Genetic algorithm-based heuristic for capacitated lotsizing problem in flow shops with sequence-dependent setups

Expert Systems with Applications 38 (2011) 7201–7207

Contents lists available at ScienceDirect

Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa

Genetic algorithm-based heuristic for capacitated lotsizing problem in flow shops with sequence-dependent setups M. Mohammadi a, S.M.T. Fatemi Ghomi b,⇑ a b

Department of Industrial Engineering, Faculty of Engineering, Tarbiat Moallem University, Karaj, Iran Department of Industrial Engineering, Amirkabir University of Technology, 424, Hafez Avenue, 15916-34311 Tehran, Iran

a r t i c l e

i n f o

Keywords: Capacitated lotsizing problem Pure flow shop Sequence-dependent setups Genetic algorithm Rolling-horizon

a b s t r a c t This paper proposes a genetic algorithm-based heuristic for the capacitated lotsizing problem in flow shops with sequence-dependent setups. The proposed heuristic combines genetic algorithm with rolling horizon approach. To evaluate the effectiveness of the proposed heuristic, some numerical experiments are conducted and corresponding results are compared with those of previously developed heuristics by authors. The comparative results indicate the superiority of genetic algorithm-based heuristic specially in solving the large-sized problem instances. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Lotsizing and scheduling are a part of operative production planning and control. In lotsizing a lot indicates the quantity of a product manufactured on a machine continuously without interruption. The task of scheduling is to determine the production sequence in which the products are manufactured on a machine. Simultaneous lotsizing and scheduling is essential when considerable sequence-dependent setup costs and setup times occur during production (Fandel & Stammen-Hegene, 2006). Lotsizing and scheduling problems have been an area of active research starting from the seminal paper of Wagner and Whithin (1958). Since then there has been a considerable amount of investigation to incorporate other important features. Among the characteristic features of the models for lotsizing and scheduling are the segmentation of the planning horizon, the time dependence of the model parameters, the information degree of the model parameters, the number of products and production stages, the production structure and the capacity restrictions (Fandel & Stammen-Hegene, 2006; Karimi, Fatemi Ghomi, & Wilson, 2003; Merece & Fonton, 2003). Models of lotsizing and scheduling are divided in the literature into small bucket and big bucket problems (Eppen & Martin, 1987). Small bucket problems break the planning horizon in small time periods which limits the number of products manufactured in a single period. Small bucket problems are divided in the literature into discrete lotsizing and scheduling ⇑ Corresponding author. Fax: +98 216413969. E-mail address: [email protected] (S.M.T. Fatemi Ghomi). 0957-4174/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2010.12.038

problem (DLSP) and proportional lotsizing and scheduling problem (PLSP). In DLSP at most one product can be manufactured in a single period, but PLSP allows up to two products (Drexl & Kimms, 1997). Consequently, if a setup is performed, the entire time interval must be devoted to the setup. That is, setups and production runs comprise of an integer number of time intervals (see, for example, Tempelmeier & Buschkuhl, 2008). Small bucket problems for multi-level multi-product production are the multi-level discrete lotsizing and scheduling problem (MLDLSP) (Kimms, 1996) and the multi-level proportional lotsizing and scheduling problem (MLPLSP) (Kimms, 1993; Kimms & Drexl, 1996). Both models enable simultaneous lotsizing and scheduling, but limit the number of products to be manufactured in a period. The multi-level capacitated lotsizing problem (MLCLSP), a big bucket problem, does not have this disadvantage, but it cannot fix the lot-sizes and schedules simultaneously. To attempt to unite the advantages of the MLPLSP and MLCLSP, Fandel and Stammen-Hegene (2006) propose a new formulation based on two-level time structure (Fleischmannand & Meyr, 1997), which enables simultaneous lotsizing and scheduling for multi-product, multi-level job shop production. However, the nonlinearity nature of the model and existing three types of binary variables make the model solution too complicated, therefore no solution approach has been presented. Recently, Mohammadi, Fatemi Ghomi, Karimi, and Torabi (in press) have proposed a mathematical formulation and four heuristics, all based on solving a succession of smaller MIPs for lotsizing in capacitated pure flow shop with sequence dependent setups. The first two heuristics were based on solving the prob-

7202

M. Mohammadi, S.M.T. Fatemi Ghomi / Expert Systems with Applications 38 (2011) 7201–7207

lem via a period by period approach. The last two heuristics were based on solving the problem via so-called relax and fix approach (Araujo, Arenales, and Clark (2007) and Beraldi, Ghiani, Grieco, and Guerriero (2008)). The first heuristic was based on the original model, but to solve larger instances of the problem, other heuristics were based on a restricted model called permutation flow shop problem which ignores majority of combinations. In this paper, more efficient algorithm base on a more simplified mathematical model are proposed to solve large scale instances of lotsizing in capacitated pure flow shop with sequence dependent setups. The paper has the following structure. Section 2 introduces a detailed description of the problem and its underlying assumptions. Section 3 provides the proposed solution approach. Section 4 reports the numerical experiments and finally Section 5 is devoted to the concluding remarks and recommendation for future studies.

2. Problem formulation 2.1. Assumptions Our assumptions and mathematical model are similar to those of Mohammadi et al. (2010) which are summarized as follows:  Several products are produced on serially-arranged machines in a pure flow shop structure.  Each machine is constrained in capacity.  When the machines are setup, sequence-dependent setup costs and times occur.  The setting-up of a machine must be completed in a period.  There must be precisely N (number of products) setups in each period on each machine, even if a setup is just from a product to itself. Since a setup time (and cost) from a product to itself is zero, note that the model does not force a machine to have exactly N positive-time (and cost) setups but rather up to N such setups. The remaining zero-time (cost) setups are modeling phantoms and do not exist in reality (Clark and Clark (2000) and Clark (2003)). This feature makes possible for a lotsize, or production run, to continue over consecutive time periods without incurring real setup for later period (setup carry over).  The required resources and parts must be ready for production.  External demand exists for final products and is satisfied at the end of each period.  There are no lead times between the different production levels for transportation or cooling the products.  Shortages are not permitted.  A component cannot be produced earlier in a period than the production of its required component is finished. In other words, production on a production level can only be started if a sufficient amount of the product from the previous production level is available; this is called vertical interaction.  To guarantee the vertical interaction, idleness between each setup and its production is defined with the help of shadow product (Fandel & Stammen-Hegene, 2006).  There are no demand and no storage costs for shadow products.  At the beginning of the planning horizon all machines are setup for a defined product.  The triangle inequality holds, i.e., it is never faster to change over from one product to another by means of a third product. In other words, a direct changeover is at least as capacity efficient as going via another product.

2.2. Mathematical model The following notations are used in the model formulation: Indices: i, j, k n, n0 , n00 m t

Index of product type Designation for a specific setup number Index of production’s level Index of planning period

Parameters: T N M bigM C m;t dj;t hj;m bj;m

pj;m;t

Si;j;m

W i;j;m

j0

Planning horizon Number of different products Number of production levels/number of machines A large real number Available capacity of machine m in period t (in time units) External demand for product j at the end of period t (in units of quantity) Storage costs unit rate for product j in level m Capacity of machine m required to produce a unit of product (or shadow product) j in time units per quantity units Production costs to produce one unit of product j on machine m in period t (in money unit per quantity unit) Sequence-dependent setup time at machine m when switching from product i to j in time units; (for i – j, Si;j;m P 0 and for i = j, Si;j;m = 0) Sequence-dependent setup cost at machine m when switching from product i–j in money units; (for i – j, W i;j;m P 0 and for i = j, W i;j;m = 0. We also assume that the relation between setup costs and times can be considered as: W i;j;m ¼ fw  Si;j;m where fw is opportunity cost per unit of setup time The starting setup configuration on machines

Decision variables: Ij;m;t yni;j;m;t

xnj;m;t qnj;m;t

Inventory level of product j at level m at the end of period t 1, if the nth setup on machine m is performed at period t when switching from product i–j; 0, otherwise Quantity of product j produced after nth setup on machine m at period t Shadow product indicating the gap (in quantity units) between nth setup (to product j) on machine m at period t and its related production in order to ensure that direct predecessor of this product (production of product j on machine m  1 at period t) has been completed. In other words, it denotes the idle time (in quantity units) before production of product j on machine m in period t in order to guarantee vertical interaction

According to the above notation, the proposed mathematical formulation for the problem can be written as follows:

M. Mohammadi, S.M.T. Fatemi Ghomi / Expert Systems with Applications 38 (2011) 7201–7207

Min

N X N X N X M X T X n¼1 j¼1

wi;j;m  yni;j;m;t þ

i¼1 m¼1 t¼1

N X N X M X T X

pj;m;t  xnj;m;t þ

N X M X T X

n¼1 j¼1 m¼1 t¼1

hj;m  Ij;m;t

7203

ð1Þ

j¼1 m¼1 t¼1

Subject to :

ð2Þ N X

dj;t ¼ Ij;M;t1 þ

xnj;M;t  Ij;M;t ;

j ¼ 1; . . . ; N; t ¼ 1; . . . ; T

ð3Þ

n¼1

Ij;m;t1 þ

N X

xnj;m;t ¼ Ij;m;t þ

N X

n¼1

0

1

N X

bigM:@

xnj;mþ1;t ;

j ¼ 1; . . . ; N; m ¼ 1; . . . ; M  1; t ¼ 1; . . . ; T

0 yni;j;m;t

 1A þ

i¼1;i–jðfor n0 >1Þ

0

n0 X N X N X

1

00 yni;j;mþ1;t A

þ

i¼1;i–jðfor n00 >1Þ N X

yni;k;m;t  Si;k;m þ

n¼1 i¼1 k¼1

N X

6 bigM:@1  00 1 nX

ð4Þ

n¼1

bk;mþ1  xnk;mþ1;t ;

n0 X N X

bk;m  qnk;m;t þ

n¼1 k¼1

n00 X N X N X

yni;k;mþ1;t  Si;k;mþ1 þ

n¼1 i¼1 k¼1

n0 X N X

bk;m  xnk;m;t

ð5Þ

n¼1 k¼1 n00 X N X

bk;mþ1  qnk;mþ1;t þ

n¼1 k¼1

j ¼ 1; . . . ; N; n0 ¼ 1; . . . ; N; n00 ¼ 1; . . . ; N; m ¼ 1; . . . ; M  1; t ¼ 1; . . . ; T

n¼1 k¼1 N X N X N X n¼1 i¼1

yni;j;m;t  Si;j;m þ

N X N X

bj;m  xnj;m;t þ

n¼1 j¼1

j¼1

N X

xnj;m;t 6 ðC m;t =bj;m Þ 

yni;j;m;t ;

N X N X

bj;m  qnj;m;t 6 C m;t ;

m ¼ 1; . . . ; M; t ¼ 1; . . . ; T

ð6Þ

n¼1 j¼1

n ¼ 1; . . . ; N; j ¼ 1; . . . ; N; m ¼ 1; . . . ; M; t ¼ 1; . . . ; T

ð7Þ

i¼1;i–jðfor n>1Þ

qnj;m;t 6 ðC m;t =bj;m Þ 

N X

yni;j;m;t ;

n ¼ 1; . . . ; N; j ¼ 1; . . . ; N; m ¼ 1; . . . ; M; t ¼ 1; . . . ; T

ð8Þ

i¼1

y1j;i;m;1 ¼ 0; N X j¼1 N X

ynj;i;m;t ¼

j–j0 ; N X

i ¼ 1; . . . ; N; m ¼ 1; . . . ; M

ð9Þ

ynþ1 i;k;m;t ; n ¼ 1; . . . ; N  1; i ¼ 1; . . . ; N; m ¼ 1; . . . ; M; t ¼ 1; . . . ; T

ð10Þ

k¼1

yNj;i;m;t1 ¼

j¼1

N X

y1i;k;m;t ; i ¼ 1; . . . ; N; m ¼ 1; . . . ; M; t ¼ 2; . . . ; T

ð11Þ

k¼1

yni;j;m;t 2 f0; 1g

ð12Þ

Ij;m;t ; xnj;m;t ; qnj;m;t P 0; 8j; m; t

ð13Þ

Ij;m;0 ¼ 0; j ¼ 1; . . . ; N; m ¼ 1; . . . ; M:

ð14Þ

Central Section

Ending Section

Iteration 1 Period 1

Beginning Section

Period 2

Central Section

Period T

Ending Section

Iteration 2 Period 1

Period 2

Beginning Section

Period T

Central Section

Iteration T Period 1

Period 2

Fig. 1. Demonstration of the iterative procedure.

Period T

7204

M. Mohammadi, S.M.T. Fatemi Ghomi / Expert Systems with Applications 38 (2011) 7201–7207

Notably, a brief explanation of constraints can be found in Mohammadi et al., 2010). 2.3. Restricted model As mentioned briefly in introduction part, to solve larger instances of the problem, Mohammadi et al. (in press) provided three (of four) heuristics base on a simplified model which assumed that products follow similar sequence in all machines. Therefore they used yni;j;t instead of yni;j;m;t . In this paper, to solve the problem through our algorithmic approach, we focus on a more simplified model. In this model in addition of sequence of products, size of products is also independent of machines. Therefore yni;j;t ; xnj;t and Ij,t are used instead of yni;j;m;t , xnj;m;t and Ij,m,t in this model. Other parameters and variables are similar to those mentioned in the former model. Therefore, our simplified model is as follows: Min

N X N X N X M X T X

wi;j;m  yni;j;t þ

n¼1 j¼1 i¼1 m¼1 t¼1

N X N X M X T X

pj;m;t  xnj;t þ

N X T X

n¼1 j¼1 m¼1 t¼1

hj;M  Ij;t ð15Þ

j¼1 t¼1

Subject to : dj;t ¼ Ij;t1 þ

N X

xnj;t  Ij;t ; j ¼ 1;...;N; t ¼ 1;...;T

ð16Þ

n¼1 n0 X N X N X

yni;k;t  Si;k;m þ

n0 X N X

n¼1 i¼1 k¼1 n0 X N X N X

bk;m  qnk;m;t þ

n0 X N X

n¼1 k¼1

yni;k;t  Si;k;mþ1 þ

n¼1 i¼1 k¼1 0

n0 X N X

bk;m  xnk;t 6

ð17Þ

n¼1 k¼1

bk;mþ1  qnk;mþ1;t þ

n¼1 k¼1

0 1 N n X X

bk;mþ1  xnk;t ;

n¼1 k¼1

n ¼ 1;...;N; m ¼ 1;...;M  1; t ¼ 1;...;T N X N X N X

yni;j;t  Si;j;m þ

n¼1 i¼1 j¼1

N X N X n¼1 j¼1

bj;m  xnj;t þ

N X N X

bj;m  qnj;m;t 6 C m;t

ð18Þ

n¼1 j¼1

and constraints with continuous variables and constraints. The approach initially adopted decomposes the model into a succession of smaller MIPs, each with a more tractable number of binary variables. For larger instances of problem, the resulting huge number of binary variables in each MIP causes great computational intractability. Rolling horizon approach has been combined with genetic algorithm in this paper. The problem would be solved through a T-iteration based heuristic. In a specific iteration k, the value of binary variables in period k would be determined by genetic algorithm. In other words, our rolling-horizon approach decomposes the planning horizon into three sections. For a given iteration k:  The first section (beginning section) is composed of the (k  1) first periods. Within this section, decision variables have been partially or completely frozen by the previous iterations according to a selected freezing strategy.  The second section (central section) includes the kth period. Value of binary variables in this period would be determined through genetic algorithm.  The third section (ending section) includes the last periods (from period k + 1 to period T). There, the model is simplified according to a selected simplification strategy. At the end of iteration k, all sections roll one period and a new iteration can then be performed. The procedure stops when there is no longer any ending section. The last iteration (T) defines all decision variables over the entire horizon. Fig. 1 demonstrates rolling horizon approach. 3.2. Genetic algorithm

m ¼ 1;...;M; t ¼ 1;...;T N X

xnj;t 6 ðbigMÞ:

yni;j;t ; n ¼ 1;...;N; j ¼ 1;...;N; t ¼ 1;...;T

ð19Þ

i¼1;i–jðfor n>1Þ

qnj;m;t 6 ðC m;t =bj;m Þ:

N X

yni;j;t ; n ¼ 1;...;N; j ¼ 1;...;N; m ¼ 1;...;M;

i¼1

t ¼ 1;...;T

ð20Þ

y1j;i;1 ¼ 0; j–j0 ; i ¼ 1;...;N N X

y1j0;i;1 ¼ 1; j0 ¼ 1;...;M

ð21Þ ð22Þ

i¼1 N X

ynj;i;t ¼

j¼1 N X

ynþ1 i;k;t ; n ¼ 1;...;N  1; i ¼ 1;...;N; t ¼ 1;...;T

ð23Þ

k¼1

yNj;i;t1 ¼

j¼1

yni;j;t

N X

N X

y1i;k;t ; i ¼ 1;...;N; t ¼ 2;...;T

ð24Þ

k¼1

¼ 0 or 1; 8i;j;t

ð25Þ

Ij;t ;xnj;t ;qnj;m;t P 0; 8j;m;t

ð26Þ

Ij;0 ¼ 0; j ¼ 1;...;N

ð27Þ

3. Solution method 3.1. Idea Rolling-horizon heuristics are usually used in dynamic lotsizing and scheduling problems, where demands are gradually revealed during the planning horizon and part types have to be allocated to machines in an on-going fashion as new orders arrive. On the other hand, a rolling-horizon approach is still suitable when all parameters are perfectly known (Araujo, Arenales, & Clark, 2008; Araujo et al., 2007; Beraldi et al., 2008; Clark, 2003; Clark & Clark, 2000; Merece & Fonton, 2003). In this case, rolling-horizon heuristics have been used to overcome computational infeasibility for large MIP problems by substituting most of the binary variables

As mentioned in Section 3.1, in each iteration k, the value of binary variables in central section (period k) would be determined by genetic algorithm. 3.2.1. Chromosome representation In this research chromosomes are represented in the form of matrices with 1  N dimensions. In Fig. 2, a sample chromosome for a specific iteration of a problem with N = 3 is depicted. The chromosome indicates the sequence of products in stages of central section period. According to this chromosome : yendk1 ;1;k ¼ y11;3;k ¼ y33;2;k ¼ 1, and other binary variables of this iteration would get value 0. endk1 is the last product in former iteration. 3.2.2. Initial population Selecting an appropriate initial population can largely increase the efficiency of a GA. For this reason, simple and effective heuristic has been used. The procedure is described as follows: 1. The products are sorted in the decreasing order of W j;m ¼ PN i¼1 W i;j;m ; j ¼ 1; . . . ; N. 2. Delete products in which Ij,k-1 > dj,k. Then, instead of each deleted products, the last remained product would be replaced. 3. Let [i] indicates the ith product in an ordered sequence in this heuristic. For [i] = 1–N: (a) Consider inserting product [i] into every position. (b) Calculate the sum of setup costs for all products scheduled so far using the actual setup costs.

1

3

Fig. 2. A sample chromosome.

2

7205

M. Mohammadi, S.M.T. Fatemi Ghomi / Expert Systems with Applications 38 (2011) 7201–7207

(c) Place product i in the position with the lowest resultant sum of setup costs. M different initial population have been made through this method (for m = 1,. . ., M), the remaining initial population have been generated randomly. 3.2.3. Fitness function The fitness value of each chromosome has been obtained by solving the corresponding whole problem consists of beginning section, central section and ending section. 3.2.4. Selection operator The tournament selection approach and roulette wheel (Gen & Cheng, 1997) have been used to undergo crossover operation. 3.2.5. Crossover operation Recombination the features of two randomly selected parents with the aim of producing better offspring is the main purpose of crossover operation. Several crossover operators have been proposed (Nagano, Ruiz, & Lorena 2008). To maintain the blocks of products and improve the algorithm, similar block order crossover (SBOX), similar job two point crossover (SJ2OX) and similar block two point crossover (SB2OX) have been used in this research. 3.2.6. Mutation operator In order to produce small perturbations on chromosomes to promote diversity of the population, mutation operator is used. Shift mutation is used in this paper. 3.2.7. Population replacement Chromosomes for the next generation are selected from the enlarged population. The best pop_size chromosomes of the enlarged population have been selected for the next generation.

This implies that in central section, the production is either zero or equal to the sum of consecutive demands for some number of periods into the future.

3.3. Calibration of parameters by means of design of experiments In this section, we study the behavior of the different parameters of the proposed heuristic. All different combinations of the aforementioned factors and parameters yield many alternative GAs. In order to calibrate the algorithms, we have choosen a full factorial design in which all possible combinations of the following factors are tested : -

Selection type: 2 levels (Tournament and Roulette wheel). Crossover type: 3 levels (SBOX, SJ2OX, SB2OX). Crossover probability (pc): 4 levels (0.5, 0.6, 0.7, 0.8). Mutation probability (pm): 4 levels (0.1, 0.15, 0.2, 0.25). Population size (psize): 3 levels (3M, 4M, 5M).

All the cited factors result in 2  3  4  4  3 = 288 different combinations and thus, 288 different genetic algorithms. Every algorithms is tested with a set of problems presented in Table 1. There is one replicate for each combination, therefore 5  288 = 1440 problems have been solved. The response variable

Table 2 Results for the objective function and computational time of the proposed heuristic. Problem size (N.M.T)

Proposed heuristic

3.3.3

(32.41) 5277.49 (102.18) 8773.12 (69.19) 8721.17 (48.53) 8839.80 (215.67) 24817.19 (1689.53) 32716.77 (491.13) 31868.72 (291.23) 32926.83 (2215.29) 65436.65 (3603.28) 47817.40 (511.58) 46181.94 (591.09) 48302.66 (4111.12) 101951.79 (3143.34) 98928.70 (2901.51) 99374.81 (5109.19) 204617.13 (7200) 323338.98 (6913.14) 310051.73 (5915.34) 301779.01 (7200) 752958.29

5.3.3 3.5.3

3.2.8. Termination criterion The termination criterion determines when GA will stop. We stop GA, if maximum number of generations, max_gen, has been executed or the pre-set number of generations without improvement in the last best solution, max_no_improve, reaches. 3.2.9. Procedure to modify variables xnj;k and Ij;k at the end of iteration k Variables of central section would be modified as follows:

dj;kþl 6

 dj;k 6

l¼1

Lj þ1 X

5.7.5 5.5.7

5.10.5 5.5.10

dj;kþl

10.7.7

l¼1

PLj The value of xnj;k would be changed to l¼1 dj;kþl þ dj;k . Lj is the last period that its demand for product j has been produced in period k. To ensure that the equation (18) holds true, Ijk would be modified as follows:

Ij;k ¼

7.5.5

10.5.5

A specific value of Lj is determined satisfying the following relation.

xnj;k

5.5.5

7.7.7

For j = 1–N For xnjk > 0

Lj X

3.3.5

N X

7.10.7 7.7.10 10.10.10 15.10.10

xnj;k þ Ij;k1  dj;k ;

ð28Þ 10.15.10

n¼1

10.10.15 15.15.15

Table 1 Dimension of problems used in calibration of parameters. 333

555

777

10  10  10

15  15  15

Note: The values inside the parentheses are the computational time in seconds.

7206

M. Mohammadi, S.M.T. Fatemi Ghomi / Expert Systems with Applications 38 (2011) 7201–7207

is based on the following performance measure: % increase over the lower bound presented by Mohammadi et al. (in press) = P5 Heusoli LBi  100Þ=5, where Heusoli is the solution obtained by a i¼1 ð LBi specific problem and LBi is the lower bound for this specific problem presented by Mohammadi et al., 2010. By following the same procedure for all combinations and all parameters results the following ‘‘best case’’ calibrations of the proposed algorithm: Selection type: tournament. Crossover type: SB2OX. Crossover probability: 0.7. Mutation probability: 0.15. Population size: 5 M.

Table 4 Comparative results for the proposed heuristic and those of provided by Mohammadi et al. (2010) (H1–H4) for large instances. Problem Size (N.M.T)

H1

H2

H3

H4

Proposed heuristic

10.5.5

– – – – – – – – – – – – – –

– – – – – – – – – – – – – –

(132.55) 22.94% (298.96) 25.43% (823.33) 26.98% (1831.27) 26.66% (1138.51) 29.38% (1383.41) 31.60% (3231.59) 25.29%

(2095.89) 19.14% (3807.97) 22.48% (5711.96) 22.11% – – (6854.35) 26.65% – – – –

(3603.28) 13.01% (4111.12) 12.91% (5109.19) 12.97% (7200) 13.11% (6913.14) 12.17% (5915.34) 12.05% (7200) 13.32%

10.7.7 10.10.10 15.10.10 10.15.10 10.10.15

4. Numerical experiments

15.15.15

For a first evaluation of the proposed heuristic, we compare its results for a set of twenty problems introduced in Mohammadi et al., 2010. Twenty different problem sizes in the range of (N, M, T) = (3, 3, 3) to (N, M, T) = (15, 15, 15) have been considered, for each problem size, five problem instances are randomly generated. The required parameters are similar to those of Mohammadi et al., 2010 and are as follows:

bj;m  Uð1:5; 2Þ; dj;t  Uð0; 180Þ; hj;m  Uð0:2; 0:4Þ; pj;m;t  Uð1:5; 2Þ; W ijm ¼ Sijm  Uð35; 70Þ; C m;t ¼ Uða; bÞ; a ¼ 200N þ 100ðm  1Þ; b ¼ 200N þ 200ðm  1Þ Our proposed heuristic is coded in Matlab programming language and are run on a personal computer with a Pentium 4 processor running at 3.4 GHZ. Table 2 shows the objective functions and CPU times of proposed heuristic. Lower bound provided by Mohammadi et al., 2010 is used to compare our proposed heuristic against heuristics provided by Mohammadi et al., 2010. Tables 3 and 4 report such comparisons for small and non-small instances, respectively.

Note: The values inside the parentheses are the computational time in seconds. The percentage values are the difference between the objective values against lower bound provided by Mohammadi et al. (2010). Also, dash character indicates that no feasible solution has been found in 7200 s.

Table 3 shows that the average difference between heuristics provided by Mohammadi et al., 2010 (H1–H4) and our proposed heuristic against mentioned lower bound for small instances are respectively, 11.74%, 13.44%, 22.83%, 18.57%, and 12.60%. Table 4 shows that the average difference between heuristics provided by Mohammadi et al., 2010 (H3–H4) and our proposed heuristic against mentioned lower bound for large instances are respectively, 26.90%, 22.59%, and 12.79%. Table 4 shows the advantages of our proposed heuristic for large instances of problem.

5. Concluding remarks The contribution of the paper is proposing and testing a new heuristic based on combining rolling horizon approach with

Table 3 Comparative results for the proposed heuristic and those of provided by Mohammadi et al. (2010) (H1–H4) for small and medium instances. Problem size (N.M.T)

H1

H2

H3

H4

Proposed heuristic

3.3.3

(2.31) 10.02% (711.11) 9.60% (98.34) 10.40% (26.26) 10.60% (7200) 11.18% – – – – (7200) 18.66% – – – – – – – – – –

(0.036) 12.77% (9.44) 12.98% (1.38) 14.25% (0.21) 13.26% (146.67) 14.74% (2147.43) 12.82% (735.58) 14.24% (318.33) 13.19% (4038.80) 12.97% (1846.93) 12.78% (734.66) 14.34% (7200) 13.76% (7200) 12.62%

(0.13) 18.11% (0.44) 18.90% (0.16) 19.90% (0.23) 19.59% (2.19) 20.33% (15.74) 23.88% (5.21) 25.23% (5.28) 25.08% (77.61) 25.48% (11.43) 24.38% (8.52) 26.48% (58.72) 24.26% (79.60) 25.22%

(0.70) 13.89% (2.98) 15.49% (1.42) 14.56% (0.91) 16.85% (24.51) 16.17% (208.22) 20.29% (58.31) 19.48% (38.81) 22.45% (1437.27) 19.35% (127.51) 21.60% 96.31)) 21.55% (1411.13) 19.42% (1983.31) 20.34%

(32.41) 12.82% (102.18) 12.53% (69.19) 12.89% (48.53) 11.86% (215.67) 13.03% (1689.53) 11.59% (491.13) 13.14% (291.23) 12.11% (2215.29) 12.89% (511.58) 12.57% (591.09) 13.67% (3143.34) 12.02% (2901.51) 12.66%

5.3.3 3.5.3 3.3.5 5.5.5 7.5.5 5.7.5 5.5.7 7.7.7 5.10.5 5.5.10 7.10.7 7.7.10

Note: The values inside the parentheses are the computational time in seconds. The percentage values are the difference between the objective values against lower bound provided by Mohammadi et al. (2010). Also, dash character indicates that no feasible solution has been found in 7200 s.

M. Mohammadi, S.M.T. Fatemi Ghomi / Expert Systems with Applications 38 (2011) 7201–7207

genetic algorithm for lotsizing multiple products in a pure flow shop with sequence-dependent setups. According to the numerical experiments, our proposed heuristic is superior specially for non-small problem sizes. Because of the expanding role of meta-heuristic approaches to solve complicated lotsizing problem (Defersha & Chen, 2008; Jans & Degraeve, 2007), the application of other meta-heuristic approaches to face this complex problem is recommended as an area for future research. References Araujo, S. A., Arenales, M. N., & Clark, A. R. (2007). Joint rolling-horizon scheduling of materials processing and lot-sizing with sequence-dependent setups. Journal of Heuristics, 13(4), 337–358. Araujo, S. A., Arenales, M. N., & Clark, A. R. (2008). Lot sizing and furnace scheduling in small foundries. Computers and Operations Research, 35(3), 916–932. Beraldi, P., Ghiani, G., Grieco, A., & Guerriero, E. (2008). Rolling-horizon and fix-andrelax heuristics for the parallel machine lot-sizing and scheduling problem with sequence-dependent set-up costs. Computers and Operations Research, 35(11), 3644–3656. Clark, A. R. (2003). Optimization approximations for capacity constrained material requirements planning. International Journal of Production Economics, 84(2), 115–131. Clark, A. R., & Clark, S. J. (2000). Rolling-horizon lot-sizing when setup times are sequence-dependent. International Journal of Production Research, 38(10), 2287–2308. Defersha, F. M., & Chen, M. (2008). A linear programming embedded genetic algorithm for an integrated cell formation and lot sizing considering product quality. European Journal of Operational Research, 187(1), 46–69. Drexl, A., & Kimms, A. (1997). Lot sizing and scheduling, survey and extensions. European Journal of Operational Research, 99(2), 221–235.

7207

Eppen, G. D., & Martin, R. K. (1987). Solving multi-item capacitated lot-sizing problems using variable redefinition. Operations Research, 35(6), 832–848. Fandel, G., & Stammen-Hegene, C. (2006). Simultaneous lot sizing and scheduling for multi-product multi-level production. International Journal of Production Economics, 104(2), 308–316. Fleischmannand, B., & Meyr, H. (1997). The general lotsizing and scheduling problem. OR Spekterum, 19(2), 11–21. Gen, M., & Cheng, R. (1997). Genetic algorithms and engineering designs. New York: Wiley. Jans, R., & Degraeve, Z. (2007). Meta-heuristics for dynamic lot sizing: A review and comparison of solution approaches. European Journal of Operational Research, 177(3), 1851–1875. Karimi, B., Fatemi Ghomi, S. M. T., & Wilson, J. M. (2003). The capacitated lot sizing problem: A review of models and algorithms. OMEGA, 31(5), 365–378. Kimms, A. (1993). Multi-level, single-machine lot sizing and scheduling (with initial } r Betriebswirtschaftslehre der inventory). Manuskripte aus den Instituten fu universität Kiel (p. 329). Kiel. Kimms, A. (1996a). Multi-level, single-machine lotsizing and scheduling (with initial inventory). European Journal of Operational Research, 89(1), 86–99. Kimms, A., Drexl, A. (1996). Some insights into proportional lotsizing and scheduling. Manuskripte aus den Instituten der universität Kiel (p. 406). Kiel. Merece, C., & Fonton, G. (2003). MIP-based heuristics for capacitated lotsizing problems. International Journal of Production Economics, 85(1), 97–111. Mohammadi, M., Fatemi Ghomi, S. M. T., Karimi, B., & Torabi, S. A. (2010). Rollinghorizon and fix-and-relax heuristics for the multi-product multi-level capacitated lotsizing problem with sequence-dependent setups. Journal of Intelligent Manufacturing, 21, 501–510. Nagano, M. S., Ruiz, R., & Lorena, L. A. N. (2008). A constructive genetic algorithm for permutation flowshop scheduling. Computers and Industrial Engineering, 55, 195–207. Tempelmeier, H., & Buschkuhl, L. (2008). Dynamic multi-machine lotsizing and sequencing with simultaneous scheduling of a common setup resource. International Journal of Production Economics, 13(1), 401–412. Wagner, H. M., & Whithin, T. M. (1958). Dynamic version of the economic lot size model. Management Science, 5, 89–96.