Modeling uncertainties in plant layout problems

European Journal of Operational Research 63 (1992) 347-359 North-Holland

347

Modeling uncertainties in plant layout problems Udatta S. Palekar Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

Rajan Batta Department of Industrial Engineering, State Uniuersity of New York at Buffalo, Buffalo, N Y 14260, USA

Robert M. Bosch Quaker Oats Inc., Chicago, IL 60610, USA

Sharad Elhence The Wharton School, University of Pennsylvania, Philadelphia, PA 19104, USA Received December 1990; revised March 1992

Abstract: The plant layout problem deals with the problem of allocation of departments to sites to minimize the total material handling cost. The problem is generally solved based on interdepartmental flows for a single period. Due to the dynamic nature of businesses - growth, fluctuating demands, changes in product mix - the optimal layout should in general, be different from period to period. A relocation cost is incurred whenever a layout is changed from one period to the next. Moreover, due to uncertainties in future predictions, system variables such as growth, demand and product mix cannot be deterministically known for the future periods. Often the interdepartmental flows for future periods can only be probabilistically predicted. In the face of these uncertainties, what should the plant layout be in future periods so that the sum of the expected material handling cost and relocation cost is minimized? This p a p e r focuses on this issue of modeling uncertainties in plant layout problems. An exact method and heuristics are suggested to solve the resulting stochastic dynamic plant layout problem. The heuristics proposed were able to generate good solutions in a reasonable amount of time for problems with up to 40 departments. Our simulation studies indicate that a rolling horizon approach yields better results instead of using a fixed horizon approach. Keywords: Plant layout; stochastic; dynamic programming

I. Introduction

1.1. Background A significant fraction of the cost of manufacturing for most products is due to the material handling required for routing raw materials, parts, sub-assemblies, assemblies and other associated materials

Correspondenceto: U.S. Palekar, Department of Mechanical and Industrial Engineering, Universityof Illinois at Urbana-Champaign, Urbana. IL 61801, USA. 0377-2217/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

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between different departments. "It has been estimated that between 20 to 50 percent of the total operational expenses within manufacturing are attributed to material handling" (Tompkins and White, 1984). The material handling cost is proportional to both the volume of flow and the distance between departments. The volume of flow is determined by the level of production of each product and the product mix, that is, the relative proportion of outgoing products. The production levels are driven by market demands and therefore are typically beyond the control of a planner. However, the distance between departments can be controlled, to an extent, through an efficient plant layout. Effective facilities planning can reduce material handling cost by at least 10 to 30 percent and thus increase productivity (Tompkins and White, 1984). It is therefore of importance that the departments be laid out to minimize the material handling cost. The study of plant layout traditionally focuses on finding layouts that minimize the material handling cost. The input information required for such problems is the interdepartmental material flow matrix (IDFM) usually in terms of pallets, trips or pounds per day for every period. Here, a period refers to an interval of time during which the input information is assumed to remain constant. The input information may change from period to period due to the following factors (Tompkins and White, 1984): • Product related: These may include changes in the design of a product, in the quantity to be produced or may involve the introduction of a new product. • Process related: These may involve changes in the process plan for some products or may relate to the introduction of new special purpose machinery. • Production related: Machine breakdowns requiring long periods of down time, changes in schedules, unavailability of materials a n d / o r tools may be included in this category. • M a n a g e m e n t related: These may include changes in the organization structure that in turn may be caused by changes in management philosophy. • Demand related: These may include seasonal fluctuations in demand and growth or decline in the demand for a product. They also may involve special one-time orders for particular items. Some of these changes are planned and hence known a priori. These may include some changes in the design, the process and management philosophy. The dynamic layout model of Rosenblatt (1986) can be used to handle such changes. On the other hand, changes in product mix, machine breakdowns, seasonal fluctuations and demand are only probabilistically known. These changes may be forecasted using market surveys or from historic trends. There is an inherent uncertainty in such forecasts. Consequently, the interactions between departments computed from these forecasts will have some uncertainty associated with them. Under these circumstances, the facilities planner must determine a master facilities plan (MFP), which consists of the layouts to be used in each period, that minimizes some measure of the material handling cost while simultaneously minimizing the disruption caused by layout changes. Typically, the MFP spans a period of between 5 to 10 years and is updated every year as new information is generated. In this paper we present a mathematical programming model and solution methods for the problem of determining the optimal master facilities plan when there is uncertainty in the forecasts. Our model considers the cost of material flow in the facility as well as costs of relocating a facility. We can obtained exact solutions for problems with up to 12 departments and 8 periods. The approximate solution methods have been successfully tested for problems with up 40 departments and 8 periods. 1.2. A t a x o n o m y o f plant layout problems

The plant layout problem is solved for a single period when the interdepartmental flow is nearly constant from period to period. Henceforth, we refer to this problem as the Static Plant Layout Problem (SPLP). If the flow is varying over time, then a Dynamic Plant Layout Problem (DPLP), which considers the flow matrices for different periods to arrive at 'optimal' layouts for the entire planning horizon, has to be solved. When solving the dynamic version of the problem, it is important to take into account the relocation costs incurred when shifting the layout from one period to the next period. Therefore, the objective in DPLP is to minimize the sum of material handling costs and relocation costs over all periods in the planning horizon.

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Plant layout problems can be broadly classified based on the length of the planning horizon, into two categories: SPLP or DPLP. SPLP focuses on finding a layout for a single period while the DPLP's objective is to find a layout schedule for all the periods in the planning horizon. The DPLP can be further classified into two classes of problems - deterministic or stochastic - based upon the degree of uncertainty with which the input information is known. Deterministic DPLP is suited for situations where the IDFM for the different periods in the planning horizon are different but known with certainty. Stochastic DPLP is the most complex as well as the most general of all cases. All other models may be viewed as special cases of this problem. An SDPLP formulation can be adopted when the IDFMs for the different periods are only probabilistically known. 1.3. Literature survey Most of the work on deterministic plant layout problems is concerned with the minimization of material handling cost. As a first step towards obtaining a layout, a quadratic assignment problem is solved to obtain a minimum cost mapping of the departments onto the available locations. Extensive efforts have been made to solve the quadratic assignment problems using either heuristics or exact solution methods. For extensive surveys on the quadratic assignment problem refer to Francis and White (1974), Mirchandani and Obata (1979), and Burkard and Derigs (1980). The quadratic assignment problem does not account for either the shape or the size of different departments and the overall plant. A visual method has been most successful for fitting departments into the constrained plant floor space. Recently, Scriabin and Vergin (1985) proposed a 3-stage algorithm which uses cluster analysis to group similar facilities to solve the plant layout problem. Several computerized packages exist which solve the plant layout problem. To mention a few, C R A F T (Buffa, Armour and Vollmann, 1964), C O F A D (Tompkins and Reed, 1973, 1976), A L D E P (Seehof and Evans, 1967), and C O R E L A P (Lee and Moore, 1967; Moore, 1971) are frequently used. A L D E P and CORELAP maximize some measure of closeness rating, whereas the other two minimize the total material handling cost. Three-dimensional plant layout packages have also been developed, for example CRAFT3D (Cinar, 1975), S P A C E C R A F T (Johnson, 1982) - for a brief comparison between these see Jacobs (1984). Other approaches to this problem have been taken by Montreuil et al. (1987) who use a b-matching problem to determine perimeter adjacency between departments and an interactive method for block layout. An exact method and heuristics for solving the deterministic version of DPLP, using dynamic programming, are presented by Rosenblatt (1986). Montreuil (1989) presents a linear programming model to determine the optimal way to transition from a given initial layout to a given final layout. In this paper, we seek to find a optimal master facilities plan using a stochastic dynamic plant layout problem. As far as we are aware, this paper presents the first model and a solution technique for a stochastic dynamic plant layout problem. 1.4. Organization of paper The emphasis of our work is on the Stochastic Dynamic Plant Layout Problem (SDPLP). After mathematically formulating SDPLP, we present an exact method and a heuristic to solve this problem. We then present our computational experience with these methods. We conclude by identifying some directions for future research.

2. Stochastic dynamic plant layout problem (SDPLP) The stochastic dynamic plant layout problem is concerned with the determination of a master facilities layout. This requires determination of the IDFMs for all periods in the planning horizon. Since the

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I D F M is calculated based on the process plans and demands for products that are to be made in the facility, it is necessary to forecast these quantities. In practice, the uncertainties associated with such forecasts are captured by determining optimistic, most likely, and pessimistic levels of production for each product and associating a probability of occurrence for these outcomes (Tompkins and White, 1984). By combining these estimates for different products with their projected process plans, it is possible to create a n u m b e r (perhaps large) of IDFMs. Furthermore, from the probabilities associated with the forecasts, it is possible to estimate the likelihood of each of these I D F M s in each period. This probability of occurrence of an I D F M in a period is most likely to be conditioned on the situation in prior periods. Such conditional changes can be easily modeled as a Markov chain. Although this may appear to restrict the model, as the state of the system in a period only depends on the state in the previous period, longer term dependencies can be easily modeled by suitably enlarging the state space. Thus, if the probability of the occurrence of an I D F M depends on the I D F M s in the previous t periods, the number of states in the model will be 2'. The MFP is then devised to minimize the expected cost of material handling over the planning horizon while simultaneously minimizing the costs of relocation due to changes in the layout from period to period. The model makes several assumptions about the availability of data. We first identify the assumptions and data requirements of the model and introduce mathematical notation that will be used to represent these quantities. Consider a problem involving n departments that have to be located at n potential sites and a planning horizon of T periods. We use the index s to represent periods with the current period being period 0 (zero). Further, we represent departments either using subscript i or k and locations using subscripts j or h. Finally, we assume that the following information is available: • A finite set of I D F M s - each I D F M representing a unique state in the Markov chain - has been identified. The actual I D F M in any period of the planning horizon will come from this set. We define S = N u m b e r of possible I D F M s or states. f sk = Work flow from department i to department k according to I D F M s. • The transition matrix containing the transition probabilities from one state to another is known. Let Ohs = Probability of one-step transition from flow matrix h to s. 4~ = The flow matrix of period 0. p~' = Probability of flow matrix s occurring in period t given the flow matrix of period 0. These probabilities can be generated by multiplying the transition probability matrix with itself t times, p~ can be efficiently evaluated by using the following relations: s

p'= E Ph'-' Ohs and p°s = 0 ¢ ~ . t1=1

• A matrix of inter-site distances is known. We define djh = Distance from location j to location h, where dj~ = O. • The costs of material handling and department relocation are known. Let a = The cost of one unit of flow to travel a unit distance. /3i = Cost per unit distance of moving department i to a new location. The objective of the S D P L P is to determine a set of layouts, one for each period, to minimize the expected cost of material handling and relocation. These layouts are represented using the following decision variables:

x[~=

1 0

if d e p a r t m e n t i is assigned to location j in period t, otherwise.

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Finally, the SDPLP can be formulated as the following mathematical programming problem: 7"

Minimize

Z=

S

~

Z

t~l

s=l

T

~

~

i=1 j=l

~ k=l

~ a - ' ~z',~j~k ~ d jr, x '~ix ' kh + E h=l

t=0

1

~ ~ i=1 j=l

~ ' ~Pdi

ih x'il~ "'+' ih

(l)

h=l

subject to the constraints x itj -_ 1

(~-)

for each location j = 1, .

. n . and . .period . . t. = . 1

T,

(2)

i=1

7= 1

Xi i

for each department i = 1. . . . . n and period

t = 1..... T,

(~)

j=l

xgit =

0 or 1

for each i = 1 . . . . , n, j = 1 . . . . . n and t = 1 . . . . . T.

The objective function (1) consists of two terms - the first being the material handling cost term and the second being the relocation cost term. Note that the subscript of summation over different periods ranges from 1 to T for the material handling cost term and from 0 to T - 1 in the relocation cost term. This is because of the implicit assumption that the current period is numbered as 0 while the planning horizon includes periods from 1 through T. Within the planning horizon, material handling costs will be incurred for each of the periods 1 through T. The relocation costs will be incurred when the layout is changed from the current state to period 1, period 1 to period 2, and so on till the last change occurs in the transition from period T - 1 to T. The above formulation assumes that the relocation costs are proportional to the distance d/i , between the old and the new location. It is easy to reformulate the problem to include a fixed charge that depends on the department for every relocation. This may be achieved by changing the relocation cost to jgidjh q-ci, where c i is the fixed cost associated with department i. The cost c i may represent the disruption cost associated with relocating department i. This change in the cost does not alter the model or the solution method that we describe later. Constraint set (2) assures that exactly one department is located in every period at every location. Similarly, constraint set (3) assures that every department is located at exactly one location in every period. Note that the constraint sets (2) and (3) are separable over time periods. The linkage between the different periods is through the objective function, and more specifically through relocation costs incurred between periods. If the costs of relocation are assumed to be negligible, the problem decomposes into T single period problems. Each single period problem can be solved using the expected flow matrix for that period to obtain the MFP. On the other hand, if the relocation costs are very high, the formulation indicates that the solution is to use the same layout for all periods. This implies that xij = xij for all t. The objective function, after suitable rearrangement, can then be written as

~ ap~fi~ dihxijxkh.

min i=1 j:=l k=l

h=l

t=l

s=l

The term in parenthesis represents a single I D F M that is the sum of the expected IDFMs in each period. In this case, the problem reduces to a single SPLP. The solutions to these extreme cases - no relocation costs and infinite relocation costs - provide lower and upper bounds, respectively, to the SDPLP. The SDPLP is a quadratic integer program. As a special case, when T = 1, the SDPLP reduces to a quadratic assignment problem, i.e. to an SPLP. The quadratic assignment problem is known to be NP-hard; as such the SDPLP is also NP-hard. Consequently, exact solution algorithms for the problem must be enumerative in nature. We next develop an exact solution algorithm based on dynamic programming.

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3. Solution strategies for SDPLP

3.1. Exact solution using dynamic programming We use dynamic programming as an optimization procedure to solve SDPLP. In our problem, a stage will be a time period and a state variable will be the layout used in this period. The layout will be represented using either A or/~. To specify a layout A we use the variables y~ that indicate the location of each department. Here Y~={10

otherwise.ifdepartmentiisassignedt°l°cati°njinlay°utA'

Given layout A and the flow matrix s, the material handling cost associated with this combination of layout and flows is expressed as:

i--I j=l

k=l

h=l

The relocation cost from layout I to layout V is calculated by summing the relocation costs of the departments that have different locations in the two layouts. Thus, the relocation cost is given by =

[3idjm YijYim" i--1 j ~ l m--1

Let G(t, A) be the minimum cost incurred from material handling and relocation in periods t through T if layout I is chosen for period t. We relate G(t, A) to G(t + 1, V) to obtain a functional equation for dynamic programming as follows: min

(s

~(p;MHC(A,s))+RC(A,.)+G(t+I,.)

)

ifO
s~l

a(t,

a) =

s

(p~MHC(A, s))

if t = T.

s--1

Since the planning horizon consists of T periods we must have G(T+ 1, V ) - - 0 , and hence the relation for G(T, A) involves no unknown quantities. Once G(T, A) is known for all A, the functional equation is recursively solved for G ( T - 1, A) and so until G(1, A) is evaluated for all layouts A in period 1. Reiterating our objective, we wish to determine the layout to use in period 1 so that the expected cost of material handling and relocation is minimized over the entire planning horizon. The optimal sequence of layouts can be obtained by tracing back the layouts that yielded the minimum cost. The difficulty with this exact dynamic programming approach is that it is computationally too expensive since there are A = n! layouts in each period, where n is the number of departments/sites. The procedure involves AZT computations as it looks at A 2 layout combinations at every stage. The number of layouts that must be considered for each period can be reduced by the use of lower and upper bounds as proposed in Rosenblatt (1986). Let Z vB and Z LB represent the upper and lower bounds respectively, and K = Z UB - Z LB. Then the only layouts that must be considered in any period are those that have a material handling cost smaller than Z°Vt+ K, where Z °vt is the optimal solution for that period. A lower bound for this problem may be obtained, as suggested in Rosenblatt (1986), by discarding the relocation costs and considering the minimum cost solution for the expected IDFM in each period. The sum of the material handling costs using this solution is a lower bound on the problem. If the relocation costs incurred due to the layout changes are added in, we get an upper bound to the overall problem (we refer to this solution as the Expflw solution). An improved upper bound may be obtained as suggested in Batta (1987) by considering infinite relocation costs. This reduces to finding an

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optimal layout for the total flow over all periods (we refer to this bound as the Batta bound in the remainder of the paper). The success of this strategy depends entirely on the quality of the upper and lower bounds. Although the use of bounds reduces the n u m b e r of layouts required, it is now necessary to find all layouts that have a cost less than a prespecified value. 3.2. Generating ranked layouts

Rosenblatt (1986) suggests the use of cut constraints to generate the r best layouts. This requires the solution of a sequence of QAPs with side constraints, one for each layout generated. This is computationally extremely expensive. An alternative approach is to modify the fathoming criteria in the branch and bound algorithm used to solve the Q A P associated with the single period problem. In a branch and bound algorithm, the best known solution, usually called the incumbent, is used as an upper bound. If the lower bound at a node is larger than the incumbent value the node is discarded or fathomed. If the optimal layout cost, Z °pt, was known, the fathoming criteria could be modified by fathoming only if the lower bound was greater than Z °Ot nt- K and storing all solutions with a value smaller than Z °pt + K. However, since Z "pt is not known, a node is only fathomed if the lower bound is greater than (incumbent value + K). As improved solutions are obtained, the incumbent value is updated and any stored layout that has a value greater than the updated value of (incumbent value + K ) is discarded. This assures that only those layouts that are necessary for the DP are generated. This approach, however, is only feasible for small problems because the time required to generate the ranked layout grows exponentially with the number of departments in the problem. Thus, finding the exact solution to the SDPLP is only feasible if the n u m b e r of departments is small. For larger problems, it is necessary to devise good heuristics. 3.3. Approximation algorithm for SDPLP

The amount of enumeration required may be reduced by two separate means. First, the number of layouts that are considered at each state of the DP can be reduced. Second, the layouts can themselves be ranked using a heuristic. To reduce the number of layouts to be considered, we first study a two-period problem. Consider a two-period SDPLP with corresponding average I D F M s F 1 and F 2. Let UB* be the set of optimal layouts for the two-period problem when the relocation costs are infinite. This has an associated cost of ZtvB for some # • UB* in stage t. Note that Z y B = zs= j(p~ MHC(/z, s)). Then the minimum cost for the S D P L P without relocation is Z ~ B + Z ~ ~. Suppose all possible layouts are ranked according to their material handling costs using F ~ and F 2, respectively. For each set of rankings, partition the layouts into two sets A, and B t such that A, = Set of all layouts with cost less than or equal to Z, w . B, = Set of all possible layouts with cost greater than Z ff~.

Proposition 1. For the two-stage problem, let A, and B t be defined as aboL~e and let A be any layout. Then the following holds: 1) I f A • A l and A ~ U B * , then A • B 2. 2 ) I f t~ • A 2 and A ~ U B * , then A • B 1. Proof. The proof is by contradiction. Consider 1). Suppose there exists a layout A such that A • A ~ , A ~ UB* and A ~ A 2. Let the material handling cost associated with A in stage t be Z~. By definition both Z~ _< Z~vB and Z~ _< Z2w . If equality holds in both stages, A • UB*, which is a contradiction. If inequality holds for the second period, using the same layout A in both stages results in a lower cost solution than any layout in UB*. This contradicts the assumption that UB* is the set of minimum cost layouts without relocation. Hence, 1) holds. Similarly it can be shown that 2) holds.

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Table 1 Example rankings for a two-stage SDPLP Stage 1

Stage 2

Set

Rank

Layout

Set

Ai

1 2 3 4 5 6 7

A B C /x ~ U B * D E F

A2

UB* B1

UB* B2

Rank

Layout

1 2 3 4 5 6 7

D /x ~ U B * B A E C F

This result can best be explained by considering Table 1 which shows a set of ranked layouts for a fictitious two-stage problem. The layouts A, B and C, which belong to A~ but not to UB* in the first period, belong to set B: in the second. Similarly, layout D, that belongs to A 2 in the second period, is in B~ in the first. The proposition does not preclude layouts such as E and F that are in both the sets B~ and B 2. Proposition 1 can be used to reduce the number of layout combinations that must be considered. If we consider a solution in which layout rn is used in period 1 and layout k is used in period 2 (m 4: k), then this solution can have a smaller cost t h a n / z ~ U B * only if Z~' + Z~ + relocation costs < Z ~ B + Z2vB. Rearranging gives the following expression: ( Z ~ B - Z ~ ) + (Z2vB - Z ~ ) > relocation costs. Clearly, if m E B 1 and k ~ B 2 , the resulting cost can only be higher than the cost o f / z ~ UB*. If either m ~B~ (or k ~B2), then the cost will be lower only if it is coupled with a suitable layout from A 2 (or A 1). Without loss of generality, let k ~ B 2. Then for a lower cost solution, necessarily, m ~ A p This also implies that Z~ a-Z2 k<0

and Z1v B - Z ~ ' _ > 0 .

Since relocation cost must be non-negative,

Iz, -zT l>lz -z l where I. I represents absolute value. Consequently, the only combinations which must be considered in the search for an optimal solution are 1. k ~ A 1 combined with any m ~ B 2 such that Z ~ B - Z1k > Z ~ - Z2vB. 2. k ~ A 2 combined with any m ~ B1 such that Z ~ B - Z k > Z~n - Z ~ B. 3. m ~ A 1 combined with any k ~ A 2. From our computational experience, we have found that only a small number of layouts from B t need to be considered and almost always the optimal layout in each period belongs to the s e t s A t and UB*. Although this result does not directly carry to more than two stages, as a heuristic we can restrict the layouts to be considered in each period to only those layouts in set A t and UB*. Further, we could decide that no more than the C best layouts will be considered for any stage. Thus, a total of only C * S layouts will be considered. The C best layouts can be obtained using the branch and bound methods described above and only saving the best C layouts. However, finding even the best layout in one period (C = 1) may take an inordinate amount of time - this is equivalent to solving a QAP. For problems with larger than 15 departments, this approach is still computationally infeasible.

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The second approach is to use a heuristic for generating the ranked layouts. The branch and bound method can be used as a heuristic if the algorithm is stopped after a fixed amount of time. This, however, does not guarantee that C solutions will be obtained. An alternate approach is to use an interchange heuristic for the QAP. The procedure can be started from several different random initial feasible layouts. The best C layouts generated during this process can then be used in the DP approach. For both the exact and heuristic algorithms we modified a F O R T R A N program for the quadratic assignment problem written by Burkard and Derigs (1980).

3.4. Extensions The SDPLP, as presented, solves the problem assuming that the flow matrix used in each period is the expected flow matrix for that period. This has two shortcomings. First, the average may not be a good predictor of the actual flow matrix that occurs in a period. This is especially true if the IDFMs for each state are markedly different. Second, for problems involving several time periods, forecasts for the later periods tend to be inaccurate. Consequently, the layouts generated for these later periods based on the SDPLP are unlikely to be good. To remedy some of these problems it is possible to consider a rolling horizon approach to this problem. The SDPLP can be solved in each period with updated information and only the layout corresponding to the current period should be implemented. This has the benefit that forecasts of future behavior are used to condition decisions in the current period but changes in data are incorporated in the model as time progresses. There is, however, a disadvantage in that frequent changes in the master facilities plan will be required resulting in less planned facility re-organization. This will add to the disruption costs associated with relocation. A second approach to remedying some of the problems associated with the use of expected costs is to use an expected value-variance criterion. In this case, the cost of the layout is a composite of the expected cost of the layout plus a multiple of the variance of this cost. This introduces a quartic term in the mathematical formulation presented earlier. Obviously, exact solution of the resulting problem would be very difficult even for a single period. However, it is possible to obtain heuristic solutions using interchange procedures. In the dynamic programming approach, the recurrence relationship can be changed as follows: c,(t,

min(C()t,t)+RC(A,l~)+G(t+l, lx)) /x

ifO
C ( a , t)

if t = T

a ) --

where S

S

C(A, t) = E (P,( MHC(A, s)) + ~ Y', ( p { ( M H C ( A , s))2). s

1

s

1

C(A, t) is the cost associated with using layout A in period t; o~ is a weight that reflects the relative importance to be attached to the 'robustness' of a layout; other terms are as defined earlier. This approach penalizes layouts that are extremely good for some high probability IDFM but relatively poor for other IDFMs. Depending on a, the importance attached to robustness, layouts which exhibit high variance become more expensive then relatively robust layouts. This allows the dynamic programming algorithm to select a higher expected cost layout if it is relatively robust.

4. Computational testing To test the solution procedures developed in the previous sections several test problems were generated and the different approaches were used to solve these problems. In the first set of experiments, a set of 8 department problems involving 2, 4 or 6 periods and 4, 6 or 8 different IDFMs were

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Table 2 Comparison between MFPs obtained using three policies and two algorithms Stages

States

2

4

2

6

2

8

4

4

4

6

4

8

6

4

6

6

6

8

MFP cost Bound

Expflw

DP Soln.

CPU time Bound

DP Soln.

Rank of Batta bound

109950 109950 121020 121020 133200 133200 223230 223230 254320 254320 262660 262660 336450 336450 386930 386930 391920 391920

109920 110115 120990 121095 133280 133390 223200 223750 254290 254750 262740 263160 336420 337370 386900 387590 392000 392710

109920 109920 120950 120950 133200 133200 223200 223200 254250 254250 262660 262660 336420 336420 386860 386860 391920 391920

1.26 3.44 0.46 3.16 0.24 3.40 1.04 3.38 0.38 3.18 0.26 3.38 1.04 3.40 0.36 3.22 0.30 3.24

3.64 5.10 1.62 4.80 0.94 5.02 5.26 6.70 2.46 6.42 1.56 6.62 7.22 8.20 3.20 8.02 2.96 9.76

24, 3 9, 3 15, 4 10, 3 4, 3 3, 3 24, 3, 3, 3 9, 3, 3, 3 15, 4, 4, 4 13, 3, 3, 3 4, 3, 3, 3 3, 3, 3, 3 24, 3, 3, 3, 3, 3 8, 3, 3, 3, 3, 3 15, 4, 4, 4, 4, 4 12, 3, 3, 3, 3, 3 4, 3, 3, 3, 3, 3 3, 3, 3, 2, 3, 3

solved using b o t h the exact m e t h o d as well as the h e u r i s t i c m e t h o d . T h e r a t i o of m a t e r i a l h a n d l i n g cost to r e l o c a t i o n cost was 0.4. F o r e a c h size, t h r e e p r o b l e m s w e r e solved using d i f f e r e n t t r a n s i t i o n matrices. T h e t r a n s i t i o n m a t r i c e s u s e d w e r e also r a n d o m l y g e n e r a t e d with t h e result t h a t the m a t r i c e s w e r e fairly dense. F u r t h e r , a m a x i m u m o f 100 r a n k e d layouts w e r e g e n e r a t e d in e a c h p e r i o d . R e s u l t s from these tests a r e p r e s e n t e d in T a b l e 2. T h e t a b l e shows t h e cost of the m a s t e r facilities p l a n ( M F P ) a n d t h e C P U time, in seconds, r e q u i r e d to o b t a i n t h e s o l u t i o n on a S U N 3 / 6 0 s t a n d a l o n e w o r k s t a t i o n if no r e l o c a t i o n is allowed ( t h a t is, t h e B a t t a b o u n d layout is u s e d in every p e r i o d - see Section 3.1), if the b e s t layout is u s e d in e a c h p e r i o d r e g a r d l e s s o f r e l o c a t i o n costs ( d e n o t e d by 'Expflw'), a n d if the D P solution is used. T h e t a b l e also shows t h e r a n k o f the B a t t a b o u n d layout at each stage. F o r e a c h c o m b i n a t i o n o f states a n d stages, the u p p e r row gives results w h e n exact r a n k i n g s w e r e used a n d the lower row gives results if heuristic r a n k i n g s a r e used. T h e results i n d i c a t e t h a t using heuristic r a n k i n g gives fairly high quality results - o f t e n the results a r e exactly t h e s a m e as with exact rankings. Surprisingly, the heuristic rankings r e q u i r e m o r e C P U time t h a n t h e exact rankings. This is e x p l a i n e d by t h e fact t h a t the heuristic rankings w e r e g e n e r a t e d using an i n t e r c h a n g e p r o c e d u r e . This r e q u i r e s several p a s s e s to g e n e r a t e d i f f e r e n t high quality layouts. T h e exact m e t h o d , which uses a m o d i f i e d b r a n c h a n d b o u n d t e c h n i q u e , is m o r e systematic and, c o n s e q u e n t l y , p e r f o r m s q u i t e well for small p r o b l e m sizes. This is not e x p e c t e d to be the case for l a r g e r p r o b l e m s . T h e cost o f t h e M F P s g e n e r a t e d a r e fairly close to each other. T h e b o u n d layout in g e n e r a l s e e m s to p e r f o r m q u i t e well, e s p e c i a l l y as t h e n u m b e r of p e r i o d s (stages) increases. T h e overall cost o f using the b o u n d layout for all p e r i o d s is c o m p e t i t i v e a n d it is r a n k e d very high in t h e l a t e r stages. This is a t t r i b u t a b l e to the fact t h a t t h e t r a n s i t i o n m a t r i c e s a r e fully d e n s e a n d r e a c h s t e a d y state very quickly. In s t e a d y s t a t e the e x p e c t e d flow m a t r i x is t h e s a m e as the I D F M u s e d for the B a t t a b o u n d layout. T h u s if the n u m b e r of p e r i o d s is large, the layouts o b t a i n e d in the l a t e r p e r i o d s by the D P m e t h o d a r e the s a m e as the b o u n d layout. This is d e m o n s t r a t e d in T a b l e 3 that shows the layouts u s e d in an 8 - d e p a r t m e n t , 8-state p r o b l e m for d i f f e r e n t n u m b e r s o f stages. T h e layouts m a r k e d by * are the s a m e as the B a t t a b o u n d layout. If a s p a r s e r t r a n s i t i o n m a t r i x is used, t h e r e is a l a r g e r d i f f e r e n c e b e t w e e n t h e d i f f e r e n t M F P s . F o r e x a m p l e , w h e n a t r a n s i t i o n m a t r i x r e p r e s e n t i n g a r a n d o m walk was u s e d for the 8-state, 6 - p e r i o d p r o b l e m , the costs were: 489560 for the B a t t a b o u n d , 486960 for the Expflw solution, a n d 486840 for t h e D P solution.

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Table 3 Actual layouts used by DP solution for an example problem (* indicates bound layout) Stage

Number of stages

1 2 3 4 5 6

Two stages

Four stages

Six stages

{1,3,6,8,2,5,7,4} {2,3,6,8,1,5,7,4}*

{1,3,6,8,2,5,7,4} {2,3,6;8,1,5,7,4}* {2,3,6,8, 1,5,7,4}* {2,3,6,8, 1,5,7,4}*

{1,3,6,8,2,5,7,4} {2,3,6,8,1,5,7,4}* {2,3,6,8, 1,5,7,4}* {2,3,6,8, 1,5,7.4}* {2,3,6,8,1,5,7,4}* {2, 3, 6, 8, 1, 5, 7, 4} *

Table 4 CPU times required for exact and heuristic methods Ranking method

Number of departments

Time for bound

Avg. time for ranking

Time to solve DP

Total C P U time

Exact

6 8 10 12

0.14 0.30 17.76 144.58

0.14 0.33 11,06 145,41

0.02 0.04 0.02 0.06

1.02 2.96 11)6.26 13(17.94

Heuristic

6 8 10 12 15 40

1.08 3.24 7.30 15.42 41.36 698.10

0.28 0.81 1.84 3.82 10.31 677.64

0.02 0.04 0.02 0.04 0.04 33.06

3.34 9.76 19.114 41.02 123.88 6152.28

Table 4 shows the CPU time required for both the exact and heuristic ranking methods to obtain the DP MFP for the 8-state, 6-period problem as the number of departments is increased. The table shows the amount of time required to obtain the bound layout, the average time to obtain rankings, the time required for the actual DP, and the total time. The time required for the exact method increases exponentially as the number of departments increases. The largest problems we were able to solve with the exact method had 12 departments while with the heuristic ranking the largest problems had 40 departments. Further, most of the time required is for obtaining the ranked layouts in each period. To show that the algorithm developed can be used for the large problems encountered in practice, we present some results for 40 department problems involving 5 states and 4, 6 or 8 periods. The transition matrices used were sparse and a maximum of 75 rankings were heuristically generated. Two different ratios of material handling cost to relocation cost were used - 2 and 10. Results of these tests are presented in Table 5.

Table 5 Results for 40-department, 5-state problems Ratio

Stages

Bound M F P

DP MFP

Cost

CPU time

Cost

CPU time

2

4 6 8

31440832 46771120 61172928

1684.22 2342.78 2924.12

31302984 46658528 61046192

3999.58 5598.76 7462.64

10

4 6 8

31431080 46766296 61191376

2705.40 2771.36 2334.16

31304368 46652408 61058328

5367.42 6406.90 7284.04

358

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Table 6 Comparison of solution quality using simulation Periods

States

Expectedcosts

2

4 6 8

DP 109920 120950 133200

Bound 109950 121020 133200

Expflw 110115 121095 133390

Actual costs from simulation DP 182577 164045 201138

Bound 182637 165075 201138

Expflw 184798 165735 202218

Roll 174950 158365 191482

4

4 6 8

223200 254250 262660

223230 254320 262660

223750 254750 263160

303078 251443 339273

303138 252473 339273

308462 257678 344263

290722 241327 323705

6

4 6 8

336420 386860 391920

336450 386930 391920

337370 387590 392710

443840 430817 493852

443900 431847 493852

454958 444570 503487

426080 415162 471222

Since the MFPs generated are based on expected flow matrices, we simulated each problem in the first set of experiments (Table 2) to observe the performance of the MFPs in a dynamic environment. Each problem was simulated using three different random seeds - all results presented in Table 6 are average results. Besides the three MFPs described earlier, we also consider a roiling horizon approach (denoted by 'Roll') in which the DP is solved in each period with updated information and only the layout for the next period is implemented. The simulation indicates that the costs obtained from all three MFPs are generally a poor estimate of the true cost of implementing the MFP. However, in all cases the DP method performed better than the other two methods. As expected, the rolling horizon method led to significantly improved results. This suggests that the DP method proposed here should be used in practice together with a roiling horizon approach.

5. Future research The heuristic in this p a p e r aimed at restricting the set of alternative layouts in each period. This can be viewed as providing 'symptomatic relief' to the problem of an excessive number of layout combinations that is faced in the dynamic programming procedure. There may exist another class of heuristic approaches which promise remedy at a fundamental level. Our ideas regarding such procedures envision a methodology for combining information from different flow matrices, with appropriate weights assigned to them, into one flow matrix. The solution obtained to the SPLP problem with such a unified flow matrix may be competitive with other heuristic solutions. The obvious advantage here is the drastic reduction in the computation time. Furthermore, the simplicity in its application may be an added factor that will ease the implementation of an SDPLP model in industry. Future research also may involve the development of a 'pre-processor' for data required by the S D P L P model. In this pre-processor, raw data such as past demand figures for each product are the inputs with the output being the flow matrices and the one-step transition probability matrix. Conversion of raw demand data into flow matrices involves knowledge about each product, its manufacturing process(es) and scheduling of production. The data about all products may be combined within each period. Then, using clustering techniques, trends in the demand may be analyzed. This will lead to the development of flow matrices from the clusters as well as the one-step transition probability matrix from the trends.

Acknowledgement The authors would like to acknowledge the insightful comments of anonymous referees and the guest editor on earlier versions of this p a p e r that have significantly improved the presentation and readability of this paper.

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