An ant colony algorithm for solving budget constrained and unconstrained dynamic facility layout problems

Omega 34 (2006) 385 – 396 www.elsevier.com/locate/omega

An ant colony algorithm for solving budget constrained and unconstrained dynamic facility layout problems夡 Adil Baykasoglu∗ , Turkay Dereli, Ibrahim Sabuncu Department of Industrial Engineering, University of Gaziantep, 27310 Gaziantep, Turkey Received 28 October 2003; accepted 10 December 2004 Available online 3 February 2005

Abstract The main characteristic of today’s manufacturing environments is volatility. Under a volatile environment, demand is not stable. It changes from one production period to another. To operate efficiently under such environments, the facilities must be adaptive to changing production requirements. From a layout point of view, this situation requires the solution of the dynamic layout problem (DLP). DLP is a computationally complex combinatorial optimization problem for which optimal solutions can only be found for small size problems. It is known that classical optimization procedures are not adequate for this problem. Therefore, several heuristics including taboo search, simulated annealing and genetic algorithm are applied to this problem to find a good solution. This work makes use of the ant colony optimization (ACO) algorithm to solve the DLP by considering the budget constraints. The paper makes the first attempt to show how the ACO can be applied to DLP with the budget constraints. In the paper, example applications are presented and computational experiments are performed to present suitability of the ACO to solve the DLP problems. Promising results are obtained from the solution of several test problems. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Dynamic facility layout; Ant colony optimization; Heuristics

1. Introduction Facility layout studies usually result from the changes that occur in the requirements for space, equipment and people. If requirements change frequently, then it is desirable to plan for change and to develop a flexible layout that can be modified, expanded, or reduced easily [1]. Flexibility can be achieved by utilizing modular equipment, general-purpose production equipment and material handling devices, etc. The change in the design of existing products, the processing sequences for existing products, quantities of production 夡 This

article was processed by Associate Editor E. Pesch. author. Tel.: +90 342 3601200x2601; fax: +90 342 3604383. E-mail addresses: [email protected] (A. Baykasoglu), [email protected] (T. Dereli), [email protected] (I. Sabuncu). ∗ Corresponding

0305-0483/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.omega.2004.12.001

and associated schedules, and the structure of organization and/or management philosophies (e.g. centralized, decentralized, hierarchical, etc.) can lead to changes in layout. When these changes occur frequently, it is important for the layout to accommodate them [2]. Impact of change on the design of the facility pointed to the need for a facility that can respond to change. An important part of the response to change is the need to rearrange workstations or modify the system structure based on changing functions, volumes, technology, product mix and so on. The dynamic layout problem (DLP) arises when the location of an existing facility is a decision variable. With the introduction of new parts and changed demands, new locations for the facilities might be necessary in order to reduce excessive material handling costs. Gupta and Seifoddini [3] concluded that one-third of USA companies undergo major dislocation of production facilities every 2 years.

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Over time the mix of parts, the volume of production for each part and the routing for each part in the system is generally subjected to change in a dynamic production environment. If everything remains constant for long period of time, a dedicated set of facilities would be more appropriate, but where this is not the case, there is a need to focus on a relayout issue. The objective function in a DLP is generally defined as the minimization of flow costs plus rearrangement cost for a series of static layout problems [4–6]. In a DLP, rearrangement costs are added whenever an area contains different departments in consecutive time periods. According to Lacksonen and Enscore [6], the DLP is required when we must balance the trade-off between increased flow cost of inefficient layouts and added rearrangement costs. Afentakis et al. [7] stated that when system characteristics change, it can cause a significant increase in material handling requirements; consequently, it shows a need to consider re-layout. They defined two cost components for relayout, i.e. cost of reconfiguration or relocation of equipment and cost of lost production. The cost of reconfiguration depends on the number of machines moved and/or the number of links in material handling changed. Traditionally, the effectiveness of layout problems has been connected to the flow of materials. Material handling cost is commonly used to evaluate alternative layout designs. The relative location of facilities in a functional layout has been determined under the criterion of material handling cost minimization. Usually, the material handling cost is assumed to be an incremental linear function of the distances between the components of the system under study. Total estimated annual material handling cost for a particular design is used to provide a quantitative measure of the flexibility of design [2]. There is a massive amount of literature available about facility layout problem [8]. But, the research effort is generally on the static facility layout problems. In recent years, research has also focused on the dynamic case. The work done by Rosenbaltt [5] has generally been accepted as the first serious approach to model and solve DLP. He developed an optimal solution methodology for DLP using a dynamic programming approach. The stages of the dynamic programming problem correspond to the periods in the planning horizon and the states correspond to specific layout arrangements. The main problem with his model is the determination of alternative layouts (states) to use in each stage. Lacksonen and Enscore [6] also studied the DLP. They modelled the problem as a modified quadratic assignment problem. Their model can be considered as a general quadratic assignment formulation of the basic DLP. They modified various static layout algorithms to solve the dynamic version of the quadratic assignment model. Lacksonen [9,10] also extended his DLP model by considering departments with unequal areas. He applied branch-and-bound routine and the cut tree algorithm to solve the mixed integer linear programming model. Urban [11] proposed a heuristic algorithm that is based on the CRAFT (steepest descent pair-wise interchange) procedure

for DLP. Conway and Venkataramanan [12] applied genetic algorithms to solve DLP. Balakrishnan and Cheng [13] also proposed a genetic algorithm for DLP. Kaku and Mazzola [14] developed a taboo search-based heuristic for the DLP. The application of simulated annealing to the DLP is shown by Baykasoglu and Gindy [2]. Erel et al. [15] also purposed several heuristics for the dynamic layout problem by using dynamic programming and simulated annealing. They plan to arrive at the optimal sequenced of layouts by implicitly enumerating over subset of all possible layouts. Given all possible layouts, they claim that the DLP can be viewed as a shortest path problem. A good survey on the DLP is published by Balakrishnan and Cheng [16] that explains the state of the research on DLP. They gave detailed explanations about some of the available algorithms on DLP alongwith their comparisons. Good heuristic techniques are necessary for solving DLP due to its high computational complexity (i.e. for an N location T period problem, (N!)T solutions are possible). Modern heuristic techniques, namely genetic algorithms taboo search, simulated annealing and ant colony optimization, can be good candidates for this problem. As discussed in the previous paragraphs, the applications of genetic algorithms, taboo search and simulated annealing to the DLP has been shown in the literature. However, we should mention here that in all of these applications budget constraints are not taken into account. It is also possible to take into account budget constraints in other modern heuristics by using some forms of penalty functions or not allowing unfeasible moves during the search, etc. This might be considered as a future work in re-implementing these algorithms for DLP. But it is a well-known fact that in some production periods due to budget limitations the reconfiguration may not be possible. Balakrishnan et al. [17] considered budget constraints in their studies of DLP and presented how this constraint can be taken into account. They add the constraint of a budget for total rearrangement costs over the entire horizon and presented a solution procedure that is based on constrained shortest path algorithms. In this research, ant colony optimization (ACO) heuristic is used for solving the unconstrained and budget-constrained DLP. The main research contribution of the present paper is to make the first attempt in the published literature to show how the ACO algorithm can be applied to DLP with the budget constraints. In the following sections of this paper, the ant colony heuristic algorithm for DLP is explained then the computational results are reported.

2. The problem statement and the mathematical program for the budget constrained DLP The DLP problem extends the well-known static layout problem where a group of departments are arranged into a layout such that the sum of the costs of flow between

A. Baykasoglu et al. / Omega 34 (2006) 385 – 396

departments is minimized under the assumption of material flows between departments is constant over time. The dynamic layout approach removes the above assumption. The DLP involves selecting a static layout for each period and then deciding whether to change a new layout in the following period. If the shifting costs are low, the layout configuration would tend to change more often to retain material handling efficiency. The opposite is true for high shifting costs [16]. However, in some periods although the reconfiguration is beneficial there might be budget limitations to carry out reconfiguration. Therefore budget limitations should be taken into consideration in solving DLP. The present paper tries to solve the DLP with the budget constraints as described in the previous section. The budgetconstrained DLP can be formally stated as follows [11,14]: p
N
N N



Min

+

Atij l Ytij l t=2 i=1 j =1 l=1 p
N
N
N N

t=2 i=1 j =1 k=1 l=1

Ctij kl Xtij Xtkl

(1)

s.t. N
i=1 N
j =1

Xtij = 1,

j = 1, . . . , N, t = 1, . . . , P ,

(2)

Xtij = 1,

i = 1, . . . , N, t = 1, . . . , P ,

(3)

Ytij l = X(t−1)il + Xtil − 2X(t−1)ij Xtil , N
N N

i=1 j =1 l=1

Atij l Ytij l
LBt−1 + Bt ,

(4)

t = 1, . . . , P , (5)

where i, k is the departments in the layout, j, l the locations in the layout, Ytij l the 0,1 variable for shifting i from j to l in period t, Atij l the fixed cost of shifting i from j to l in period t, Xtij the 0,1 variable for locating i at j in period t, Ctij kl the cost of material flow between i located at j and k located at l in period t, LB the left over budget and B the budget. The objective function minimizes the total of the cost of layout rearrangements and the cost of material flow between departments during the planning horizon. Constraint set (2) requires every department assigned to a location in every period and (3) requires every location to have a department assigned to it in every period. Constraint (4) adds shifting costs to the material flow cost if a department is shifted between locations in a period. Constraint (5) prevents the shifting of facilities until budget has accumulated the necessary resources.

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3. The ant colony heuristic for the budget constrained DLP Ant colony algorithms are becoming popular approaches for solving combinatorial optimization problems in the literature. They were first introduced by Dorigo et al. [18]. Several refined versions have been proposed [19–21] to improve its performance. For a comprehensive review on ant algorithms, refer to Dorigo et al. [22,23]. The ant system has been applied to the job shop scheduling problem by Colorni et al. [24], to the graph colouring problem by Costa and Hertz [25], to the quadratic assignment problem by Maniezzo [26] and to the vehicle routing problem by Bullnheimer et al. [21], and Dereli et al. [27]. The fundamental idea of ant heuristics is based on the behaviour of natural ants that succeed in finding the shortest paths from their nest to food sources by communicating via a collective memory that consists of pheromone trails. Due to ant’s weak global perception of its environment, an ant moves essentially at random when no pheromone is available. However, it tends to follow a path with a high pheromone level when many ants move in a common area, which leads to an autocatalytic process. Finally, the ant does not choose its direction based on the level of pheromone exclusively, but also takes the proximity of the nest and of the food source, respectively, into account. This allows the discovery of new and potentially shorter paths. This process is depicted in Fig. 1. The behavioural mechanism as explained above can be used to solve combinatorial optimization problems by simulation, with artificial ants searching the solution space instead of real ants searching their environment. In addition to this, the objective values correspond to the quality of the discovered food and an adaptive memory is the equivalent of the pheromone trails [18]. To guide their search through the set of feasible solutions, the artificial ants are furthermore equipped with a local heuristic function. The ant colony heuristic approach, like, genetic algorithms and simulated annealing, is attractive since its optimization scheme is based on natural metaphors. But we should mention here that for successful application of the ant algorithms to the optimization problems, it should be easy and/or possible to define (or present) the solution space as a network (or graph). Luckily solution spaces of most of the combinatorial optimization problems especially the ones with binary decision variables like the DLP can be described (and/or mapped) graphically. In Fig. 2, a graphical example is given to show how a two periods three departments dynamic layout problem’s solution space can be shown on a network. In the network only one ant (i.e. one feasible solution) is presented. The ant travelled through locations 2–1–3 to locate departments 1–2–3 in the first demand period and travelled through locations 3–2–1 to locate departments 1–2–3 in the second demand period. The solution string and the corresponding mapping are therefore: 2–1–3–3–2–1 and 1–2–3-1–2–3. The solutions are represented as strings in

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FOOD

FOOD

3.1. Notation used in the ant colony heuristic N M S i, z j t m r F Lit BLit

NEST

(a)

NEST

(b)

Pij t

FOOD

FOOD

tf cp(i) kp(I ) Q R W


NEST

(c)

NEST

(d)

Fig. 1. Dynamics of pheromone trail: (a) initial state, (b) time 1, (c) time 2, (d) final state: ants converge to the shortest path.

number of departments and locations ant colony size (number of ants) number of periods departments in the layout (i, z = 1, . . . , N ) locations in the layout (j = 1, . . . , N ) periods (t = 1, . . . , S) ants (m = 1, . . . , M) counter for the unsuccessful iteration number objective function value location of i in period t (Lit=j ) in the best solution location of i in period t (BLit=j ) pheromone quantity (desirability of assigning department i to location j in period t) total pheromone quantity chosen possibility of the department i cumulative chosen probability of the department i the parameter that is used when updating the pheromone unsuccessful iteration limit swapping number. Every ant perform the swapping procedure for W times parameter that is used when updating the pheromone (0 <
< 1) this parameter denotes that the departments without any pheromone can be chosen with a probability.

3.2. The procedure of the ant algorithm for the budget constrained DLP The ACO heuristic is adapted to the budget constrained DLP as described below:

Period-1

Departments 1 2 3

1

2

2

2

1

• Initial solution: Firstly, an initial solution is generated. A good initial solution is desired in order to converge to a solution in shorter computational time. The steps for generating an initial solution is as follows:

3

1

3

3

3

2

1

3

3

1

1

2

Ant-1 the route in the 1st demand period

Period-2

2

1

3

2

2

3

1

Ant-1 the route in the 2nd demand period

3 2

1

Fig. 2. Network representation of DLP solution space.

the computer implementation. But they can be visualized as trees for better understanding as shown in Fig. 2. In the next sub-sections, we will present the application of the ant algorithm to the DLP.

◦ Repeat the following steps for every ant (M times). Start with first ant (m = 1) and Fbest = a big number. ◦ Step 1: Find a random solution for the first period. For every department, select a location randomly (all locations has the same probability for selection) and assign the departments to that locations. Use this arrangement for every period so the rearrangement cost zero (Li1 = Li2 = · · · = Lis , i = 1, . . . , N ). ◦ Step 2: Improve on this solution using the improvement procedure. Repeat the improvement procedure until objective function cannot be improved after R iterations. – The improvement procedure: Randomly select a period, and two different departments in that period

A. Baykasoglu et al. / Omega 34 (2006) 385 – 396

for swapping. Before swapping the departments calculate the objective function (F1 ). After swapping of the locations of the departments calculate the objective function (F2 ) again and check if the budget constraints are satisfied. If a better solution has been found (F1 > F2 ) and budget constraints are satisfied then accept the swap and set r = 0, else if (F1 < F2 ) or budget constraints are not satisfied try another swapping and set the unsuccessful iteration counter r = r + 1. ◦ Step 3: Update the best solution found. If the objective function of the mth ant’s solution (Fm ) is better than the best solution (Fbest ), then update the best solution (Fbest = Fm ; BLit = Lit , i = 1, . . . , N, t = 1, . . . , S), else ignore mth ant’s solution. ◦ Step 4: Clear the decision variables (Lit = 0). Because, these decision variables will be used by the next ants. Update the ant number (m = m + 1) and go to Step 1. After completing the improvement procedure for every ant, the best ant’s initial solution is recorded by updating the best solution (BLit = Lit , i = 1, . . . , N, t = 1 . . . , S) in each iteration. Before continuing to the next section (Starting the ACO procedure) deposit pheromone for the best initial solution (j = Lit , Pij t = Pij t + Q/Fbest , i = 1 . . . , N, t = 1 . . . , S). (Q is a heuristic parameter, in the test problems Q is set as the target objective function value of the problem. Target objective value can be taken as lower bound, if this is not known a value that is lower than the best known solution can be set as target.) • ACO procedure: Repeat this procedure until the iteration number reaches to the iteration limit or the best solution’s objective function value is equal to or smaller than the target objective function (Fbest
Ftarget ). Starting parameters: (m = 1, r = 0). This steps of the procedure are as follows: ◦ Step 1: At the beginning of the iteration, update initial solution (Lit = BLit , i = 1, . . . , N, t = 1, . . . , S). At the first iteration use the solution that is obtained previously. ◦ Step 2: Start the ant solution procedure. Repeat this procedure until ant number reaches the ant colony limit or the best solution’s objective function value is equal to or smaller than the target objective function (Fbest
Ftarget ). Ant solution sub procedure is given below: – Step 2.1: Update the initial solutions for mth ant as the best solution (Lit = BLit , i = 1, . . . , N, t = 1, . . . , S).

389

– Step 2.2: Start the swapping procedure. In this procedure a new solution is generated randomly according to the pheromone. Repeat the swapping procedure W times. • Step 2.2.1: First, choose a period randomly (all periods has same selection probability). Then choose a department (u) randomly (all departments has same selection probability). • Step 2.2.2: Choose department (v) such that the selection probability of the department is proportional with the related pheromone values (Pu,lc(v),t + Pv,lc(u),t ). This issue will be described in detail at the following section. The second department (v) can be same as the first chosen department (u). This makes possible not to change the location of the department. • Step 2.2.3: Lastly swap the departments and update the locations of the departments (first location = Lut , second location = Lvt . After swapping Lut = second location, Lvt = first location). – Step 2.3: Improve mth ant’s solution. Improvement procedure is the same as in the initial solution procedure. – Step 2.4: Update the pheromone. First evaporate the pheromone (Pij t =Pij t ∗
, i=1 . . . , N, j =1, . . . , N, t = 1 . . . , S). Then deposit the pheromone for mth ant’s improved solution (j = Lit , Pij t = Pij t + Q/Fbest ∗
, i = 1 . . . , N, t = 1 . . . , S). – Step 2.5: Update the best solution found. If the objective function of the mth ant (Fm ) is better than the best solution (Fm < Fbest ), update the best solution (Fbest =Fm ; BLit =Lit , i=1, . . . , N, t =1 . . . , S) so the next ant can use this solution as the initial solution. Update pheromone again. Again evaporate the pheromone (Pij t =Pij t ∗
, i=1, . . . , N, j =1 . . . , N, t = 1, . . . , S). Deposit pheromone for mth ant’s solution but this time deposit more pheromone than the previous step (j = Lit , Pij t = Pij t + Q/Fbest , i = 1 . . . , N, t = 1 . . . , S). By cancelling out the
parameter from the formula the pheromone can be increased more. In this way the selection probability of the better solutions can be increased. Clear the unsuccessful iteration counter (r = 0). – Step 2.6: Update ant number (m = m + 1) and go to Step 2.1. ◦ Step 3: If the best solution cannot be improved increase the unsuccessful iteration counter (r = r + 1). If r = R then clear all pheromones (Pij t = 0, i = 1 . . . , N, j = 1, . . . , N, t = 1, . . . , S) and deposit the pheromone for the best solution (j = BLit , Pij t = Pij t + Q/Fbest , i = 1 . . . , N, t = 1 . . . , S). ◦ Step 4: Increase the iteration number and go to Step 1 for the next iteration.

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A. Baykasoglu et al. / Omega 34 (2006) 385 – 396

3.3. Random selection procedure according to the pheromone in ant colony procedure During the swapping procedure, the first chosen department denoted with u. This procedure is used to find the second department (v) for swapping. Determine all departments to be chosen with probability according to pheromone quantities. tf =

N
i=1

Pu,lc(i),t + pi,lc(u),t ,

u = [1/(tf + N)],

(6) (7)

cp(i) = [(pu,lc(i),t + pi,lc(u),t ) + ]u, ⎞ ⎛ n
cp(i) = 1⎠ . i = 1, . . . , N and ⎝

(8)

i=1

Choose randomly a department, according to the selection probability. kp(i) = kp(i − 1) + cp(i), i = 1, . . . , N (kp(0) = 0, kp(N) = 1).

(9)

Randomly generate q ∈ (0, 1). Choose the ith department whose cumulative probability satisfies the following inequality: kp(i − 1)
q < kp(i).

(10)

Denote the selected department with v.

4. Computational study The proposed ant colony algorithm was tested on a Pentium-III model PC at 1400 MHz (128 MB RAM). An extensive computational work is carried out with the data set obtained from Balakrishnan and Cheng [13], which contains problems for 6, 15, 30 departments for 5 and 10 periods, respectively. Balakrishnan and Cheng [13] used this data set to test their genetic algorithm (NLGA). They also compared their results with the Conway and Venkataramanan’s [11] genetic algorithm (CONGA). The results of their comparison between NLGA and CONGA showed that NLGA performs better. Erel et al. [15] also used the same data set to test their eight dynamic programming (DP) implementations and two simulated annealing (SA) implementations. In the first part of the comparison study the budget constraints are not taken into account. This is mainly due to the unavailability of published result for making a direct comparison of the constrained DLP algorithms. The results obtained from the proposed ant colony algorithm and the comparison of NLGA, CONGA and ten implementations of the Erel et al. [15] for unconstrained DLP are presented in Tables 1–3.

The results show that in small size problems, there is no significant difference between the qualities of the solutions. For larger size problems the proposed ant colony algorithm performs better than NLGA and CONGA. However, the proposed ant colony algorithm cannot perform better than the DP and SA heuristics of Erel et al. [15] for larger size problems. In Tables 1–3 the comparison of the mean CPU times of the all algorithms except NLGA and CONGA (because their CPU times are not available) are presented. The CPU times of the DP implementations [15] are smaller than the proposed ACO algorithm’s CPU times. But for the larger size problems the CPU times of the present ACO algorithm are better than the SA heuristics of Erel et al. [15]. In the second part of the comparison study, the budget constraints are added to the problems. The same data set is used in the both constrained and unconstrained DLP problems [13]. The shifting cost values that are obtained from the solutions of the unconstrained problems are used to set the budget constraints. Three tests are carried out with three different levels of budget constraints. Results obtained from all of these three tests are presented in Table 4 . The first rows (i.e. Test no. 1) in Table 4 represents the unconstrained problem results. In the first test (Test no. 2), the budget constraints are fixed in every period and the value of these constraints are found by dividing the total shifting cost of the unconstrained problem to the number of transitions between periods. In the second test (Test no. 3 in Table 4), the value of the budget constraints for each period are found by taking the half of the shifting costs of the unconstrained problem. In the third test (Test no. 4 in Table 4), the value of the budget constraints for each period are found by adding 10% more to the shifting costs of the unconstrained problem. It should also be noticed that the unused budgets in a period can be transferred to the following periods. Table 4 presents a compressed version of the results obtained for the budged constrained DLP. Due to space constraints the detailed results are located in the following web page: http://www1.gantep.edu.tr/∼baykasoglu/bcdlp.htm. The ACO algorithm’s parameters must be chosen heuristically, since there is no known optimal strategy for this. By trial-and-error, we found that the following parameter settings that gave good solutions: Q = the target objective function value of the problem, R = (N ∗ S)/2, W = (N ∗ S)/2,
= 0.5,  = 1. The results we report were obtained using these settings.

1 2 3 4 5 6 7 8 CPU Time

10

214,313 212,134 207,987 212,741 211,022 209,932 214,252 212,588 206

106,419 104,834 104,320 106,399 105,628 103,985 106,439 103,771 52

BEST

217,251 216,055 208,185 212,951 211,076 210,277 215,504 214,621 1326

106,419 104,834 104,320 106,509 105,628 104,053 106,439 103,771 70

ACO

214,397 212,138 208,453 212,953 211,575 210,801 215,685 214,657 NA

106,419 104,834 104,320 106,515 105,628 104,053 106,978 103,771 NA

NLGA

218,407 215,623 211,028 217,493 215,363 215,564 220,529 216,291 NA

108,976 105,170 104,520 106,719 105,628 105,605 106,439 104,485 NA

CONGA

214,313 212,134 207,987 212,741 211,022 209,932 214,252 212,588 <1

106,419 104,834 104,320 106,509 105,628 103,985 106,447 103,771 <1

DP_10

N: number of departments and locations, S: number of periods, CPU time: mean CPU time as seconds.

1 2 3 4 5 6 7 8 CPU time

5

6

Prob. no.

S

N

Table 1 Results for six departments unconstrained DLP

214,313 212,134 207,987 212,741 211,022 209,932 214,252 212,588 <1

106,419 104,834 104,320 106,509 105,628 103,985 106,447 103,771 <1

DP_10I

214,313 212,138 208,246 213,117 211,022 210,000 214,252 213,022 <1

106,419 104,834 104,320 106,885 105,737 104,053 106,447 104,185 <1

DP_5

214,313 212,138 208,060 212,747 211,022 210,000 214,252 213,022 <1

106,419 104,834 104,320 106,515 105,737 104,053 106,447 104,185 <1

DP_5I

214,313 212,134 207,987 212,747 211,076 210,000 214,823 212,588 215

106,419 104,834 104,520 106,399 105,737 103,985 106,439 103,771 55

SA_EG_1

214,313 213,015 208,351 212,747 211,072 209,932 214,438 212,588 206

106,419 104,834 104,320 106,399 105,628 103,985 106,439 103,771 52

SA_EG2 A. Baykasoglu et al. / Omega 34 (2006) 385 – 396 391

1 2 3 4 5 6 7 8 CPU

10

982,298 973,179 985,364 974,994 938,846 968,323 977,410 985,041 7

481,378 478,816 487,886 481,628 484,177 482,321 485,384 489,072 2

BEST

1,017,741 1,016,567 1,021,075 1,007,713 1,010,822 1,007,210 1,013,315 1,019,092 2407

501,447 506,236 512,886 504,956 509,636 508,215 508,848 512,320 167

ACO

1,047,596 1,037,580 1,056,185 1,026,789 1,033,591 1,028,606 1,043,823 1,048,853 NA

511,854 507,694 518,461 514,242 512,834 513,763 512,722 521,116 NA

NLGA

1,055,536 1,061,940 1,073,603 1,060,034 1,064,692 1,066,370 1,066,617 1,068,216 NA

504,759 514,718 516,063 508,532 515,599 509,384 512,508 514,839 NA

CONGA

986,811 985,154 989,081 979,139 986,029 976,917 985,535 990,844 712

484,054 489,322 491,310 487,884 491,617 490,205 490,544 494,994 111

DP_10L

984,344 984,779 988,635 976,456 938,846 974,436 982,790 990,372 724

483,568 489,322 491,310 487,275 491,346 489,847 490,051 493,577 119

DP_10LI

991,093 987,453 993,799 983,208 989,680 979,297 992,897 992,962 55

484,972 491,102 493,632 489,929 494,040 490,782 491,984 496,811 20

DP_5L

N: number of departments and locations, S: number of periods, CPU time: mean CPU time as seconds.

1 2 3 4 5 6 7 8 CPU

5

15

Prob. no.

S

N

Table 2 Results for 15 departments unconstrained DLP

988,322 985,147 993,318 982,632 985,966 978,683 989,272 988,959 67

482,123 488,840 493,632 489,480 494,040 490,782 490,251 496,672 28

DP_5LI

986,592 984,601 990,218 978,726 984,975 976,610 987,019 990,247 206

484,369 487,274 491,790 487,956 491,178 490,305 490,161 494,954 14

DP_10S

983,070 983,826 990,153 977,548 983,053 975,290 986,325 988,584 218

483,708 485,702 491,790 486,851 491,178 489,947 489,583 494,534 22

DP_10SI

995,319 988,396 992,824 982,270 987,963 981,406 992,807 993,902 7

484,369 489,819 493,224 489,698 493,097 492,275 492,430 496,990 2

DP_5S

991,801 985,360 990,794 982,112 982,893 979,731 988,870 990,376 19

483,708 488,382 492,597 489,698 491,738 492,202 489,155 496,473 10

DP_5SI

982,298 973,179 985,364 974,994 975,498 968,323 977,410 985,041 6470

481,378 478,816 487,886 481,628 484,177 482,321 485,384 489,072 1635

SA_EG_1

984,013 983,550 988,465 980,045 982,191 973,199 985,270 989,520 3867

481,792 488,592 492,536 485,862 489,946 488,452 487,576 493,030 946

SA_EG2

392 A. Baykasoglu et al. / Omega 34 (2006) 385 – 396

S

1 2 3 4 5 6 7 8 CPU

1,223,124 1,231,151 1,230,520 1,200,613 1,210,892 1,239,255 1,248,309 1,231,408 8663

604,408 604,370 603,867 596,901 591,988 599,862 600,670 610,474 2061

ACO

1,228,411 1,231,987 1,231,829 1,227,413 1,215,256 1,221,356 1,212,273 1,245,423 NA

611,794 611,873 611,664 611,766 604,564 606,010 607,134 620,183 NA

NLGA

1,362,513 1,379,640 1,365,024 1,367,130 1,356,860 1,372,513 1,382,799 1,383,610 NA

632,737 647,585 642,295 634,626 639,693 637,620 640,482 635,776 NA

CONGA

1,174,773 1,175,323 1,174,023 1,155,879 1,128,136 1,144,030 1,143,814 1,168,142 7008

581,805 574,657 581,030 571,730 561,079 567,202 572,262 575,445 1324

DP_10L

1,171,853 1,169,138 1,174,023 1,152,684 1,128,136 1,143,824 1,142,494 1,167,900 7358

579,741 570,915 581,030 569,874 561,079 567,154 568,196 575,445 1499

DP_10LI

1,180,120 1,179,022 1,175,920 1,157,918 1,131,518 1,147,517 1,147,016 1,170,929 595

583,082 576,592 581,691 575,024 561,424 570,435 573,878 576,091 222

DP_5L

1,171,413 1,174,421 1,170,019 1,156,016 1,131,518 1,147,517 1,145,934 11,170,929 945

581,942 571,563 580,549 574,070 561,424 570,435 571,254 576,091 297

DP_5LI

N: number of departments and locations, S: number of periods, CPU time: mean CPU time as seconds.

1,171,178 1,169,138 1,165,525 1,152,684 1,128,136 1,143,824 1,142,494 1,167,163 63

579,741 570,906 577,402 569,596 561,078 567,154 568,196 575,273 23

Prob. no. BEST

10 1 2 3 4 5 6 7 8 CPU

30 5

N

Table 3 Results for 30 departments unconstrained DLP

1,172,434 1,175,551 1,175,240 1,155,998 1,129,143 1,144,539 1,143,788 1,167,163 1477

581,805 575,004 581,170 571,749 561,078 568,554 572,706 575,273 131

DP_10S

1,171,178 1,170,747 1,165,525 1,153,981 1,128,784 1,144,092 1,143,183 1,167,163 1827

579,741 570,906 577,402 569,596 561,078 568,554 571,580 575,273 306

DP_10SI

1,181,743 1,177,212 1,176,997 1,158,507 1,132,926 1,149,893 1,147,041 1,171,658 63

582,858 576,106 581,262 574,110 562,857 570,356 572,797 576,149 23

DP_5S

1,180,087 1,170,810 1,173,529 1,156,517 1,132,926 1,149,893 1,146,987 1,171,428 413

581,369 572,511 580,186 573,001 562,857 573,156 569,145 576,149 182

DP_5SI

1,175,756 1,173,015 1,166,295 1,154,196 1,141,738 1,158,322 1,157,505 1,179,888 87,200

583,081 573,965 580,102 572,139 563,503 574,805 573,361 581,614 21,710

1,174,815 1,177,743 1,171,932 1,154,945 1,140,116 1,158,227 1,163,761 1,177,565 46,152

583,227 574,116 577,787 573,446 565,735 570,905 571,499 581,966 10,691

SA_EG_1 SA_EG2 A. Baykasoglu et al. / Omega 34 (2006) 385 – 396 393

Test no.

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2

P.N.

1

2

3

4

5

6

7

8

103,771 108,603

106,439 108,114 108,114 107,521

104,053 108,551 108,588 109,568

105,628 108,188 108,669 106,834

106,509 110,284 109,474 109,995

104,320 107,650 108,113 106,977

104,834 105,731 106,802 105,755

1,06,419 106,991 106,419 106,419

O.F.

19

18

17

16

15

14

13

12

P.N.

1 2

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

Test no.

512,886 524,745

506,236 519,860 523,397 525,379

501,447 512,046 517,302 512,481

214,621 220,144 220,055 221,839

215,504 219,788 219,788 219,788

210,277 217,739 217,597 217,397

211,076 217,096 217,142 219,835

212,951 217,350 217,350 217,350

O.F.

Table 4 Computational results of DLP for different budget constraints

30

29

28

27

26

25

24

23

P.N.

1 2

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

Test no.

1,007,210 1,032,072

1,010,822 1,039,268 1,059,084 1,034,928

1,007,713 1,034,417 1,048,028 1,030,672

1,021,075 1,049,928 1,068,886 1,036,409

1,016,567 1,030,560 1,041,725 1,028,099

1,017,741 1,039,960 1,061,535 1,032,807

512,320 527,590 530,135 522,490

508,848 511,240 521,767 515,327

O.F.

41

40

39

38

37

36

35

34

P.N.

1 2

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

Test no.

1,223,124 1,218,971

610,474 615,788 618,022 614,882

600,670 610,371 605,413 608,664

599,862 603,965 605,752 599,782

591,988 605,348 607,299 599,439

596,901 611,799 611,421 608,004

603,867 608,768 614,621 613,072

604,370 605,655 608,461 607,954

O.F.

48

47

46

45

P.N.

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

Test no.

1,231,408 1,240,367 1,245,145 1,240,958

1,248,309 1,251,610 1,249,389 1,247,708

1,239,255 1,242,314 1,241,012 1,245,548

1,210,892 1,239,687 1,228,490 1,237,846

O.F.

394 A. Baykasoglu et al. / Omega 34 (2006) 385 – 396

A. Baykasoglu et al. / Omega 34 (2006) 385 – 396

395

44

1 2 3 4

1,200,613 1,224,249 1,216,632 1,219,502

Ant colony heuristic is a promising optimization technique for solving complex combinatorial optimization problems like the DLP. If the solution space of the problem can be defined in a network form efficiently then it has the ability to find good solutions to the problem. In the past, ACO has been applied to several combinatorial optimization problems successfully. In this study its application to constrained and unconstrained DLP is presented. DLP is a complex combinatorial optimization problem. Several attempts have been done in the literature in order to solve this problem using analytical and heuristic procedures. In this study, an ant colony algorithm with a simple but effective data structure and solution generation mechanism is proposed for its solution. The proposed ant colony algorithm is applied to test problems from the available literature that results are known. In these tests the proposed algorithm found not all of the best solutions but competitive solutions. An extensive computational work is performed on a data set obtained from Balakrishnan and Cheng [13], which contains problems for 6,15,30 departments for 5 and 10 periods. The results of this study are compared with Balakrishnan and Cheng’s [13] genetic algorithm (NLGA), Conway and Venkataramanan’s [12] genetic algorithm (CONGA), Erel et al.’s [15] dynamic programming and simulated annealing heuristics. The comparisons presented that proposed ant colony algorithm performs better than NLGA and CONGA but not as good as Erel et al.’s [15] heuristics for larger size problems. The ant colony algorithm is also applied to budget constrained DLP problem successfully. Due to unavailability of published results for budget constrained DLP no comparisons is made. The research on DLP can be extended in the following directions:

1 2 3 4

P.N.: problem no., O.F.: objective function value.

1 2 3 4 11

208,185 219,024 219,024 219,024

22

1 2 3 4

508,215 514,277 520,721 517,139

33

604,408 610,903 606,465 612,039

1,230,520 1,227,438 1,229,156 1,224,383 1 2 3 4 43 1 2 3 4 1 2 3 4 10

216,055 217,412 217,412 217,412

21

1 2 3 4

509,636 521,971 520,556 519,407

32

1,019,092 1,035,031 1,062,510 1,040,019

1,231,151 1,221,882 1,229,328 1,222,370 1 2 3 4 1,013,315 1,042,420 1,049,658 1,038,597 1 2 3 4 31 504,956 518,223 526,131 519,095 1 2 3 4 217,251 223,017 220,776 221,010 1 2 3 4 9

3 4

108,603 108,603

20

3 4

529,629 523,918

3 4

1,061,751 1,036,017

42

3 4

1,228,403 1,223,520

5. Conclusions

• The present algorithms for DLP consider only single objectives. Multiple objective cases of DLP can also be modelled and solved. • Budget constraints can be incorporated into the other modern heuristics (genetic algorithms, tabu search and simulated annealing) and results can be compared with the results obtained in this paper. • Graph theoretical formulation of DLP problem can be developed and solved as an alternative solution and modelling strategy. • “Unequal department areas” and “multiple floors” cases of DLP can be modelled and solved. • DLP problem can also be extended to the other domains like dynamic formation and layout of manufacturing and assembly cells. • Detailed statistical comparisons of the algorithms that are developed for solving DLP can be carried out. • In the case of modern heuristics, alternative solution representation, neighbour generation mechanisms can be devised to improve the effectiveness of these heuristics in terms of solution quality and computational time requirements.

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