation (Heragu and Kusiak, 1991). Although the MIP ormulation was proven to be a powerul tool in solving an SFLP, due to its associated computational intensity, it only solved a small-sized (less than six departments) problem optimally (Meller and Gau, 1996) SFLP Approach As shown in the previous section, it is evident that the conventional optimization approaches are not capable o solving a large-sized FLP. In addition, these optimization approaches have also been proven to be NP-Hard, which implies that computational complexities increase as the number o departments increase (Kusiak and Heragu, 1987). Alternatively, heuristics approaches, such as simulated annealing (SA) (Meller and Bozer, (1996); Montreuil et al., (1993)), genetic algorithms (GA) (Tam and Chan, 1998), and tabu search (TS) (Chiang and Kouvelis, 1996), have been developed to overcome this computational diiculty. Other heuristics, such as pair-wise interchange methods, computerized relative allocation o acilities technique (CAFT), layout optimization using guillotine induced cuts (LOGIC), SHAPE, nonlinear optimization layout technique (NLT), qualitative layout analysis using automated recognition o patterns (QLAAP), MULTI-loor plant layout evaluation (MULTIPLE), and FLEXible bay structure (FLEX-BAY) (Meller and Gau, 1996), have also been used to generate an SFLP solution. 13

29 2.2.3 Additional Considerations and Applications in SFLP One o the most signiicant issues o the FLP is how to consider or compare the qualitative and quantitative attributes o the FLP in the problem ormulation simultaneously. An earlier attempt to address such a problem was conducted by osenblatt (1979). He presented a heuristic or combined quantitative and qualitative attributes, which addresses the multiple purposes o the FLP. The heuristic aims to minimize the quantitative attributes (material handling cost) while maximizing the qualitative attributes (closeness ratio). Likewise, an analytical hierarchy process (AHP) was proposed to quantiy the qualitative attributes o the layout (Shang 1993). The relative weight that was generated rom the AHP could be used or combining qualitative and quantitative attributes o acility layout in a multiple-objective QAP problem (Shang 1993). Similarly, the dominant index (DI) concept was also introduced to resolve the issues o unit and comparability o qualitative and quantities criterion considered in the FLP (Chen and Sha, 1999). Many other methods such as the expert system (Kumara et al., 1988; Malakooti and Tsurushima, 1989; Heragu and Kusiak, 1990) and the uzzy set (Evan et al., 1987) also can be used to ormulate the multi-criteria acility layout problem. Another important constraint in the acility layout design is space allocation (Tam, 1992). One o the objectives is minimizing the interdepartmental low between cells considering the geometric constraints (Tam, 1992). To ormulate the space-allocation problem, both space allocation and geometric properties o the cell is deined. A slicing tree structure was used to deine the space allocation, whereas the geometric properties o the cell were deined by the associated aspect ratio (Tam, 1992). In addition, another aspect o the space-allocation problem was pointed out by Bozer et al. (1994), andhawa and West (1995), and Kochhar and Heragu (1998). For certain industries 14

30 like semiconductor manuacturing, steel production, artillery combat vehicle abrication, etc., the acility is required to be located vertically due to the horizontal space limitation (andhawa and West, 1995). Thereore, material handling has become a major concern, since a traditional acility layout design algorithm can only address a single-loor problem. To solve the multiple loors FLP, Bozer et al., (1994) introduced the space-iling curves concept to address both single and multiple loor problems. Similar to the multi-loor FLP, the multi-bay FLP was ormulated as an MIP, which can be solved using the dynamic programming to determine optimal layouts (Meller, 1997). In addition to the two approaches mentioned, a MULTI-HOPE algorithm, which is an extension o the HOPE algorithm, can also be used to solve such a problem (Kochhar and Heragu, 1998). As shown, the ormulations and solution approaches to solve the SFLP have been well established. However, with the recent market trend where product demand and level o product mix changes continuously, the existing SFLP approaches may all short o providing an eective solution in such an environment. Numerous researchers ventured into a new area o FLP trying to address and resolve the shortcoming o the SFLP. An overview o such research is provided in the ollowing section. 2.3 Dynamic Facility Layout Problem In an environment where changes in product mix, introduction o new products, and discontinuation o existing products are presented, the acility layout is obligated to be redesigned in order to maintain the eiciency o the acility. According to Aentakis et al. (1990), changes in product demand can render the current acility layout ineicient and increase material handling costs. To maintain or redesign an eicient acility layout, it is necessary to continuously evaluate the changes in product demand, the low between departments/machines, 15

31 and the existing layout. Although many models have been developed to address the FLP with multiple relative constraints, current volatile changes in the market will demand that the layout be robust and agile to adapt to any dynamic change DFLP Formulation Similar to the SFLP, the DFLP was commonly ormulated as an extension o the QAP, considering time and cost associated with alteration o the layout design. Two types o cost are introduced when the layout is altered: (1) loss in production time and (2) cost o moving departments rom one location to another (Kochhar and Heragu, 1999). Most DFLP approaches use these costs to evaluate the worthiness o redesigning the layout. Hence, the objective is to ind a balance between these two costs and the beneit o redesigning the layout (Kochhar and Heragu, 1999). A comprehensive review o the DFLP related literatures is given by Koren et al. (1999). As in the traditional QAP, the computational intensity required to solve a large-sized problem is a challenge. To eectively solve the DFLP, numerous heuristics were developed, and the overview o these heuristics is provided in the ollowing section DFLP Approach An earlier attempt to address the FLP dynamically was conducted by osenblatt (1986), who showed that the problem can be ormulated using dynamic programming (DP). However, by considering the total number o layout combinations at each period, the problem became very large. To overcome this problem, a simpliying procedure using the concept o upper-bound and lower-bound heuristics was proposed. However, one o the major drawbacks o this approach was the inhabited nature o DP, which can not solve the problem with more than twelve departments (curse o dimensionality) (Urban, 1993). Hence, a more eective algorithm with moderate computation intensity was required. Many other researchers based their work on 16

32 improving the algorithm to solve the DFLP eectively and eiciently. In 1993, Urban proposed a heuristic based on the steepest-descent pair-wise interchange procedure to resolve this issue. The procedure generated the layout rom the varying time window with an additional consideration o rearrangement cost. As an attempt to develop an eective algorithm to solve the DFLP, the modiication o ive existing QAP algorithms was proposed (Lacksonen and Enscore, 1993): CAFT (Armour and Bua, 1963), cutting planes (Burkard and Bonniger, 1983), branch-and-bound (Pardalos and Crouse, 1989), dynamic programming (osenblatt, 1986), and cut trees (Gomory and Hu, 1961). Among these, the cutting plane algorithm was ound to be the eicient approach to solve the DFLP. In addition, an improved pair-wise exchange heuristic o Urban s (1993) heuristic was proposed to solve the DFLP (Balakrishnan et al., 2000). This heuristic was perormed in two steps: (1) perorm the backward pass rom the solution obtained by Urban s (1992) heuristic and (2) combining the Urban s (1992) heuristic and DP, and solve or solutions. Since the Urban s (1993) heuristic is a orward pass in nature, the quality o the solution o the later state depended on the quality o the solution in the preceding state, which can be perceived as a disadvantage. Thereore, incorporating the backward pass helped to improve the quality o the generated solution (Balakrishnan et al., 2000). To improve the solution quality while reducing the computational time, numerous researchers have used the GA to solve a DFLP. One o these earlier attempts was conducted by Conway and Venkataramanan (1994). Similarly, a new algorithm called DHOPE was developed (Kochhar and Heragu, 1999), which was the extension o the previously developed algorithm or static layouts known as MULTI-HOPE. DHOPE was designed to generate layouts or two periods. The generated layouts were evaluated based on the reduction in rearrangement costs 17

33 and material handling costs. The roulette wheel selection method was proposed or the selection o the most likelihood string (parent). Experimental results indicated that by using the developed methodology, DHOPE could provide a signiicant reduction in cost when compared to the usual methods. Subsequently, research aimed to enhance GA procedure proposed by Conway and Venkataramanan (1994) was conducted, and the resulting improved GA procedure was developed (Balakrishnan and Cheng, 2000). This GA procedure used a new crossover operator to increase the search space with the addition o a mutation process and a generational replacement approach to diversiy the solutions. This GA also used nested loop, which is comprised o an inner loop and an outer loop. The inner loop substituted the most unit individual rom each generation, while the outer loop replaced the large number o unlucky individual in a generation (Balakrishnan and Cheng, 2000). Likewise, the concept o a hybrid GA was proposed in 2003 to urther improve the eiciency o the existing GA algorithms (Balkrishnan et al., 2003). This hybrid GA algorithm adopted a new crossover operator and a mutation process to improve the quality o solutions. Dynamic programming was used as the crossover operator, while the CAFT was employed or the mutation process. In addition, Urban s (1993) pair-wise interchange method was used to generate the initial population. This proposed GA was tested against the GA algorithm proposed by Balakrishnan and Cheng (2000) and the SA procedure proposed by Baykasoglu and Gindy (2001). The results showed that the proposed hybrid GA provided better solutions when compared with other GAs and SAs in most cases. Simulated annealing (SA) is one o the heuristic approaches, which is well known to be an eicient and eective tool to solve a combinatorial optimization problem (Baykasoglu and Gindy, 2001). In 2001, an attempt to apply SA to solve the DFLP was conducted by Baykasoglu and Gindy (2001). The proposed SA algorithm was tested and compared with the DP proposed 18

34 by osenblatt (1986) and the GA algorithm proposed by Conway and Venkataramanan (1994). The results indicated that the proposed SA perormed better than the other evaluated algorithms (Baykasoglu and Gindy, 2001). Another eective heuristic procedure that can be used to solve the DFLP is the tabu search (TS). In the study o Kaku and Mazzola (1997), a two-staged TS was employed. The irst stage executed the diversiied search o the solution space using a constructive approach to obtain the initial solution, while the second stage perormed the intensiication around the number o promising solutions identiied in the irst stage. This method proved to be very eective in obtaining good solutions when compared with other evaluated heuristic approaches namely the algorithm by Lacksonen and Enscore (1993) and Urban (1993). Through literature surveys, DFLP approaches proved to be eective tools to solve an FLP, where the product demand and product mix luctuates continuously. Numerous approaches and ormulations were developed to provide an FLP solution in such environment. Nevertheless, in the environment where the product demand and product mix are varied and unpredictable rather than changing but certain, a new FLP approach is needed. The ollowing section discusses this type o FLP approach. 2.4 Stochastic Facility Layout Problem As shown in the two previous sections, both the SFLP and DFLP approaches proved to be eicient in solving an FLP where the production requirements are priory known and certain. However, in the manuacturing environment, where the production volume and the level o product mix are highly unpredictable, the need or a dierent FLP approach has emerged. Traditionally, most FLP approaches assume that the production volume and level o product mix are deterministic or priory known, based on either a orecasting technique or an historical trend. 19

35 However, the inherent uncertainties or variations o the resulting orecast are inevitable. Thereore, the layout design based on such a orecast is also subjected to uncertainty, which consequently compromises the layout eectiveness. In the literature, there are two distinct type o approaches handle the uncertain production environment in the FLP: (1) the probabilistic scenario-based approach and (2) the uzzy approach. The ollowing sections provide an overview o research in the stochastic SFLP and DFLP, and the FLP uzzy approaches Stochastic FLP Method (Static) An earlier eort to address the uncertainty in production volume and product mix was put orward by osenblatt and Lee (1987). As a result, a robust approach to a single-period plant layout problem under a stochastic environment was developed. In the situation where there are many possible states o demand to be considered, assigning equal probabilities to each state may not be the most appropriate option (osenblatt and Lee, 1987). Based on this argument, the proposed approach categorized the probability o occurrence o production scenarios into three levels (high, medium, and low), which aimed to improve the eiciency o generated layouts. Subsequently, a more advance probabilistic method was proposed to solve an FLP in a single-period stochastic layout problem (osenblatt and Kropp, 1992). This method simultaneously considered all anticipated production scenarios by assigning each production scenario with its associate probability o occurrence. The problem was ormulated as a QAP, which was used to generate the optimal layout or each production scenario. In addition, the ollowing assumptions were made: (1) a inite number o possible product mixes scenarios can occur, (2) each scenario can be represented by a unique rom-to matrix, (3) each scenario has a predetermined probability o occurrence, and (4) the probabilities are mutually exclusive and jointly exhaustive (osenblatt and Kropp, 1992). In order to simpliy the problem to consider all 20

36 possible production scenarios, a weight-average low matrix was used. The purpose o this matrix was to compress numerous low matrices into a single low matrix, which can be calculated as F S = psfs (2.1) s = 1 where p s represents the probability o occurrence or scenario s and F s is the associated low matrix or scenario s. This weight-average low matrix is used as input to generate the best layout, which minimizes the total expected material handling cost over all scenarios Stochastic FLP Method (Dynamic) To improve the range o applications o the existing stochastic FLP approaches, several studies were conducted in the stochastic DFLP area. One study developed an approach similar to that o osenblatt and Kropp (1992) with an extension o the multi-period case or an unequalarea FLP (Yang and Peters, 1998). The approach aimed to minimize the expected material handling cost and machine rearrangement cost. Two types o layout were taken into an account: robust and dynamic. For robust layouts, the entire planning horizon was considered as one planning window. On the other hand, dynamic layouts constituted several sets o planning windows. The machine rearrangement cost was used to assist in selecting the type o layout. In cases where the machine rearrangement costs were high, the robust layout was preerred. On the other hand, i the machine rearrangement costs were low, the adaptable layout was avored. The expected material handling cost was obtained based on the expected low density (EFD) between two departments i and j, and was calculated as k, T k + T = [ π ] [ π ] (2.2) F ( i, j) P ( t) ( t) t = k π ( t) ( t) 21

37 where ( t) π represents the possible production scenarios in period t, [ ( t) ] low matrices o ( t) π, and P[ ( t) ] π is the associated π denotes the probability o occurrence o π ( t). Two years later, a similar study was conducted to address the stochastic DFLP. Similar to the approach by osenblatt and Kropp (1992) and Yang and Peters (1998), this approach aimed to solve an FLP where multiple inite demand scenarios are considered with the additional consideration o product demand distributions (Benjaaar and Sheikhzadeh, 2000). This approach utilized the compress low matrix concept to simpliy the problem. This matrix was also used as the input to develop the best layout. Similarly, another method to solve the stochastic DFLP or stochastic dynamic plant layout problem (SDPLP) was proposed (Palekar et al., 1992). The objective o this method was to minimize the expected material handling cost and relocation cost or each period. In their study, the Markov chain theory was used to estimate the likelihood (probability o occurrence) o each o the demand scenarios to occur and assumed that the probability o occurrence o a product demand in a period is most likely to be conditioned to the situation in the previous period. Four assumptions were made regarding the availability o the product inormation: (1) each low matrix represents a unique state in the Markov chain, and there is a inite set o low matrices, (2) the transition probabilities rom one state to another are known, (3) the distance matrix is known, and (4) The material handling cost and relocation cost are known. The probability o each demand scenario to occurring in each period is calculated as S t t 1 0 s = h hs s = s h= 1 p p O and p O φ (2.3) 22

38 where S is the number o possible low matrix or state, and O hs represents the probability o one step transition rom state s to h. These probabilities are then used to calculate the expected material handling cost. Further, Montreuil and Laorge (1992) attempted to address the FLP problem under uncertainty using the scenario tree o probable utures. This method can help to provide a set o layouts, which oers a robust master plan to accommodate the potential utures. Figure 2.1 depicts o the typical scenario tree o probable utures. F F6 F F7 F1 0.3 F3 0.6 F8 0.5 F4 0.1 F F F11 F12 Figure 2.1. Typical scenario tree o probable utures (Montreuil and Laorge, 1992) Fuzzy Approach in the FLP In recent years, the uzzy approach has been used to incorporate the uncertain nature o product demand into the FLP in the research o Evans et al. (1987), Grobenlyn (1987), Dweiri and Meier (1996), Aiello and Enea (2001), Deb and Bhattacharyya (2003), and Enea et al. (2005). To solve the block design layout problem, the uzzy linguistic heuristic based on uzzy 23

39 relation and uzzy ranking was applied to determine the location o each department in a manually generated layout (Evans et al., 1987). In addition, the uzzy implication and uzzy consistency method was employed to solve the stochastic FLP by deining a grade o satisaction or the optimality criterion to identiy the most suitable location or each department (Grobenlyn, 1998). In addition, the AHP and uzzy decision-making procedure were also used collectively to generate a acility activity chart (Dweiri and Meier, 1996). According to Aiello and Enea (2001), the previous stochastic FLP algorithm such as osenblatt and Lee (1987) robust approach did not consider the limitation in the capacity o the acility. Consequently, in some instances, the obtained solutions may be ineasible. To overcome this problem, a uzzy approach that uses a constrained arithmetic operator to mimic the uncertainty nature o the product demand was developed (Aiello and Enea, 2001). This approach also claimed to provide a reduction in computational complexity when compare with the other stochastic models. To handle the uncertain and vague data, another methodology based on uzzy decisionmaking system was developed to solve the FLP (Deb and Bhattacharyya, 2003). A multi-actors uzzy inerence was used or selecting the location o the department in an open continual plane considering both quantities and qualitative actors. To generate the robust layout that is eective over a wide range o production scenarios, Enea et al. (2005) proposed the use o uzzy number and constrained arithmetic operators to ormulate the FLP with the objective o minimizing the uzzy material handling cost. In the previous sections, all three types o FLP approaches and ormulations were discussed. As shown, most o these approaches were ormulated as a QAP, which was categorized as an NP-hard problem. To reduce the computational intensity required to solve a 24

40 QAP, numerous heuristics were used, such as GA, TS, and SA. Among these, GA is one o the most widely used heuristics to solve an FLP. The next section provides a survey o GA application in the FLP research area. 2.5 Genetic Algorithm In the early 1970s, Holland (1975) developed a heuristic inspired by Darwin s theory o evolution by natural selection known as the genetic algorithm. The main advantage o using a GA is due to the randomness induced in the algorithm, which consequently broadens the search space. A larger search space helps to increase the possibility that the algorithm will obtain a better solution. Based on the general GA procedure, the candidate solutions are represented in orm o a chromosome, which is randomly generated. The GA evaluates all solution candidates in a easible region, and selects the best solution. The best solutions are identiied using a itness unction, which is developed based on the objective o the problem. Once the best solutions (parents) are selected, the GA repopulates another set o candidate solutions (ospring) by the reproduction process called crossover and mutation, which are applied to the parents. The GA repeats these processes until a speciic number o iterations is met, and the best solution is selected among the last set o solution candidates in the last iteration based on the itness unction Application o GA in FLP Due to a reduction in computational time oered by the GA and the combinatorial nature o the FLP problem, numerous researchers have employed the GA as an alternative approach to solve an SFLP. For instance, the GA was used to obtain the optimal/near optimal solution or an unequal area FLP (Tate and Smith, 1995). It has also been used to solve a cell layout problem to obtain a path skeleton design (Banerjee et al., 1997). In the ollowing year, a GA called HOPE 25

41 was developed and applied to solve both an equal and unequal-area FLP (Kochhar et al., 1998). Further, ajasekharan et al. (1998) proposed a GA-based decomposition strategy to solve the lexible manuacturing system (FMS). This proposed methodology comprised a two-step hierarchical decomposition approach, which demonstrated an ability to generate a good result while using a minimum amount o time. In addition, Balakrishnan et al. (2003) extended the existing GA algorithm to a hybrid GA to address the DFLP. To urther improve the eiciency o the GA, Tam and Chan (1998) employed a parallel GA or solving the FLP. The layout was represented using a slicing tree approach and the chromosomes were generated such that the internal and external nodes are considered as substrings, which help to broaden the search space or the GA. Mak et al. (1998) proposed a mathematical model and new crossover operator or the GA to solve the FLP. Experimental results indicated that the eectiveness o the proposed approach was competent. In 1998, Islier proposed the use o modeling chromosomes in three parts to solve the FLP. The irst part was modeled as the sequence o departments, the second part represented the area or these departments, and the third part was used or representing the concatenated cells in a string. Three actors are considered in the objective unction, which are load, shape and deviation. An improved GA to acilitate the design o eicient layouts or the DFLP was proposed by Balakrishnan and Cheng (2000). A new crossover operator along with the use o a mutation and generation replacement strategy was proposed to increase population diversity and eiciency in computation time. Based on the work o Tam and Chan (1998), Hakim (2000) urther modiied their slicing tree representation in the GA algorithms to solve the FLP. He also pointed out the problems o using the existing crossover and mutation operators and presented a new transplanting method, which can generate eicient o-springs. A new slicing tree structure in the 26

42 GA was proposed by Shayan and Chittilappilly (2004). This method removes the repairing procedure constraint and induces the penalty unction to generate easible layout. 2.6 esearch Motivation Although, extensive research on methodologies to solve the FLP considering many dierent type o conditions and constraints has been conducted, research on uncertain production environment is limited. The current research on the development o acility layouts involving uncertain production environment used multiple rom-to scenarios. Assigning a probability value to the rom-to chart or a scenario assumes that every low in the rom-to chart is dependent on other lows within the chart. In other words, all product demands are dependent. For instance, Yang and Peters (1998) considered multiple scenarios or a single period by assigning the probability o occurrence or each scenario. Similarly, Benjaaar and Sheikzadeh (2000) identiied the scenario in the same manner as Yang and Peters (1998) with additional consideration o the probability distribution. In addition, Krishnan et al. (2008) developed models or the multi-scenario-single-period problem and the multi-period-multi-scenario problem, aiming to minimize the maximum losses or both problems. However, in the environment where each product demand by itsel demonstrates high variability and is independent rom one another, the scenario-based method may not be suitable or addressing overall uncertainty o demands. In the ollowing chapters, the stochastic FLP ramework called the risk-based acility layout approach is presented. This approach aims to provide a more elaborate variability or uncertainty assessment o product demand. It also includes the FLP ormulation and solution approaches to the stochastic FLP in a single-period case and a multi-period case. 27

43 CHAPTE 3 FACILITY LAYOUT ISK ASSESSMENT METHOD 3.1 Introduction In most FLP approaches, layout design or allocation o departments is directly driven by low intensity, which is the total low between a pair o departments or all products. A change in the demand o any products will modiy the strength o the relationship between the relevant departments. When the change is large enough to cause the low intensity between one pair o departments to exceed or all below that o another pair o departments, the layout could be compromised. When there is uncertainty associated with product demand, it is necessary to take into consideration the associated risk. Speciically, risk is deined as a measure o the potential loss occurring due to natural or human activities (Modarres, 2006). In the context o acility layout design, risk can be viewed as potential loss caused by perturbations in projected product demand. Two undamental issues must be addressed when dealing with uncertain product demands in FLPs. The irst is determining the best layout or the projected demand. The second is the assessment o risk associated with the layout with respect to variability in product demands. Trade-o between the optimal layout or the projected demand and the potential loss associated with possible variations in product demand must be analyzed. One o the most desirable characteristics o a layout design is the ability o maintaining its lexibility and eiciency against uncertainty in product demands. 3.2 Notations The ollowing notations are used in the acility layout risk assessment method: 28

44 N Total number o departments in layout Flow intensity between departments i and j Flow intensity between departments k and l x h Demand per period or product h, where h = 1 H C D Material handling cost o moving one product per unit distance Distance between departments i and j o the current layout p Probability that low intensity between departments i and j exceeds that between departments k and l g(x h ) Probability density unction or the demand o product h ϕ ( ) Probability density unction or the low intensity between departments i and j E [ ] Expected relative change in with respect to when > F Projected low intensity between departments i and j F Projected low intensity between departments k and l G( ) Cumulative density unction or low intensity isk that low intensity between departments i and j exceeds that between departments k and l 3.3 Facility Layout isk Assessment Procedure In this research, the inluence o demand uncertainty on acility layout is assessed using the risk assessment method. Mathematically, risk can be expressed as (Modarres, 2006) 29

45 ISK= u. ici (3.1) i where u i is the probability o event i and c i denotes the associated consequence. For the FLP, the probability o occurrence is deined as the probability that low intensity between departments i and j exceeds that between k and l. The consequence can be viewed as the resulting increase in the material handling cost. The risk assessment o the acility layout is conducted using the ollowing our steps: Identiy all signiicant low intensities, which are at risk. Assess the relative increase o the signiicant low intensity. Assess the layout risk. ank the risk Identiication o Signiicant Flows Traditionally, product demands are determined rom historical data or orecasting techniques. In this research, these demands are reerred to as projected demand. Due to luctuation in the current market demand, when the projected demand values are obtained through orecasting techniques, these demand values may or may not be the same as the expected demand, which is typically obtained based on the demand distribution. In this research, the projected demand values, which are dierent rom the expected demand values are analyzed. andom product demands and sequence data or nine-department ive-product case are shown in Table 3.1, and the associated rom-between chart is presented in Table

46 TABLE 3.1 SUMMAY OF PODUCT DEMAND AND POCESS SEQUENCE Product Product Demand Distribution Projected Demand Product Sequence 1 Uniorm (100, 200) Uniorm (275, 320) Uniorm (200, 310) Uniorm (50, 150) Uniorm (40, 400) TABLE 3.2 FOM-BETWEEN CHAT Dept For the data provided, the demand or product 2 is 280 units and the one or product 5 is 50 units. Based on probability density unctions (pd) o the product demands, there is a possibility that the demand or product 5 is higher than that or product 2. I this scenario occurs, the layout design based on the projected demand values (Table 3.1, Column 3) may no longer be the best layout. Figure 3.1 shows two layout designs. The irst is generated based on projected demand values, and the second is generated based on the demand scenario in which low intensities consisting o product 3 are equal to those o low intensities consisting o product 2. 31

47 This case is an extreme case and can be used to demonstrate the impact o uncertainty in product demand (a) (b) Figure 3.1. Facility layout based on projected demand values (a) and the scenario where low intensity consisting o product 3 is equal to product 2 (b). Based on this example, the low intensities that are exposed to risk can be identiied by checking the ollowing conditions: 1. Is the probability that the low intensity between departments i and j will exceed the low intensity between any other departments k and l equal to zero? 2. Are departments i and j adjacent to one another in the layout generated using projected demand values? The irst condition is used to check i there is any possibility that the low intensity under consideration can be altered due to uncertainty, which in turn will change the relationship between its corresponding pair o departments. The second condition checks i departments i and j are adjacent. I they are adjacent, the material handling costs cannot be reduced urther, even i the low intensity between departments i and j increases. Flow intensities that do not satisy both conditions are considered to be subject to risk. 32

48 For the data provided in Tables 3.1 and 3.2, the best layout or the projected demand is shown in Figure 3.1a. A total o thirteen low intensities ( 12, 13, 18, 27, 35, 45, 46, 47, 56, 57, 68, 78, and 79 ) is considered in the rom-between chart (Table 3.2). By evaluating each o these low intensities through the risk conditions, only 18 and 45 are at risk Assessment o elative Increase in Flow Intensity (Mathematical Formulation Method) The previous section presented the method o identiying those low intensities that are at risk. In this section, the assessment o the probability o occurrence or low intensity between departments i and j exceeding that o departments k and l procedure, along with the method to quantiy these relative changes are detailed. In order to calculate the probability o occurrence o low intensity between departments i and j exceeding that o departments k and l ( p ), the pd o each low intensity is needed. In general, the low intensity between two departments may either be the result o a single-product demand or multiple-product demands. In the case o one product, the pd o the low intensity is the pd o the product demand itsel. When multiple products low between the two departments, the pd o the low intensity is attributed to all o those products. To estimate the pd o a combined-product demand, the cumulative distribution unction (cd) o combined-product demands can be calculated using convolution theory and it can be calculated as ϕ( ) = P( X + X < x) = g(x - x )g(x )dx (3.2) where g(x 1 ) denotes the pd o the demand o the irst product, and x is the variable or the total low intensity level o the combined product demand. It is assumed that the distributions o these 33

49 demands are known and independent, and they are o continuous types. Once the pd o each low is determined, p can be calculated using p = Pr( ) = ϕ( ) ϕ( ) d d 0 (3.3) To illustrate the proposed method, a uniorm distribution is assumed or all product demands due to its useulness in dealing with limited inormation regarding product demands. Similar derivations using other distributions can also be developed. The pd o the uniorm distribution is expressed as 1 g(x)=, a x b b- a Otherwise, g(x)=0 (3.4) where a and b are the lower and upper bounds, respectively, o the distribution. From equation (3.2), the pd o a low intensity consisting o two product types can be expressed as ollows: x -ax 2 x - ax - a 1 x2 g(x - x 1 )g(x 1 )dx =, where(a x + a 1 x ) x < (b 2 x + a 1 x ) a 2 x 1 (bx - a 1 x )(b 1 x - a 2 x ) 2 x -bx 1 ax b 2 x1 g(x - x 1 )g(x 1 )dx =, where(b x-a x + a 1 x ) x (a 2 x + b 1 x ) 2 ϕ( x ) = < 2 (bx - a 1 x )(b 1 x - a 2 x ) 2 bx 2 b x + b 1 x - x 2 g(x - x x-b 1 )g(x 1 )dx =, where(a x + b x 1 (b 1 x ) x (b 2 x + b 1 x ) 2 x - a 1 x )(b 1 x - a 2 x ) 2 0, elsewhere (3.5) where a x h and b x h denote the lower and upper bounds, respectively, o the distribution o product h. For example, considering the data provided in Tables 3.1 and 3.2, the low intensity between departments 3 and 5 comprises product 1 and 3, which ollow uniorm distribution with upper and lower bounds o 100 and 200, and 50 and 150, respectively. By using equation (3.5), the pd o the low intensity between departments 3 and 5 ( ϕ ( 35 )) is derived as 34

50 35 - ax - a 1 x4, < 250 (bx - a 1 x )(b 1 x - a 4 x ) 4 ϕ( 35 ) = Pr( X 1 + X 4 < 35 )= b x + b 1 x , (bx - a 1 x )(b 1 x - a 2 x ) 2 Once the pd o low intensity is determined, and then p can be obtained rom equation (3.3). 35 For example, based on the inormation provided in Tables 3.1 and 3.2, p can be calculated as < 250, then I 35 p = ( ) ( )d d - a - a 1 d d b b b b x1 x2 a ϕ ϕ = a (bx -a 1 x )(b 1 x - a 2 x ) (b 2 - a ) 3 3 b +(a x +a 1 x ) b a (b - a )- - b (a x +a 1 x )(b 2 - a ) 2 3 = (b - a )(b - a )(b - a ) x1 x1 x2 x ( ) = (150-50)( )(400-40) ( ) ( )( ) = I , then p = ( ) ( )d d b + b - 1 d d b b b b x1 x2 a ϕ ϕ = a (bx -a 1 x )(b 1 x - a 2 x ) (b 2 - a ) b +(b x + b 1 x ) b (b x + b 1 x )(b 2 - a )- (b - a )+ (b - a ) 2 3 = (b - a )(b - a )(b - a ) x1 x1 x2 x2 35

51 = 400 +( ) (150-50)( )(400-40) ( )( )- ( ) + ( ) where = a and intensity, and b are the lower and upper bounds, respectively, o the distribution o low a low intensity. and b are the lower and upper bounds, respectively, o the distribution o Traditionally, the material handling cost is the product o the low intensities, distance between corresponding pairs o departments, and material handling cost per unit distance. And, as deined previously, the risk o acility layout is the potential increase in material handling cost caused by the uncertainty o product demand. Thereore, to estimate this increase in material handling cost, the expected relative changes in low intensity between departments i and j with respect to that o departments k and l ( E[ ] ) is needed and can be obtained using the ollowing equations: b b (3.6) E [ ] = ( - +Q) ϕ ( ) ϕ ( )d d, where Q= F - F a For demonstration, the estimate o 35 E[ 18 ] can be calculated using equation (3.6) as 150 < 250, then I 35 b b a ϕ ϕ E [ ] = ( - +Q) ϕ ( ) ϕ ( )d d,where Q= F - F = b b x1 x2 a ( - +Q)( - a - a ) d d (bx - a 1 x )(b 1 x - a 2 x )(b 2 - a ) = 2 b b x1 x2 a - +Q -(a + a )( - +Q) d d (bx - a 1 x )(b 1 x - a 2 x )(b 2 - a ) 36

52 Let (bx - a 1 x )(b 1 x - a 2 x )(b 2 - a ) = M, then - +Q (a +a )( - +Q) E[ ] d d d d 2 b b b b x1 x2 = a a M M Let - +Q 2 b b = a d d M E[ ] 1 2 b Qb 2 2 b 3 3 Q (b a ) + (b a ) + + (b a ) = (b - a )(b - a )(b - a ) x1 x1 x2 x2 = (160)(400) ( ) + - ( ) + + ( ) ( )(150-50)(400-40) = And let (a + a )( - +Q) b b x1 x2 = a d d M E[ ] 2 2 b 2 2 b Q (a x + a 1 x ) (b 2 - a ) +b Q -(b - a ) + + (b - a ) = (b - a )(b - a )(b - a ) x1 x1 x2 x ( ) ( ) + (400*160) -( ) + + ( ) = ( )(150-50)(400-40) = Thereore, E[ ] = E[ ] - E[ ] = E[ 35 ] - E[ 35 ] = =

53 , then I 35 b b a ϕ ϕ E [ ] = ( - +Q) ϕ ( ) ϕ ( )d d,where Q= F - F = b b x1 x2 a ( - +Q)(b + b - ) d d (bx - a 1 x )(b 1 x - a 2 x )(b 2 - a ) = 2 b b x1 x2 a Q +(b + b )( - +Q) d d (bx - a 1 x )(b 1 x - a 2 x )(b 2 - a ) Let (bx - a 1 x )(b 1 x - a 2 x )(b 2 - a ) = M. Then Q (b + b )( - +Q) E[ ] d d + d d 2 b b b b x1 x2 = a a M M Let Q 2 b b = a d d M E[ ] 1 2 b +Q Qb 3 3 b (b - a )- (b - a ) - (b - a )- (b - a ) = (b - a )(b - a )(b - a ) x1 x1 x2 x (160)(400) ( )- ( ) - ( ) - ( ) = ( )(150-50)(400-40) = And let (b + b )( - +Q) b b x1 x2 = a d d M E[ ] 2 2 b 2 2 b Q (b x + b 1 x ) (b 2 - a ) +b Q -(b - a ) + + (b - a ) = (b - a )(b - a )(b - a ) x1 x1 x2 x2 38

54 ( ) ( ) + (400*160) -( ) + + ( ) = ( )(150-50)(400-40) = Thereore, E[ ] = E[ ] + E[ ] = E[ 35 ] + E[ 35 ] = = Hence, E[ 18 ] is = Table 3.3 is a summary o the or low intensities that are exposed to risk. SUMMAY OF E[ ] TABLE 3.3 E[ ] values E[ ] VALUES (MATHEMATICAL FOMULATION METHOD) 12 E[ ] 13 E[ ] 27 E[ ] 35 E[ ] 45 E[ ] 46 E[ ] E[ 18 ] E[ 45 ] E[ ] 47 E[ ] 56 E[ ] 57 E[ ] 78 E[ ] 79 E[ ] 86 E[ ] E[ 18 ] E[ 45 ] Assessment o elative Increase in Flow Intensity (Simulation Method) In a real manuacturing setting, the distribution that the product demands ollow can come in many orms. The previous section presented the mathematical ormulation or the acility layout risk assessment. Although, the method is proven to be useul in quantiying layout risk, the time required to derive the necessary ormulations can be very extensive when the product demands ollow a more complex distribution (more variables) and when more than two products are combined in a low intensity. To provide an alternative method o estimating E[ ], the simulation method or acility layout risk assessment model was 39

55 developed. Figure 3.3 demonstrates the general steps to estimate method. E[ ] using the simulation Identiy all, at risk Stop and calculate the average o ( E[ ] ) Yes Speciy number o iteration (I) Set iteration i = 1 i = I No Set iteration i = i + 1 andomly generate low intensity ( ) based on its distribution Calculate - ( - ) > 0 No Yes =0 =( - )+(F -F ) Figure 3.2. Simulation process low chart. To illustrate the usage o the simulation method, the estimation o 13 E[ 18 ] is simulated or 50 iterations sotware. esults o the simulation are presented in Table 3.4. Based on these results, the average o ( E[ 18 ] ) is , and its corresponding 95 percent conidence interval is (102.69, ). 40

56 ESULTS OF TABLE E[ 18 ] VALUES USING SIMULATION (50 ITEATIONS) Iteration Iteration Iteration Iteration Iteration For a more accurate estimation o E[ ] values, the simulation is run or 10,000 iterations, and the resulting values are shown in Tables 3.5 and 3.6 is the summary o the calculated 95 percent conidence interval o E[ ] values. SUMMAY OF THE TABLE 3.5 E[ ] VALUES (SIMULATION METHOD) E[ ] 12 E[ ] 13 E[ ] 27 E[ ] 35 E[ ] 45 E[ ] 46 E[ ] E[ 18 ] E[ 45 ] E[ ] 47 E[ ] 56 E[ ] 57 E[ ] 78 E[ ] 79 E[ ] 86 E[ ] E[ 18 ] E[ 45 ]

57 TABLE 3.6 SUMMAY OF CALCULATED 95 PECENT CONFIDENCE INTEVAL OF E[ ] VALUES USING SIMULATION METHOD E[ ] 95 Percent Conidence Interval 12 E[ 18 ] [78.79, 82.87] 13 E[ 18 ] [155.24, ] 27 E[ 18 ] [78.79, 82.87] 35 E[ 18 ] [96.77, ] 45 E[ 18 ] [105.63, ] 46 E[ 18 ] [78.79, 82.87] 47 E[ 18 ] [78.79, 82.87] 56 E[ 18 ] [105.63, ] E[ ] 95 Percent Conidence Interval 57 E[ 18 ] [96.77, ] 78 E[ 18 ] [150.12, 155.1] 79 E[ 18 ] [155.24, ] 86 E[ 18 ] [150.12, 155.1] 12 E[ 45 ] [6.00, 6.45] 27 E[ 45 ] [6.00, 6.45] 46 E[ 45 ] [6.00, 6.45] 47 E[ 45 ] [6.00, 6.45] To provide more statistical insight on these two methods, the E[ ] values acquired by both methods were examined and compared. Through observations, all o the E[ ] values obtained by mathematical ormulation method (Table 3.3) all within the range o the 95 percent conidence interval o concluded that E[ ] acquired via the simulation method (Table 3.6). Hence, it can be E[ ] values generated by both methods are not dierent Facility Layout isk Calculation In the previous two sections, the method o quantiying the impact o uncertainty in product demand was proposed in term o relative increase in low intensity ( E[ ] ). Once all E[ ] values are obtained, the risk associated with the acility layout can be determined. The risk value ( ) is an increase in material handling cost as a result o the probable increases in demand and is calculated using equation (3.7). 42

58 = E[ ] C D (3.7) These calculated risk values provide an estimate o probable increases in material handling cost due to the probability that low intensity between departments i and j surpasses that between departments k and l. A risk-low table, comprises o values, is used to represent the estimated risk. Table 3.7 shows the ormat o a risk-low table. TABLE 3.7 ISK-FLOW TABLE j For example, equation (3.7) is used to calculate, which is obtained as = E[ ] C D = 87*1*100.23= $8, where E[ ] value is calculated previously in section 3.2.1, C is assumed to be $1/unit 18 distance, and D 18 is the distance between departments 1 and 8 o the layout generated based on projected demand (Figure 1a). The remaining values are calculated and are presented in Table

59 TABLE 3.8 ISK-FLOW TABLE FO NINE-DEPATMENT CASE STUDY $7,180 $14,072 $7,180 $8,730 $9,577 $7, $340 - $ $ $7,180 $9,577 $8,730 $13,576 $14,072 $13, $ isk-anking To generate the ranking o each low intensity, a risk-low table is used, and the signiicant low can be identiied by calculating the maximum risk or the low rom departments i to j to surpass the others ( * ). Thus, * can be expressed as * = max ( ) (3.8) The values are sorted and ranked in descending order in this step. These rankings are * generated to assist in indicating which low intensities contribute to a signiicant increase in risk value, when product demand is altered with respect to the others. The interpretation o these risk values can be subjective since it depends upon the risk attitude o the decision maker. Based on the example shown in the previous section, by using equation (3.8), is $14,072 and * 18 is * 45 $340, and the risk ranking is * 18 * ollowed by Conclusion In this chapter, the deinition o risk in the context o acility layout design is established. isk here is deined as the potential increase in material handling cost due to uncertainty in 44

60 product demand. Subsequently, the development o a acility layout risk assessment method is detailed. The proposed method aims to quantiy the impact o the uncertainty in product demand to the layout in terms o an increase in material handling cost. To calculate this increase, two methods or the estimation o expected relative increase in low intensity ( E[ ] ) are oered: (1) a mathematical ormulation method and (2) a simulation method. The estimated is later used to calculate the risk associated with the layout. The next chapter presents an E[ ] value approach or solving an FLP utilizing these risk values. The goal o this approach is to reduce the impact o the product demands uncertainty to the acility layout. 45

61 CHAPTE 4 ISK-BASED FACILITY LAYOUT APPOACH FO SINGLE-PEIOD POBLEM 4.1 Introduction In Chapter 3, the acility layout risk assessment method was presented. This method aims to quantiy the impact o the uncertainty in product demand to the eiciency o the acility layout in terms o risk. Once the risk o a acility layout is measured, it can be incorporated with the design o the acility layout process. In this chapter, the development o a risk-based acility layout approach or a single-period case is discussed. The objective o this approach was to minimize the risk o the layout. In other words, the acility layout was designed to minimize the maximum expected increase in material handling cost o the layout. A risk-based acility layout approach was perormed using the three-step procedure: 1. Develop the best layout based on projected demand values. 2. Perorm acility layout risk assessment. 3. Develop the layout to minimize the maximum risk. The ollowing sections describe the development o a risk-based acility layout approach, and detail the mathematical ormulation and GA procedure required or layout generation. 4.2 Notations The notions used in the risk-based acility layout approach or a single-period case are the ollowing: Z P N Material handling cost or layout based on projected demand values Total number o locations in layout; p = 1 P and q = 1 P Total number o departments in layout where i and j are indices or departments 46

62 isk o acility layout D Distance between any two locations p and q pq X Binary variable, 1 i department i is allotted to location p or location p is allotted ip to department i, 0 otherwise without considering risk Y G GA population size GA number o generation * D Distance between any two locations p and q or risk-based layout pq * X Binary variable, 1 i department i is allotted to location p or location p is allotted ip to department i, 0 otherwise or the risk based layout y y th population o g th generation o s th risk scenario in t th period gst y gt y th population o g th generation in t th period; y = 1 Y g st g th generation o s th risk scenario in t th period g t g th generation o t th period; g = 1 G Z i Cost o best solution in i th population Z* Cost o best solution in any population i isk o best solution in i th population * isk o best solution in any population F Projected low intensity between departments i and j * F Flow intensity between departments i and j considering risk 47

63 4.3 Layout Generation or Projected Demand As mentioned earlier, the risk associated with a acility layout is deined as the potential increase in material handling cost caused by uncertainty in product demand. Hence, to estimate the additional material handling cost, the best layout based on the projected demand values was irst determined. This layout problem can be ormulated as Min N-1 N P P Z = F CD X X (4.1) i=1 j=i+1 p=1 q=1 pq ip jq Subject to the ollowing: N X ip=1 p (4.2) i=1 P X ip=1 i (4.3) p=1 { i=1...n-1,j=i+1...n,p=1...p,and q=1...p } where constraint equation (4.2) was used to ensure that there is only one department per location, and constraint equation (4.3) was used to guarantee that each location is allotted to only one department. 4.4 GA Procedure Since the mixed-integer programming shown in section 4.3 is categorized as a classical quadratic assignment problem (QAP) or acility layout design, a GA approach was employed. A one-dimensional array chromosome was used to represent the order o departments to be placed in a layout. The chromosomes were represented by numerical representation (e.g., 01, 02, 03, etc.) o a string placement scheme or the layout generation. The departments were placed in successive rows rom let-to-right and then rom right-to-let. The width and height o the 48

64 acility were speciied or placement o the departments. For example, the placement o departments or the string is shown in Figure 4.1. Figure 4.1. Department placement scheme. The GA cost unction is provided in equation (4.1). This cost unction attempts to minimize the material handling cost or the projected demand. The itness unction is given in equation (4.4). The GA algorithm was veriied using several existing cases to veriy the code. Zi Z* -(( ) β ) α -1 Z* i Fitness Value = v( i) = K( Z ) e (4.4) where = 0.5, and is a dynamic actor that is continuously modiied as time increases. For problems in the single-period scenario, ater experimentation, the ollowing ranges o values are used or : 0.002Z*, when 0<t<T/ Z*, when T/5 t<2t/5 β = 0.006Z*, when 2T/5 t<3t/ Z*, when 3T/5 t<4t/5 0.01Z*, when 4T/5 t T where i is the current generation, Z* is the cost o the best solution in any population, T is the total run time, and t is the current time. The value o used in the itness unction is dependent on time as well as minimum cost. This itness unction was designed such that as the cost 49