Combining Wang–Landau sampling algorithm and heuristics for solving the unequal-area dynamic facility layout problem

Accepted Manuscript

Combining Wang-Landau sampling algorithm and heuristics for solving the unequal-area dynamic facility layout problem Jingfa Liu , Dawen Wang , Kun He , Yu Xue PII: DOI: Reference:

S0377-2217(17)30314-4 10.1016/j.ejor.2017.04.002 EOR 14367

To appear in:

European Journal of Operational Research

Received date: Revised date: Accepted date:

8 April 2016 18 January 2017 2 April 2017

Please cite this article as: Jingfa Liu , Dawen Wang , Kun He , Yu Xue , Combining Wang-Landau sampling algorithm and heuristics for solving the unequal-area dynamic facility layout problem, European Journal of Operational Research (2017), doi: 10.1016/j.ejor.2017.04.002

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ACCEPTED MANUSCRIPT Highlights A model of an unequal-area dynamic facility layout problem is described. A heuristic Wang-Landau (WL) sampling algorithm is put forward. Vacant point strategy, pushing strategy and pressuring strategy are proposed to update layout. The proposed algorithm is applied to solve the unequal-area dynamic facility layout problem.

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Combining Wang-Landau sampling algorithm and heuristics for solving the unequal-area dynamic facility layout problem Jingfa Liua,b, Dawen Wanga,b, Kun Hec, Yu Xuea,b a

School of Computer & Software, Nanjing University of Information Science & Technology, Nanjing 210044, China

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Jiangsu Engineering Center of Network Monitoring, Nanjing University of Information Science & Technology, Nanjing 210044, China

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School of Computer Science & Technology, Huazhong University of Science and Technology, Wuhan 430074, China

ABSTRACT

Article history:

The dynamic facility layout problem (DFLP) is the problem of placing facilities in a certain

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ARTICLE INFO

plant floor for multiple stages so that facilities do not overlap and the sum of the material handling and rearrangement costs are minimized. We describe a model, where the facilities have

Global optimization

unequal-areas and the layout for each stage is produced on the continuous plant floor. The most

Dynamic facility layout

difficulty of solving this problem consists in the lack of a powerful optimization method. The

Unequal-area

Wang-Landau (WL) sampling algorithm is an improved Monte Carlo method, and has been

Wang-Landau sampling algorithm

successfully applied to solve many optimization problems. In this paper, we combine the WL

Group decision-making

sampling algorithm and some heuristic strategies to solve the unequal-area DFLP. In the WL

Heuristic strategies

sampling algorithm, a vacant point strategy is applied to update layout at one stage. To prevent

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Keywords:

overlapping of facilities and reduce the empty space among facilities, a pushing strategy and a pressuring strategy are applied. We have tested the proposed algorithm on four groups of cases and the computational results show that the proposed algorithm is effective in solving the

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unequal-area DFLP.

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1. Introduction

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Because of the important significance of saving the actual production costs, the facility layout problem (FLP) is very early to be studied. In the early time, the objective of the FLP is to minimize the material handling costs by arranging facilities in a static production environment. As material flows among facilities will not change, the layout problem is called the static facility layout problem (SFLP) which has been studied by many scholars. Mir and Imam (2001) employed a hybrid optimization approach which contained a simulated annealing method and an analytical method. Bozer and Meller (1997) researched the distance-based facility layout problem, and proposed an alternative distance measure basing on the expected distance between two facilities. Meller et al. (1999) proposed some general classes of valid inequalities for a mixed-integer programming (MIP) model of SFLP. They used these inequalities in a branch-and-bound algorithm and moderately increased the range of solvable problems. Motivated by the work of Meller et al. (1999), Sherali et al. (2003) presented an improved MIP model and several effective solution strategies for the SFLP. Liu and Meller (2007) researched the continuous–representation-based SFLP, and proposed a genetic-algorithm-based heuristic which combined the sequence-pair representation with the MIP model. Anjos and Vannelli (2006) presented a mathematical-programming framework based on the combination of two new mathematical-programming models. Komarudin and Wong (2010) proposed an ant system which used slicing tree representation to easily represent the SFLP without too restricting the solution space. Ulutas and Kulturel-Konak (2012) proposed a clonal selection algorithm which had a new encoding and a 

Corresponding author.

E-mail address: [email protected] (Dawen Wang)

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novel procedure to copy with dummy facilities to fill the empty space in the plant floor. Gonçalves and Resende (2015) developed a hybrid approach which combined a biased random-key genetic algorithm, a novel placement strategy and a linear programming model. Singh and Sharma (2006) and Drira et al. (2007) presented a literature review of the SFLP. In today‟s market, with increasing global competition and short life cycle of production, material flows among facilities change during the planning horizon, so that the problem becomes the dynamic facility layout problem (DFLP). The dynamic facility layout problem generally includes two categories: equal-area dynamic facility layout problem and unequal-area dynamic facility layout problem, which correspond to the facilities with equal-area and facilities with unequal-area, respectively. Rosenblatt (1986) is the first one to study the equal-area DFLP and proposed a heuristic optimization scheme based on dynamic programming. Afterwards, some scholars have also studied it and put forward all kinds of heuristic methods. Urban (1993) proposed a pair-wise interchange procedure which made use of „forecast windows‟ to find different sets of good layout plans for the planning horizon. Baykasoglu and Gindy (2001) proposed a simulated annealing algorithm with an effective data structure and neighborhood generation mechanism. Şahin and Türkbey (2009) presented a new hybrid heuristic based on the simulated annealing approach supplemented with a tabu list. Pourvaziri and Naderi (2014) presented a novel hybrid multi-population genetic algorithm. Unlike the available previous genetic operators, they designed operators to search only the feasible space. Hosseini-Nasab and Emami (2013) proposed a hybrid particle swarm optimization algorithm which combined a simple and fast simulated annealing process. Their algorithm further explored the continuous solution space by using a coding technique. Mckendall Jr and Liu (2012) presented three tabu search heuristics. The first is a simple tabu search heuristic, the second adds diversification and intensification strategies to the first, and the third is a probabilistic tabu search heuristic. Hosseini et al. (2014) proposed a robust and simply structured hybrid technique based on three heuristics: imperialist competitive algorithm, variable neighborhood search, and simulated annealing. Bozorgi et al. (2015) applied a data envelopment analysis method and a tabu search-based algorithm which used a dynamic tabu list and a diversification strategy to find the most efficient layout. Ulutas and Islier (2015) studied the equal-area DFLP in footwear industry and proposed a clonal selection algorithm based on affinity maturation and receptor editing processes. Kouvelis et al. (1992) addressed this problem by using the concept of robust layouts, and proposed a branch and bound method to generate the robust layouts for the manufacturing systems. Li et al. (2015) considered the remanufacturing equal-area DFLP with uncertainties and proposed a simulated annealing algorithm. For the unequal-area DFLP, Montreuil and Venkatadri (1991) first presented a proactive strategic methodology sustained by a linear programming model. Yang and Peters (1998) put forward a heuristic procedure based on the construction type layout design algorithm. Dunker et al. (2005) proposed a hybrid algorithm which combined dynamic programming and a genetic algorithm. For each stage the genetic algorithm evolved a population of layouts while the dynamic programming provided the evaluation of the fitness of the layouts. McKendall and Hakobyan (2010) developed a boundary search heuristic which placed facilities along the boundaries of already placed facilities, and used a tabu search method to improve the obtained solutions. Jolai et al. (2012) presented a multi-objective particle swarm optimization method which found the optimal scheme using a particle best position and a swarm best position. Dong et al. (2009) considered the unequal-area DFLP under dynamic business environment and proposed a simulated annealing algorithm based on a shortest path model. Mazinani et al. (2013) considered a new kind of DFLP with flexible bay structure, in which facilities are assigned to parallel bays in a plant floor, and proposed a genetic algorithm. Ma et al. (2015) studied the unequal-area DFLP with uncertain product demands and proposed an improved genetic algorithm which combined triangular fuzzy number operation and adaptive local search with genetic algorithm. For a review of solution methodologies for the DFLP, one can see Balakrishnan and Cheng (1998) and Kulturel-Konak (2007). Although some approaches mentioned above have been applied to solve the unequal-area DFLP, their

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efficiency still needs to be improved. The Wang-Landau (WL) sampling algorithm (Wang and Landau, 2001; Landau et al., 2004; Liu et al., 2016) is an improved Monte Carlo algorithm. Unlike conventional Monte Carlo simulations that generate a probability distribution at a given temperature, the WL sampling method can estimate the density of states accurately via a random walk, which produces a flat histogram in energy space. There have been many improvements and applications on the WL sampling algorithm. For example, Schulz et al. (2003) proposed a simple modification of the WL sampling algorithm. This modification removed the systematic error that occurred at the boundary of the range of energy over which the random walk took place. Seaton et al. (2010) used the WL sampling algorithm to describe the thermodynamic behavior of a continuous homopolymer. Liu et al. (2014) proposed an improved WL sampling method which incorporated the generation of initial conformation based on the greedy strategy and the neighborhood strategy based on pull-moves into the WL sampling method to predict the protein structures on the face-centered-cube (FCC) hydrophobic-hydrophilic (HP) lattice model. The most challenge of solving the DFLP is that the function to be optimized is characterized by a multitude of local minima separated by high-energy barriers. The WL sampling method can visit the all accessible states of the system, which means it can jump out of these high-energy barriers. Therefore, the WL sampling method is an ideal global search algorithm for solving the DFLP. In this paper, by combining the WL sampling algorithm and some heuristics, we put forward a heuristic WL (HWL) sampling algorithm. The numerical results show that the proposed algorithm is an effective method for the unequal-area DFLP. The structure of this paper is as follows. In the second section, we introduce the mathematical model of the DFLP. In the third section, we introduce the WL sampling algorithm and the heuristic strategies. In the fourth section, we list the computational results and make a comparison. The fifth section is the conclusion of this paper. 2. Problem Formulation

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In this section, we provide a mathematical model of the unequal-area DFLP. First, some notations for describing the mathematical model are presented.

Facility index, i, j = 1,..., N, where N = the number of facilities Stage index, t = 1,..., T, where T = the number of stages

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Material handling cost per unit distance between facility i and facility j in stage t Rearrangement cost of facility i in the beginning of stage t Material flow between facility i and facility j in stage t Shorter edge length of facility i in stage t Longer edge length of facility i in stage t Length of plant floor Width of plant floor Interior of facility i

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Ctij Rti Ftij Wti Lti L W Ii

Variables xti, yti Centroid coordinate of facility i in stage t lti, wti Length and width of facility i in stage t dti Orientation of facility i in stage t (0: Vertical, 1: Horizontal) rti

1  0

If facility i is rearranged in stage t ( xt 1,i  xti or yt-1,i  yti or d t-1,i  dti ) Otherwise

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Minimize: total cost E(X) =

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t  2 i 1

Constraints:

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Ii  Ij =   i, j, (2) xti − 0.5lti ≥ 0  t, i, (3) xti + 0.5lti ≤ L  t, i, (4) yti − 0.5wti ≥ 0  t, i, (5) yti + 0.5wti ≤ W  t, i, (6) xti, yti, lti, wti ≥ 0  t, i. (7) Here X represents a configuration and X = (x11, y11, d11, ..., x1N, y1N, d1N ; ...; xt1, yt1, dt1, ..., xtN, ytN, dtN ; ...; xT1, yT1, dT1, ..., xTN, yTN, dTN), where (xti, yti) is the pick-up/drop-off location of facility i in stage t and (xt1, yt1, dt1, ..., xtN,

lti = Ltidti + Wti(1 − dti)  t, i, wti = Lti(1 − dti) + Wtidti  t, i. 3. Solution procedure for the DFLP

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ytN, dtN) is a layout in stage t, 1tT, 1iN. Objective (1) is to minimize the sum of the material handling and rearrangement costs. Constraint (2) indicates that there is no overlapping between any two facilities. Constraints (3) to (6) make sure that the facilities are not placed out of the plant floor. Constraint (7) is the constraint of variables‟ range, and lti and wti are calculated as follows: (8) (9)

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In this section, we propose a heuristic Wang-Landau (HWL) sampling algorithm for solving the DFLP with unequal-area facilities. First, we introduce the Wang-landau sampling algorithm. Then, some heuristic configuration updating strategies are put forward to update the current configuration. Finally, the framework of the proposed HWL is presented.

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3.1. Wang-Landau sampling algorithm

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The Wang-Landau (WL) sampling algorithm is a novel Monte Carlo (MC) algorithm and was proposed by Wang and Landau (2001). It gets a flat histogram of energy by a random walk in the whole energy space following a certain acceptance criterion, so as to estimate the density of states g(E(X)), i.e., the number of all possible states (configurations) for an energy level E(X) of the system. The total cost E(X) represents the energy in this paper. The algorithm samples the energies of the whole energy space comprehensively while obtaining the density of states g(E(X)) for the range of possible energies. Because of the comprehensive sampling, we can find the approximate global optimal value of the energies, and this is the theoretical principle we apply the WL sampling algorithm to solve the DFLP. Before introducing the WL sampling algorithm to the DFLP, we first give a partition for the energies. Considering that the values of energies in range are positive real numbers in this paper, we divide all possible energies of the configurations into numerable intervals. For example, we divide [0, 1000] into 1000 energy intervals [0, 1), [1, 2), ... , [999, 1000), and the numbers that are bigger than 1000 will be divided into the single energy interval [1000, 1000+). The energy interval [E, E) is denoted by [E], where E rounds E down to its nearest integer, E rounds E up to its nearest integer. For example, E = 704.36 falls into energy interval [704, 705), which is denoted by [704.36]. The basic process of applying the WL sampling algorithm to the DFLP is as follows. Since at the beginning of

ACCEPTED MANUSCRIPT the algorithm all possible energies and the density of states g(E(X)) are unknown, we set respectively the density of states g(E(X)) and the corresponding histogram to 1 when accessing to a new energy. In the simulations on the DFLP, the new configurations can be generated by a heuristic configuration updating procedure (see section 3.2), but not all the new configurations can be accepted. The acceptance probability of a new configuration is proportional to the reciprocal of the density of states g(E(X)) associated with the new configuration. Assume that the original configuration is X1, and the new configuration is X2. The acceptance probability P from X1 to X2 is as follows:  g ( E ( X 1 ))  p( X1  X 2 )  min  , 1  min(eln( g ( E ( X1 ))) ln( g ( E ( X 2 ))) , 1) . g ( E ( X ))  2 

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If X2 is accepted, then g(E(X2)) will be multiplied by a modification factor fi, and the histogram function H([E(X2)]) will add 1, that is, g(E(X2)) = fi * g(E(X2)) (ln(g(E(X2))) = lnfi + ln(g(E(X2)))), H([E(X2)]) = H([E(X2)]) + 1 (the initial value of i is 0); otherwise, g(E(X1)) and H([E(X1)]) will be updated as follows: g(E(X1)) = fi * g(E(X1)) (ln(g(E(X1))) = lnfi + ln(g(E(X1)))), H([E(X1)]) = H([E(X1)]) + 1. Because the value of the density of states g(E(X)) will become so large that it will lead to the data overflow, we update g(E(X)) with the formula that contains logarithm. The initial value of the modification factor fi is set differently in different model. If the value of f0 is too small, it will take a long time to find all the possible energies; conversely, if the value of f0 is too large, there will be a statistical error. A value of e (2.7181...) is chosen for f0 in this paper. The flatness of the histogram determines the convergence of the WL sampling algorithm. In the simulations, we check the flatness of the histogram every 103 MC steps, where a MC step represents a new configuration generated by the algorithm. While the histogram is flat, the density of states g(E(X)) will converge to its true value by proportioning to the accuracy of the modification factor ln(fi). After this, we reduce the modification factor fi by a modified formula: fi+1 = f i ,

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and reset the values of all entries of H([E(X)]) to 0. Then we start a new round of iteration. However, in reality, it is hard to get the absolute flat histogram. The so-called “flat histogram” in the WL sampling algorithm refers to the values of all entries of H([E(X)]) are greater than the value that the average of all entries of histogram multiplies k (0
ACCEPTED MANUSCRIPT given first. The envelope rectangle is the rectangle that could envelop all facilities exactly. The vacant point is the point that is inside envelope rectangle, but not inside any facility. As shown in Fig. 1, the dotted rectangle represents an envelope rectangle, and points in the shaded region of the envelope rectangle can be selected as the vacant points. The vacant point strategy of updating the layout is descripted as follows. First, generate randomly 100 vacant points in the envelope rectangle. Then, choose the facility with the largest unit material handling cost (UMHC) and place temporarily its centroid at every vacant point with vertical and horizontal directions of facility. Here, the formula for calculating the unit material handling cost of facility i in stage t is as follows: UMHCti 

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Next, calculate the unit material handling cost of the chosen facility associating with each vacant point, and place formally the centroid of the chosen facility at the vacant point where the unit material handling cost of the chosen facility is the smallest. With the positions of other facilities unchanged, finally we gain two new layouts due to the two placing directions.

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Fig. 1. The envelope rectangle of a layout.

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3.2.2. Pushing strategy It is inadvisable that facilities overlap with each other after executing the vacant point strategy. To make sure that constraint (2) is satisfied, the pushing strategy is applied. In order to avoid repetitive movement, a principle of to-up or to-right is used in the pushing strategy, that is, the facility can only move to the up or to the right. Suppose facilities i and j overlap each other. As shown in Fig. 2, Atij labeled in Fig. 2(a)-(d) represents the overlapping distance in the direction of the X-axis between facilities i and j in stage t. Btij labeled in Fig. 2(a)-(d) represents the overlapping distance in the direction of the Y-axis between facilities i and j in stage t. Suppose we need to move facility j. The movement rules are as follows. If facilities i and j overlap each other as Fig. 2(a), we first check whether Atij is smaller than Btij or vice versa. If Atij is smaller, facility j will be moved to the right by Atij value. If Btij is smaller, facility j will be moved to the up by Btij value. If facilities i and j overlap as Fig. 2(b), and if Atij is smaller, facility j will be moved to the right by Atij value; otherwise, facility j will be moved to the up by Btij value. If facilities i and j overlap as Fig. 2(c) or Fig. 2(d) and Atij is smaller, facility j will be moved to the right by Atij value; otherwise, facility j will be moved to the up by Btij value. The process of the pushing strategy is as follows. First, choose the facility with the largest overlapping area. Then, move the chosen facility according to the movement rules. If the chosen facility overlaps with more than one facility, the facility which has the minimum index is the reference of the chosen facility. If there are other facilities that have the overlapping, this procedure is repeated. We use an example to describe the process of the pushing strategy. Suppose we have a layout as Fig. 3(a) in

ACCEPTED MANUSCRIPT stage t. Facility 2 overlaps with facilities 1 and 3 and has the largest overlapping area, so we choose facility 2 to move. Facility 1 has the minimum index between facilities 1 and 3, and the overlapping between facilities 2 and 1 is considered. Because Bt12 is smaller than At12, facility 2 is moved to the up. The result of this movement is shown in Fig. 3(b). As you can see, this movement leads to the overlapping between facilities 2 and 4. We continue this pushing procedure and solve the overlapping between facilities 2 and 4. The final layout is shown in Fig. 3(c). Y

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3.2.3. Pressuring strategy After using the pushing strategy to prevent overlapping among facilities, we may gain a layout where facilities are placed out of the plant floor. To make the layout compact, we apply the following pressuring strategy. First, calculate the material handling cost (MHC) of each facility in each stage by the formula: N

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Then, choose the facility with the largest material handling cost and name it the basic facility. Move the candidate facility towards the basic facility. Here, the candidate facility is the facility which is nearest to the basic facility. There are two ways of the movement: first vertically move and then horizontally move, and first horizontally move and then vertically move. We choose the way by which we could get more improvement in the material handling costs of the layout. In each way of the movement, we move the candidate facility until it touches another facility or its centroid is aligned with the centroid of the basic facility. If these two cases may happen, we choose the former in priority. If there are still other facilities that can be moved, this procedure is repeated. We use an example to describe the process of the pressuring strategy. Suppose we have a layout as Fig. 4(a), and facility 4 is the basic facility. It is obvious that facility 3 is nearest to facility 4, so we move facility 3 towards facility 4. No matter which way we choose for the movement, the movement will lead to a layout as Fig. 4(b). The termination condition of the movement is that the down edge of facility 3 touches the basic facility, and its right edge touches facility 2. The next facility nearest to the basic facility is facility 1. This facility can be first vertically moved until its up edge touches the basic facility, and then be horizontally moved with no obstruction until its centroid is aligned with the centroid of the basic facility (see Fig. 4(c)). Facility 2 is vertically moved until its down edge touches the basic facility. The final layout is shown in Fig. 4(d).

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Fig. 4. An example of pressuring strategy.

3.2.4. Computation of rearrangement costs We define the facility group as the combination of some facilities. If the centroids of all facilities in the facility group are unchanged relative to each other in two successive stages, we consider the relative positions among these facilities are unchanged. As long as the relative positions among facilities are unchanged in two successive stages, we think the all facilities which are contained in the facility group don‟t generate rearrangement costs. We use an example to describe this computational process of rearrangement costs. As shown in Fig. 5(a)-(b), because the positions of facilities 1, 2, 3 and 4 change in two successive stages, there are four rearrangement costs by expression (1). In fact, the relative positions between facility 3 and facility 4 are unchanged. We take facilities 3 and 4 as a facility group. If we move the entire layout upward shown in Fig. 5(b) until facilities 3 and 4 reach

ACCEPTED MANUSCRIPT the positions as same as the positions shown in Fig. 5(a), we can obtain the layout shown in Fig. 5(c). Obviously, Figs. 5 (b) and (c) are isomorphic. If the layout in Fig.5(c) is still a feasible layout which satisfies constraints (2)-(9), Fig. 5(c) will lead to only two rearrangement costs compared with Fig. 5(a). In the computational process of rearrangement costs, we always choose the maximum facility group that contains more facilities. This will reduce the rearrangement costs of the layout. Y

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Fig. 5. An example of computation of rearrangement costs in two successive stages.

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3.2.5. Configuration updating process We update a configuration by updating the layouts at different stages in the configuration. For the first stage, we apply the vacant point strategy on the original layout to get two temporary layouts where the picked facility has two different placing directions. Hereafter, we apply the pushing strategy on these two temporary layouts to eliminate the overlapping among facilities, and use the pressuring strategy to further get two compact layouts. Finally, we calculate respectively the material handling costs of these two compact layouts, and choose the layout with the smaller material handling cost as the updated layout. For other stages, first we copy the updated layout of the previous stage as the original layout of this stage. Then we apply the vacant point strategy on the original layout to get two temporary layouts, and apply the pushing strategy and the pressuring strategy on these two temporary layouts to get two compact layouts without overlapping. At this time, we calculate the material handling costs and the rearrangement costs of these two compact layouts, respectively. Here, the rearrangement costs are generated by comparing the newly produced compact layouts with the original layouts and computed by the strategy of section 3.2.4. The sums of the material handling costs and rearrangement costs are the total costs for these two compact layouts. We choose the smaller total cost and compare it with the material handling cost of the original layout. If the smaller total cost is less than the material handling cost of the original layout, we set the compact layout with the smaller total cost as the updated layout. Otherwise, repeat this updating process starting from the vacant point strategy. If this process is repeated more than 103 times and at every time we do not gain an accepted layout, we set the original layout as the updated layout. 3.3. Description of WL combining with heuristics By incorporating the heuristic configuration updating procedure into the WL sampling algorithm, a heuristic Wang-Landau (HWL) layout algorithm for the unequal-area DFLP is presented. The detailed computational procedure is outlined below. Step1: Generate an initial configuration X1 (where the layout for each stage is as same as that in Yang and Peters (1998)). Calculate E(X1). Let the optimal configuration Xmin = X1, and the total cost of optimal configuration Emin = E(X1). Initialize the set of intervals containing visited energies (i.e., total costs) S = {[E(X1)]}, the state density function g(E(X1)) = 1, and the histogram function H([E(X1)]) = 1. Let i = 0, l = 0, k = 0.7, f0 = e.

ACCEPTED MANUSCRIPT Step2: Update current configuration X1 by the heuristic configuration updating procedure, and gain a new configuration X2. Compute E(X2). Let g(E(X2)) = 1, H([E(X2)]) = 1, and l = l + 1. Step3: If [E(X2)]  S, let S = S ∪ {[E(X2)]}. Step4: If random (0, 1) < min(eln( g ( E ( X1 ))) ln( g ( E ( X 2 ))) , 1) , accept X2, and let X1 = X2, E(X1) = E(X2), ln(g(E(X2))) = lnfi + ln(g(E(X2))), H([E(X2)]) = H([E(X2)]) + 1, go to step5; otherwise, do not accept X2 and let ln(g(E(X1))) = lnfi + ln(g(E(X1))), H([E(X1)]) = H([E(X1)]) + 1, go to step6. Step5: If E(X2) < Emin, let Xmin = X2, Emin = E(X2). Step6: If l % 1000 = 0, go to step7; otherwise, go to step2.

Step8: Let fi+1 =

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Step7: If H([E(X)])  k for all visited energy intervals [E(X)] ∈ S, then go to step8; otherwise go to step2.

f i , i = i + 1.

Step9: If fi < 1.0001, then the algorithm stops, output Xmin and Emin; otherwise, for each [E(X)] in S, let H([E(X)])=0, but g(E(X)) remains unchanged, and go to step2.

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4. Computational results and discussion

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In order to evaluate the performance of the HWL sampling algorithm, we apply the algorithm on four groups of cases. The first group contains two classical dynamic facility layout cases: P6-6 and P12-4 (Yang and Peters, 1998). The second group contains two optimality comparison test cases created in this paper. The third group contains a dynamic facility layout case with practical application and the fourth group contains three single-stage facility layout cases. The algorithm is compiled by using JAVA language and all experiments are carried out on a PC with an Intel Core 2 Duo, 2.94 GHz CPU and 2.0 GB RAM. For each case, we run the HWL sampling algorithm ten times independently.

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4.1 Computational results of two classical dynamic facility layout cases

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The first classical dynamic facility layout case is to place 6 facilities in 6 stages (P6-6), and the second is to place 12 facilities in 4 stages (P12-4). These two cases were first presented in Yang and Peters (1998), and were further calculated in Dunker et al. (2005), McKendall and Hakobyan (2010), Jolai et al. (2012), and Asl and Wong (2015). Jolai et al. (2012) used 50 units and 200 units for rearrangement cost of each facility (the following brief write down for ReCost) on the first and second cases, respectively, as in Yang and Peters (1998); but it was 19 and 50 units for the first and second cases in Dunker et al. (2005), Mckendall and Hakobyan (2010), and Asl and Wong (2015). In this paper, in order to compare the results with the above calculations, we use not only 19 and 50

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units for P6-6, but also 50 and 200 units for P12-4. The 3030 and 5050 plant floors are chosen for P6-6 and P12-4, respectively, and the material handling cost per unit distance is set to 1. The computational results by the HWL sampling algorithm for case P6-6 are shown in Table 1. As shown in the table, the best objectives and running times obtained by the HWL are uniformly better than results by other four algorithms for both 19 and 50 ReCost. In fact, for ReCost of 19, the best objective obtained by the HWL is reduced by (6569.0 − 6523.5) / 6569.0 = 0.69% compared with that by the hybrid genetic algorithm (HGA) in Dunker et al. (2005), reduced by (6648.3 − 6523.5) / 6648.3 = 1.88% compared with that by the tabu search and boundary search heuristic (TS/BSH) in McKendall and Hakobyan (2010), and reduced by (6763.7 – 6523.5) / 6763.7 = 3.55% compared with that by the modified particle swarm optimization (MPSO) in Asl and Wong (2015). The layouts for objective 6523.5 are shown in Fig.6. Facility 2 is replaced in the beginning of stage 4, and others are consistent with the original, which leads to one rearrangement in the layouts. For ReCost of 50, the best objective obtained by the HWL is reduced by (6659.4 − 6597.0) / 6659.4 = 0.94% compared with that by the

ACCEPTED MANUSCRIPT multi-objective particle swarm optimization (MOPSO) in Jolai et al. (2012). The layouts for objective 6597.0 are shown in Fig. 7. These 6 layouts keep the same positions in different stages, which means there are no rearrangement costs. The statistics of the computing results for case P6-6 are shown in Table 2. Table 2 shows the best and worst objectives, the average objectives, and the standard deviations of these objectives over 10 independent runs. The standard deviation is calculated by the following formula: n

a i 1

i

 a

2

(13) n Here, ai represents the objective by the HWL sampling algorithm in the ith run; a represents the average of these

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objectives and n is the running times. For ReCost of 19, the deviation between the best and worst objectives is 96 and the standard deviation is small. For ReCost of 50, the deviation between the best and worst objectives is 86 and the standard deviation is small too. This means the objectives obtained by the HWL sampling algorithm are not scattered for both ReCosts of 19 and 50.

Fig. 6. P6-6 layouts by the HWL for stages 1-6(from left to right, from top to bottom) with objective 6523.5.

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Fig. 7. P6-6 layouts by the HWL for stages 1-6(from left to right, from top to bottom) with objective 6597.0.

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Table 1 Comparison of computational results by five algorithms for case P6-6. TS/BSH (2010)

ReCost

Cost

Time(min)

Cost

Time(min)

19

6569.0

29.4

6648.3

27.7

50

―

―

―

―

MOPSO (2012)

MPSO (2015)

HWL (in this paper)

Cost

Time(min)

Cost

Time(min)

Cost

Time(min)

―

―

6763.7

39.2

6523.5

4.6

6659.4

8.4

―

―

6597.0

5.2

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HGA (2005)

Table 2

Statistics of results by the HWL sampling algorithm for case P6-6. Best objective

Worst objective

Average objective

19

6523.5

6619.5

6582.1

26.1

50

6597.0

6683.0

6648.3

29.6

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ReCost

Standard deviation

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The computational results by the HWL sampling algorithm for case P12-4 are shown in Table 3. From the table, the objective obtained by the HWL is a little worse than that by TS/BSH for ReCost of 50. But the objectives are better than those by HGA, MPSO and MOPSO for ReCosts of 50 and 200, respectively. In fact, for ReCost of 50, the best objective obtained by the HWL is increased by (26950.5 − 26845.5) / 26950.5 = 0.39% compared with that by TS/BSH, but reduced by (27748.0 − 26950.5) / 27748.0 = 2.88% compared with that by HGA and (28826.6 – 26950.5) / 28826.6 = 6.51% compared with that by MPSO. In terms of computation time, the HWL overmatches other algorithms for ReCost of 50. The layouts for objective 26950.5 are shown in Fig. 8. Ten facilities are replaced in the beginning of stage 2 except facilities 10 and 11. Besides, positions of all facilities are kept invariant in other stages. There are 10 rearrangements in all layouts. For ReCost of 200, the best objective obtained by the HWL is reduced by (28115.5 − 27924.5) / 28115.5 = 0.68% compared with that by MOPSO, while the computational time of the former is a litter longer than that of the latter. The layouts for objective 27924.5 are shown in Fig. 9. There is none rearrangement in all layouts, which is similar to the layouts of objective 6597.0 for case P6-6. The statistics of the computing results for case P12-4 are shown in Table 4. Table 4 shows the best and worst objectives, the average objectives, and the standard deviations of these objectives over 10 independent runs. For ReCost of 50, the deviation between the best and worst objectives is 630.5, and the standard deviation is small.

ACCEPTED MANUSCRIPT This means the objectives obtained by the HWL are not scattered. For ReCost of 200, the deviation between the best and worst objectives is 396.5, which is much smaller than that for ReCost of 50. The standard deviation is small too, it means the objectives obtained by the HWL are not scattered as the former. Table 3 Comparison of computational results by five algorithms for case P12-4. TS/BSH (2010)

MOPSO (2012)

ReCost

Cost

50

27748.0 160.0

26845.5

82.0

―

200

―

―

―

28115.5 50.5

Time(min) Cost

―

Time(min) Cost

MPSO (2015)

HWL (in this paper)

Time(min)

Cost

―

28826.6 185.1

26950.5 57.8

―

27924.5 63.4

Table 4 Statistics of results by the HWL sampling algorithm for case P12-4.

Time(min) Cost

―

Time(min)

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HGA (2005)

Best objective

Worst objective

Average objective

Standard Deviation

50

26950.5

27581.0

27374.4

176.1

200

27924.5

28321.0

28057.5

93.2

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ReCost

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Fig. 8. P12-4 layouts by the HWL for stages 1-4(from left to right, from top to bottom) with objective 26950.5.

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Fig. 9. P12-4 layouts by the HWL for stages 1-4(from left to right, from top to bottom) with objective 27924.5.

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4.2 Optimality comparison test

Two optimality comparison test cases are created to illustrate the deviations between the results by the HWL and the optimal results. The first case, which places 3 facilities in 3 stages, is denoted as P3-3. The second case,

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which places 4 facilities in 2 stages, is denoted as P4-2. For P3-3, we define a 129 plant floor, i.e., length of the plant floor along the X-axis is 12 and width of the plant floor along the Y-axis is 9. The ReCost is 15 units and the dimensions of facilities and the material flows among facilities in each stage are presented in Appendix 1. For

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P4-2, a 1011 plant floor is defined, and the ReCost is 9 units. The dimensions of facilities and the material flows among facilities in each stage are presented in Appendix 2. For P3-3, we can construct an optimal configuration by referring equations 1 to 9. The objective of this optimal configuration is 1510.5 and the corresponding layouts are shown in Fig. 10, where two facilities are reset in the beginning of stage 2 except facility 3, and positions of all facilities in stage 3 are consistent with those in stage 2. This case is also solved by the proposed HWL sampling algorithm. Over 10 independent runs, the best objective obtained by the HWL is equal to that of the optimal configuration by referring equations 1 to 9, and the best layouts by the HWL are the same as layouts of the optimal configuration, which means the HWL sampling algorithm can find the optimal configuration for case P3-3. For P4-2, we can also construct an optimal configuration by referring equations 1 to 9. The objective of the optimal configuration is 827.5 and the corresponding layouts are shown in Fig. 11. Furthermore, we use the proposed HWL sampling algorithm to solve case P4-2. By generating a legitimate initial layout, the best layouts with the objective 848.5 are obtained by the HWL over 10 independent runs. The best layouts by the HWL are shown in Fig. 12. The deviation between the best result by the HWL and the objective of the optimal layouts is 21. We can see from Fig. 11, in the beginning of stage 2 we reset facility 4 so that its centroid is aligned with the centroid of facility 1, and an optimal configuration (Fig. 11) is obtained. However, it is hard for the HWL to search this optimal configuration. Note that in the beginning of stage 2, as facility 3 has the largest UMHC according to equation 11, the centroid of facility 3 is reset within vacant region and thereafter the pushing strategy and the pressuring strategy are executed. Obviously, it could not yield the optimal layout in stage 2 as shown in Fig. 11, so the layout in stage 1 remains invariant according to the configuration updating process described in Section 3.2.5. But the configuration by the HWL is sub-optimal, demonstrating the effectiveness of the HWL in solving hard case for the unequal-area DFLP.

Fig. 10. P3-3 optimal layouts for stages 1-3 (from left to right) with objective 1510.5.

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Fig. 11. P4-2 optimal layouts for stages 1-2 (from left to right) with objective 827.5.

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Fig. 12. P4-2 layouts by the HWL for stages 1-2 (from left to right) with objective 848.5.

4.3 Computational results for a dynamic facility layout case with practical application We give a dynamic facility layout case with practical application. There is a plant which is mainly responsible

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for production of automotive battery start-up line. The dimension of the plant is 2422 square meters. In accordance with the principle of similar facilities placed together, this plant is divided into 8 regions (i.e., facilities). The names and dimensions of all regions are presented in Appendix 3. The production is divided into three stages, and plans of each stage are as follows: product 4300 class-A batteries, 1000 class-B batteries, 2400 class-C batteries in the first stage; product 1600 class-A batteries, 0 class-B batteries, 5500 class-C batteries in the second stage; product 200 class-A batteries, 5800 class-B batteries, 500 class-C batteries in the third stage. The material flows among all regions and the matrix of material handing cost per unit distance at each stage are summarized in Appendix 3. It costs 500 dollars if one region is reset. The sum of costs by the HWL for this case is 236821.5 dollars, and the output layouts are shown in Fig. 13, where positions of all regions in the second stage are consistent with the original and seven regions are reset except region 1 in the third stage, which leads to 7 rearrangements in the layouts. To test the difference of the effectiveness of the HWL on the DFLP relative to the SFLP, we amount the material flows of three stages in the plant to the material flows of single stage, that is, we convert the DFLP into a SFLP which is a special DFLP with one stage. The sum of costs by executing the HWL for the SFLP is 257622.0 dollars. The former is reduced by (257622.0 – 236821.5) / 257622.0 = 8.1% in product cost compared with the latter, which shows the significance of the DFLP. The optimal configuration by the HWL over ten independent runs for the SFLP is shown in Fig. 14.

Fig. 13. Layouts by the HWL for stages 1-3 (from left to right) with objective 236821.5 in dynamic case with practical application.

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Fig. 14. Layout by the HWL with objective 257622.0 in single-stage case with practical application.

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4.4 Computational results for three single-stage facility layout cases To check further the effectiveness of the HWL, the computational results for three single-stage facility layout (i.e., SFLP) cases are presented in this subsection. These three cases which include 8 facilities, 11 facilities, and 20 facilities are denoted as P8, P11, P20, respectively (Asl and Wong (2015)). The material handling cost per unit

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distance is set to 1 for each of these cases. The dimensions of the plant floors are chosen as 1212, 1515, and 1414, respectively. The computational results by the HWL sampling algorithm for the three single-stage cases are shown in Table 5. The objectives obtained by the HWL are better than the results by the modified particle swarm optimization (MPSO) in Asl and Wong (2015) for both P8 and P11. In fact, it decreases (208.7 - 193.1) / 208.7 = 7.5% for P8 and (1335.6 - 1280.1) / 1335.6 = 4.2% for P11 compared with those by MPSO. The improvements obtained by the HWL for P20 are (1333.8 - 1264.6) / 1333.8 = 5.2% and (1287.3 - 1264.6) / 1287.3 = 1.8%, in contrast to FLOAT in Imam and Mir (1993) and HOT in Mir and Imam (2001), respectively. It only increases (1264.6 - 1264.2) / 1264.6 = 0.03% for P20 compared with that by MPSO. The optimal layouts by the HWL over 10 independent runs for P8, P11, and P20 are shown in Fig. 15, Fig. 16, and Fig. 17, respectively. The statistics of the computing results are shown in Table 6. In the table, the best and worst objectives, the average objectives, and the standard deviations over 10 runs are presented. For all the three single-stage cases, the standard deviations are small indicating that the objective values obtained by the HWL are not scattered. Table 5

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Comparison of computational results by four algorithms for three single-stage cases. FLOAT (1993)

HOT (2001)

MPSO (2015)

HWL (in this paper)

P8

―

―

208.7

193.1

P11

―

―

1335.6

1280.1

P20

1333.8

1287.3

1264.2

1264.6

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Table 6

Statistics of results by the HWL sampling algorithm for three single-stage cases. Case

Best objective

Worst objective

Average objective

Standard deviation

P8

193.1

198.0

195.9

1.3

P11

1280.1

1302.4

1288.6

6.1

P20

1264.6

1313.8

1289.7

16.9

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Fig. 16. P11 layout by the HWL with objective 1280.1.

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Fig. 15. P8 layout by the HWL with objective 193.1.

Fig. 17. P20 layout by the HWL with objective 1264.6.

5. Conclusions

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In this paper, we have discussed the dynamic facility layout problem (DFLP) with unequal-area facilities which aims at minimizing the sum of the material handling and rearrangement costs. Firstly, we present a mathematical model to describe this problem. Then we combine the Wang-Landau (WL) sampling algorithm and some heuristics to solve the problem. The WL sampling algorithm obtains a flat energy histogram by walking in the energy space randomly, which can estimate the density of states of the system accurately. It will reach the near lowest energy when estimating the density of states. Four groups of cases are used to test the proposed algorithm, and the results indicate the effectiveness of the algorithm. Furthermore, the proposed method will be beneficial for the DFLP with variable shapes and areas of facilities throughout the time horizon, and can be generalized to the research and application on multi-objective group decision-making.

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Acknowledgments

This work is supported by the National Natural Science Foundations of China (Grant Nos. 61373016, 61472147, 61373064, 61403206), the Major Program of the National Social Science Foundation of China (Grant No. 16ZDA047) and the Natural Science Foundation of Jiangsu Province (Grant No. BK20141005). Appendix 1. Parameters of optimality comparison test case P3-3 Dimensions of facilities in P3-3 Facility

1

2

3

Dimension(Li, Wi)

(6, 4)

(6, 4)

(9, 6)

Material flows of stage 1 for P3-3

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1

2

3

1

0

50

1

2

50

0

40

3

1

40

0

Facility

1

2

3

1

0

50

40

2

50

0

1

3

40

1

0

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Material flows of stage 2 for P3-3

Facility

1

2

3

1

0

20

50

2

20

0

1

3

50

1

0

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Material flows of stage 3 for P3-3

Appendix 2. Parameters of optimality comparison test case P4-2 Dimensions of facilities in P4-2. 1

2

3

Dimension(Li, Wi)

(8, 3)

(11, 4)

(6, 3)

Material flows of stage 1 for P4-2. 1

2

3

4

1

0

10

1

1

2

10

0

10

3

1

10

0

4

1

10

10

(7, 3)

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Material flows of stage 2 for P4-2.

1 2

2

3

4

0

1

10

80

1

0

10

10

10

10

0

10

80

10

10

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Appendix 3. Parameters of the case with realistic implementation Names and dimensions of machine regions No.

Name

Dimension(mm)

No.

Name

Dimension(mm)

1

Raw material region

4.04.0

5

Tin solder region

3.22.5

2

Clip assemble region

3.62.0

6

Punching region

3.83.6

3

Coil wire region

3.02.5

7

Testing region

4.03.5

4

Clamping region

4.84.0

8

Packaging region

4.03.6

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Material flows among machine regions Direction of

Transporting material / item Stage 1

Stage 2

Stage 3

1-3

4300

1600

200

1-4

1000

0

5800

2-4

2400

5500

500

3-4

4300

1600

200

3-5

0

0

700

3-7

3400

7100

6300

4-5

7700

7100

6500

4-6

4300

1600

5-4

3400

5500

5-6

4300

1600

6-3

3400

5500

6-7

4300

1600

7-8

7700

7100

2

3

1

0

0.5

0.2

2

0.5

0

0.6

3

0.2

0.6

0

4

0.3

0.5

0.4

5

0.3

0.5

0.3

6

0.5

0.4

0.2

7

0.3

0.3

8

0.4

0.4

200

6300 200

6500

4

5

6

7

8

0.3

0.3

0.5

0.3

0.4

0.5

0.5

0.4

0.3

0.4

0.4

0.3

0.2

0.6

0.2

0

0.5

0.4

0.2

0.3

0.5

0

0.5

0.5

0.6

0.4

0.5

0

0.5

0.3

0.6

0.2

0.2

0.5

0

0.5

0.2

0.3

0.6

0.3

0.5

0

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Matrix of material handing cost per unit distance (Cij).

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