This paper addresses the deterministic time-varying batch ordering lot sizing problem with backorders. The

3. replenishment is instantaneous,

* Tel.: +202 797 5355; fax: +202 795 7565.E-mail address: gaafar@aucegypt.edu

Computers & Industrial Engineering 51 (2006) 433444

www.elsevier.com/locate/dsw0360-8352/$ - see front matter 2006 Elsevier Ltd. All rights reserved.objective is to determine the optimum ordering plan to satisfy a set of known demands over a specic planninghorizon. The optimum ordering plan is the one that minimizes the total cost. Cost elements include the order-ing cost (P) charged every time an order is made regardless of the quantity, the holding cost (h) per piece perperiod (charged on any quantity left at the end of the period), and the backorder cost (g) charged as a shortagepenalty per piece per period. In the case addressed in this paper, it is assumed that any order must be an inte-ger multiple of a xed quantity (Q, Q > 1). Other assumptions of the problem include:

1. the replenishment quantity is unconstrained,2. cost factors are time independent,In this paper, genetic algorithms are applied to the deterministic time-varying lot sizing problem with batch orderingand backorders. Batch ordering requires orders that are integer multiples of a xed quantity that is larger than one.The developed genetic algorithm (GA) utilizes a new 012 coding scheme that is designed specically for the batch orderingpolicy. The performance of the developed GA is compared to that of a modied Silver-Meal (MSM) heuristic based on thefrequency of obtaining the optimum solution and the average percentage deviation from the optimum solution. In addi-tion, the eect of ve factors on the performance of the GA and the MSM is investigated in a fractional factorial exper-iment. Results indicate that the GA outperforms the MSM in both responses, with a more robust performance. Signicantfactors and interactions are identied and the best conditions for applying each approach are pointed out. 2006 Elsevier Ltd. All rights reserved.

Keywords: Genetic algorithms; Lot sizing; Fixed quantity; Batch ordering; Silver-Meal

1. IntroductionApplying genetic algorithms to dynamic lot sizingwith batch ordering

Lot Gaafar *

The American University in Cairo, 113 Kasr el Aini Street, P.O. Box 2511, Cairo 11511, Egypt

Abstractdoi:10.1016/j.cie.2006.08.006

434 L. Gaafar / Computers & Industrial Engineering 51 (2006) 4334444. backorders are only allowed to make up for quantity discrepancies that result from batch ordering, and5. an order must be placed in the rst period.

As an example, assume that the demand for a planning horizon of six periods is 10, 7, 6, 9, 11, and 5 forperiods one through six, respectively, and that P = $30, h = $2, g = $4, and Q = 6. This demand may be cov-ered by orders of 24, 18, and 6 in periods, 1, 4, and 6, respectively. The total cost of such a plan is $158.

A dynamic programming algorithm to obtain the optimum solution to a simpler version of this problem(assuming a variable order quantity with no backorders) was developed by Wagner and Whitin (1958). Severalextensions of the original WagnerWhitin (WW) algorithm were developed over time (Elsayed & Boucher,1993). Of relevance to this paper are the extensions by Vander Eecken (1968), Elmaghraby and Bawle(1972), Webster (1989), Li, Hsu, and Xiao (2004). Vander Eecken (1968) extended the WW algorithm tothe case where orders must be integer multiples of a xed quantity (Q > 1), with the restriction of no backor-ders. Elmaghraby and Bawle (1972) extended Vander Eeckens work to the case where backorders are allowed,but with the restriction that the ordering cost is constant over time. Webster (1989) developed a backorderversion of the WW algorithm using dynamic programming. Several authors provided extensions to Elmagh-raby and Bawles work (Bitran & Matsuo, 1986), but the most general extension was developed by Li et al.(2004) to the case where all cost elements are time varying with allowance for backordering.

Although the WW algorithm and its extensions provide the optimum solution for various versions of the lotsizing problem, they are not widely used. Instead, practitioners prefer other algorithms that are simpler, eventhough they may generate suboptimal solutions for the following reasons:

1. Exact algorithms are often dicult to understand by practitioners due to their complexity (Silver & Peter-son, 1985).

2. Heuristics are easily tailored for various extensions of the lot sizing problem, for which no optimizationalgorithm has been developed yet (Jans & Degraeve, 2004).

3. In a rolling horizon situation, an optimization algorithm becomes a heuristic too, and is often surpassed bysimple heuristics (Stadtler, 2000).

One of the most commonly used suboptimal algorithms is the Silver-Meal (Silver & Peterson, 1985). Theoriginal Silver-Meal (SM) algorithm nds the number of periods for which the total inventory costs per periodis minimized and then orders the exact quantity to cover the demand for those periods. In this paper the SM ismodied (MSM) so that whenever the total demand for a group of periods is not an integer multiple of thexed quantity (Q), the closest two integer multiples are tried out. In the previous example, when testing thecost of an order to cover the rst three periods that have a total demand of 23, the two quantities of 18and 24 are considered. The MSM solution for the example given above is to order 18 and 30 in periods 1and 4, respectively, for a total cost of $140.

This paper investigates the applicability of genetic algorithms (GAs) to the targeted problem and comparesits performance to that of the MSM. The research is motivated by the fact that GAs have been successfullyapplied to several production planning problems including lot sizing. GAs have several desirable featuresincluding: generality, ease of modication and parallelism (Reeves, 1997). Aytug, Khouja, and Vergara(2003) attribute the popularity of GAs to several factors. First, GAs only require a computable objective func-tion with no requirements of linearity, convexity, or dierentiability. GAs are also easy to implement as theyrequire no bookkeeping mechanism or theoretical bounds. Furthermore, empirical evidence shows that GAsare quite successful on computationally intractable problems, mostly with discrete solution spaces, which isusually the case with problems addressed using dynamic programming (Papadimitriou & Steiglitz, 1998).GAs generally provide good solutions to large problems in a reasonable time (Aytug et al., 2003).

Several authors have applied GAs to various versions of the lot sizing problem. As an example, Ozdamarand Birbil (1998) utilized GA, simulated annealing (SA), and tabu search to solve the capacitated lot sizingproblem on parallel machines. Hung and Chien (2000) also utilized the three approaches (GA, SA, and tabusearch) to solve the multi-class multi-level capacitated lot sizing problem. Khouja, Michalewicz, and Wilmot(1998) investigated the use of GAs to solve the economic lot size scheduling problem using the basic period

approach. Prasad and Krishnaiah Chetty (2001) applied GAs to multilevel lot sizing and observed that under

L. Gaafar / Computers & Industrial Engineering 51 (2006) 433444 435a rolling horizon environment, the performance of GAs is superior to the popular heuristics. Dellaert and Jeu-net (2000) developed a hybrid GA for solving large unconstrained multilevel lot sizing problems. Theirapproach utilizes GAs to improve initial solutions generated from the part period algorithm, period ordermethod, and the Wagner-Within approach. Also, Dellaert, Jeunet, and Jonard (2000) proposed a GA to solvethe general multi-level lot sizing problem with time-varying costs. Staggemeier, Clark, Aickelin, and Smith(2002) presented a hybrid GA to solve a lot sizing and scheduling problem by minimizing inventory and back-log costs of multiple products on parallel machines with sequence-dependent set-up times. Basnet and Leung(2002) presented a multi-period inventory lot sizing scenario, where there are multiple products and multiplesuppliers. Sarker and Newton (2002) develop GA code with three dierent penalty functions to determineoptimal batch sizes for products and the purchasing policy of associated raw materials. GA results were com-pared to optimal solutions, and the GA with a static penalty function obtained the global optimum 100% ofthe time. Moon, Silver, and Choi (2002) developed a hybrid genetic algorithm to address the lot schedulingproblem with time-varying lot sizes. A numerical experiment showed that the hybrid genetic algorithm outper-formed Dobsons heuristic for the problems considered. Dimopoulos and Zalzala (2000) and Aytug et al.(2003) provide a general survey on the use of GAs for manufacturing and operations management optimiza-tion. Dellaert and Jeunet (2003) provide an overview of various heuristics and propose a new heuristic for themulti-level lot sizing problem. Jans and Degraeve (2004) provide a wider survey of meta-heuristics applica-tions in dynamic lot sizing.

In all cited applications of GAs to the lot sizing problem, the quantities under consideration were variable.However, Shittu (2003) used GAs to address the lot sizing problem with batch sizing and compared their per-formance to that of the MSM. Comparisons were based on the best resulting solution. Shittu used a 01 cod-ing for the GAs where a 0 indicates no order in the corresponding period and a 1 indicates an order in thecorresponding period for its demand and the demand of all subsequent periods with a 0 code. Order quan-tities on the nal GA solution were rounded to the nearest multiple of Q randomly in a post processing algo-rithm. This coding imposes a restriction that may have led to the poor performance that Shittu (2003) reported(a 4.5% average deviation from the best solution). Shittus results may also contain an error as the reportedperformance of the Silver-Meal heuristic (8.1% average deviation from the best solution) is out of sync withestablished results in the literature.

This paper investigates the performance of GAs in the batch ordering case, with the assumptions stated atthe beginning of this section, using a 012 coding that will be explained in Section 2. The performance of theGAs is compared to that of the MSM. A 251V fractional factorial experiment is used to compare the perfor-mance of the two approaches and identify the signicant factors that aect the frequency of obtaining the opti-mum solution and the average percentage deviation from the optimum solution. As the experiment implies,the eects of ve factors and some of their interactions on the performance of the GA and the MSM areinvestigated.

In evaluating the performance of the GAs, optimum solutions are chosen as obvious benchmarks. Compar-isons against the MSM provide a good indication of the GAs performance. The MSM was chosen because itis one of the most commonly used methods with reasonable accuracy. The choice of the MSM is also justiedby the simplicity/accuracy scale comparing various approaches to the lot-sizing problem developed by Zhiwei,Heady, and Lee (1994).

In the reminder of this paper, Section 2 introduces the research approach and describes the development ofthe GA. Section 3 describes the experiment used for performance evaluation. Section 4 discusses the researchresults. Finally, Section 5 presents the research conclusions and points out directions for future research.

2. Approach: The genetic algorithm

Genetic algorithms (GAs) were rst introduced by Holland (1975) as optimization search techniques thatgenerate and evaluate solutions until a stopping criterion is satised. In searching the solution space, GAs,simulate evolutionary processes observed in nature. In GAs, a solution is represented as a chromosome, whichin our case is the string of T digits (where T is the number of periods). Each digit may take the value of 0, 1, or2. A digit may also be called a gene or a bit. The question of how many future periods to cover with an order is

explicitly answered by the zero-one-two coding where a value of one or two in any digit indicates that an order

should be placed in the period corresponding to that digit for its demand and the demand in all subsequentperiods with a code of zero. When the total demand for the periods covered by the order is not an integermultiple of the xed quantity (Q), a digit value of one indicates an order quantity that is the closest integermultiple of Q lower than the total demand, while a digit value of two indicates an order quantity that isthe closest integer multiple of Q higher than the total demand. For example, a chromosome of 200101 forthe demand pattern of 10, 7, 6, 9, 11, and 5 for periods one through six, respectively, indicates an order of24 in period one to cover the demand of the rst thee periods with an extra unit as a result of batch ordering,an order of 18 in period four to cover the demands in periods four and ve, and an order of 6 in period six tocover its demand and the shortage of one unit created by the second order. The total cost of the plan repre-sented by this chromosome, based on the data given in the example of Section 1, is $158.

A set of such chromosomes is generated randomly to form an initial population. The tness of each chro-mosome in the population is evaluated based on the total cost of the plan it represents. The lower the cost, thebetter (higher) the tness. A new population of the same size is generated from the original population bymanipulating its chromosomes using genetic operators. The new population is evaluated and another popu-lation is generated from it. This process continues until a stopping criterion is met. Genetic operators generate

436 L. Gaafar / Computers & Industrial Engineering 51 (2006) 433444new chromosomes (children) from existing ones (parents) by manipulating the values assigned to the genes(digits) of some chromosomes (that are chosen randomly) in two ways: crossover and mutation. Every cross-over operator is applied to two chromosomes (parents) and results in two new ones (children). Every mutationoperator is applied to one chromosome and results in a dierent chromosome.

A large number of mutation and crossover operators were tested through pilot runs to choose the appro-priate ones for this research. In these pilot runs, the eect of each operator on improving the objective functionis tracked and the most eective operators are selected. As a result, two crossover operators and six mutationoperators were chosen. These operators are listed in Table 1 along with a brief description of each operator.Figs. 1 and 2 further explain the two crossover operators.

In Fig. 1, a simple crossover is performed by choosing two parents (P1 and P2) and a position (4) randomly.Starting from the chosen position (4) and moving to the end of both chromosomes, genes are swappedbetween the two parents to produce the two children (C1 and C2). Alternatively, a simple crossover maybe performed by swapping genes starting from the chosen position and moving backward to the beginningof the parent chromosomes. The direction of the swap is chosen randomly.

In Fig. 2, a uniform crossover is performed by choosing two parents (P1 and P2) randomly. For each geneposition, a decision (based on a random number) is made on whether to swap the genes of the two parents atthat position (Y) or not (N). The two children (C1 and C2) are produced as a result of these decisions.

Table 2 summarizes the parameters of the developed GA. The initial population consists of 200 randomlygenerated chromosomes. Subsequent populations are generated by applying the crossover and mutation oper-ators, as listed in Table 2, to the preceding generation. The crossover and mutation rates are both set to 1.0.The selection scheme is the roulette wheel with elitist selection (Goldberg, 1989). The algorithm stops when

Table 1The genetic operators

Operator Description

Simple crossover Swaps the genes of two randomly chosen chromosomes starting from a randomly chosenposition (Fig. 1)

Uniform crossover Randomly swaps the genes of two randomly chosen chromosomes (Fig. 2)Random mutation Replaces a randomly chosen chromosome with a new one generated randomlyBoundary mutation Randomly switches genes (to either 0, 1, or 2) starting from a randomly selected position till the

end of the chromosomeOrdering mutation Maintains the number of orders in a randomly chosen chromosome, but distributes them randomly

in the new chromosomeChange ordering mutation Randomly increases or decreases the number of orders in a randomly chosen chromosome and

distributes the new orders randomly in the new chromosomeRandom swapping mutation Swaps two randomly chosen genes in randomly chosen chromosomeNeighbor swapping mutation Swaps a randomly chosen gene with the gene before it or the gene next to it in a randomlychosen chromosome

L. Gaafar / Computers & Industrial Engineering 51 (2006) 433444 437Randomlychosen position

P1: 2 C1: 2 2 0 1 2 0

P2: 1 C2: 1 1 1 0 2 1

00

010210201

1201100

Fig. 1. An example of the simple crossover operator.

Y N N Y N Y Y NP1: 2 C1: 1 2 1 2

P2: 1 C2: 2 1 0 2

11001201100

0210201 0001

Fig. 2. An example of the uniform crossover operator.

Table 2Parameters of the genetic algorithm

Parameter Value/type

Population size 200Number of generations 100100 generations have been used or when the optimum solution is reached, whichever comes rst. This meansthat at most 20,000 evaluations of the objective function were permitted per instance. Parameters in Table 2were chosen based on several pilot runs. They were also guided by the experiment conducted by Shittu (2003).Shittu used the rst ve genetic operators listed in Table 1 and a population size of 150 with 200 generationsfor a total of 30,000 objective function evaluations per instance. Shittus comparisons are based on best, notoptimum, solutions.

To maintain solution feasibility, the rst bit of any chromosome is always forced to be a 1 or a 2 at thegeneration time or after mutation. The Bit_Mod heuristic (Prasad & Krishnaiah Chetty, 2001) was used toprevent orders in zero demand periods. The Bit_Mod heuristic involves a simple check of the demand in everyperiod corresponding to a 1 or 2 bit. If the demand is zero, the bit is turned o (set to zero) and the rstfollowing bit corresponding to a demand period is set to either 1 or 2.

3. The evaluation experiment

An experiment was planned to investigate the eect of ve factors on the heuristics: demand pattern (A),batch size (B), the ratio of the ordering cost to the carrying cost (C, C = P/h), the ratio of the carrying cost tothe backorder cost (D, D = h/g), and the length of the planning horizon (E). Each factor is investigated on twolevels, i.e., A (seasonal and constant), B (6 and 24), C (100 and 800), D (0.125 and 1.0), and E (12 and 30).Table 3 summarizes the factors and their chosen levels. The experimental design followed the guidelines pro-vided by Aytug et al. (2003). Factors and their levels were chosen based on similar work in the literature (Gaa-far & Choueiki, 2000; Shittu, 2003).

Number of simple crossovers 10Number of uniform crossovers 5Number of random mutations 100Number of boundary mutations 10Number of ordering mutations 10Number of change ordering mutations 10Number of random swapping mutation 10Number of neighbor swapping mutation 10

Crossover rate 1.0Mutation rate 1.0Selection scheme Roulette wheel with elitist selection

3200the pfrom

438 L. Gaafar / Computers & Industrial Engineering 51 (2006) 433444mum solution, nSM and nGA.Optimum solutions were generated using a modied version of the WW algorithm developed by Webster

(1989). Backorders were limited to making up for quantity discrepancies that result from batch ordering.All algorithms were coded in MATLAB 6.5 (The MathWorks, Inc., 2002). The Design-Expert 6 software

(STAT-EASE, 2001) was used to analyze the results. Experiments were executed on a personal computerequipped with a Pentium 4 processor working at a speed of 1.70 GHz. The MSM heuristic required an exe-cution time of about 0.1 s per instance, while the GA required an average of about 54 s per instance. The mod-ied WW algorithm required an average of about 117 s per instance. Each run (200 instances) required about9 h to execute all algorithms.4. Re

Th16 rulookof thintera

BadGAcost tinstances were executed (16 runs times 200 instances per run). Two responses were chosen to evaluateerformance of the MSM and GA algorithms. The two responses are the average percentage deviationthe optimum solution, dSM and dGA, and the percentage of time each algorithm reached at the opti-A full factorial experiment would require 25 = 32 runs, but this research employs one-half fractional facto-rial experiment, i.e. 251V requiring only 16 runs. The subscript V indicates that the experiment is of resolutionve, where main eects and two-way interactions are only aliased with three-way and higher order interactions(Montgomery, 2001).

Using information from the literature (Gaafar & Choueiki, 2000), the constant demand pattern was gen-erated using the model: dt = a + et, 1 6 t 6 T, where, dt is the demand in period t, T is the number of periodsin the plan (factor E), a is a constant generated from an exponential distribution with a mean of 100, and et is anormally independently distributed error component with a mean of 0 and a constant variance of r2

(r = 0.1a). The seasonal demand pattern was generated using the following model:

dt a1 a2 sin 2pt mT et; 1 6 t 6 T

where, a1 is a constant generated from an exponential distribution with a mean of 100, a2 is the amplitude ofthe sinusoidal curve (a2 = 0.5a1), m is a constant generated from a discrete uniform distribution ranging be-tween 0 and T 1 to randomly vary the starting point of the demand pattern, and et is a normally indepen-dently distributed error component with a mean of 0 and a constant variance of r2 (r = 0.1a2). In bothpatterns, a check is made to ensure that demand in the rst period is always greater than zero.

For this experimentation, four distinct sets (of two hundred demand patterns each) were generated; one foreach of the seasonal and constant demand patterns and for each of the 12 and 30 period horizons. Overall,

Table 3Factor levels for the 251V fractional factorial experiment

Factor Levels

Low High

A: Demand pattern Seasonal ConstantB: Batch size 6 24C: The ratio of the ordering cost to the carrying cost 100 800D: The ratio of the carrying cost to the backorder cost 0.125 1.0E: The planning horizon 12 30sults

is section presents and discusses the main research results. Table 4 summarizes the response values for allns. As the overall averages of Table 4 show, the GA outperforms the MSM in both responses. A closerat Table 4 reveals that the GA outperformed the MSM in all runs. Fig. 3 displays the graphs of the eectse various factors on the deviation responses. These graphs help to visualize the eect of the factors andctions on the displayed responses.sed on Fig. 3, Factors A (demand pattern) and E (planning horizon) have the most inuential eects onand dSM. Their eects on dSM are more pronounced. In addition, Factor C (the ratio of the orderingo the carrying cost), has a signicant eect on dSM only.

L. Gaafar / Computers & Industrial Engineering 51 (2006) 433444 439Factor eects are within logical expectations. It is expected that seasonal data would be harder to optimizethan constant data, explaining the eect of Factor A. Factor B (batch size) is also expected to have a positiveimpact on the deviation from the optimum solution. The larger the batch size the more the cost will deviatewhen the wrong decision is made. It seems, however, that a bigger range of Factor B is needed to show itseect. A low ordering cost to carrying cost ratio (Factor C) is expected to lead to more frequent orders

Table 4Summary of the 251V experiment results

Run Factor Responses (based on 200 instances per run)

A B C D E % Deviation fromoptimum

% Optimumsolutions

dSM dGA nSM nGA

1 C 6 100 0.125 30 0.514 0.093 7 45.52 S 6 100 0.125 12 0.607 0.006 59.5 98.53 C 24 100 0.125 12 0.723 0.014 39.5 964 S 24 100 0.125 30 1.083 0.599 29.5 34.55 C 6 800 0.125 12 0.617 0.000 73 1006 S 6 800 0.125 30 2.728 0.269 9.5 42.57 C 24 800 0.125 30 2.113 0.104 4.5 518 S 24 800 0.125 12 3.014 0.025 29.5 96.59 C 6 100 1.000 12 0.633 0.002 31 99

10 S 6 100 1.000 30 1.045 0.199 29.5 60.511 C 24 100 1.000 30 1.014 0.136 3.5 38.512 S 24 100 1.000 12 0.728 0.008 57.5 9713 C 6 800 1.000 30 2.337 0.317 2 3114 S 6 800 1.000 12 3.219 0.000 28 10015 C 24 800 1.000 12 0.297 0.000 82.5 10016 S 24 800 1.000 30 2.877 0.556 4.5 40

Overall averages 1.472 0.146 30.7 70.7and the optimum plan will usually call for just in time ordering (an order in every period for its require-ments). Such a plan is easily obtainable by the MSM. On the other hand, when Factor C is at its high level,fewer orders will be placed and nding the optimum solution is harder. Therefore, the MSM performance isexpected to vary signicantly as Factor C varies.

Factor D (the ratio of the carrying cost to the backorder cost) is expected to have an eect similar to FactorC, but because backorders were limited to less than Q (for rounding the order to the nearest lower multiple ofQ), the eect of Factor D was minor. Finally, increasing the planning horizon (Factor E) is expected toincrease the deviations from the optimum solution as it leads to a larger solution space.

Results in Fig. 3 indicate that the most inuential factors on dSM are Factors C, A, and E, respectively.These results are consistent with the results reported by Gaafar and Choueiki (2000). The MSM has its worstperformance when Factor C is at its high level. On the other hand, the most inuential factor on dGA is Fac-tor E. The GA has its worst performance when Factor E is at its high level. This is a logical result as the highlevel of Factor E leads to a larger solution space that is expected to degrade the performance of the GA. Allother factors seem to have a minor eect on dGA which indicates the robustness of the GA.

Factor eects on the frequency responses (nSM and nGA) are similar to those displayed in Fig. 3, but are inthe opposite direction. In other words, a factor that increases the deviation from the optimum solution (dSMor dGA) decreases the frequency of obtaining the optimum solution (nSM or nGA).

A more formal analysis of the data is summarized in Table 5, which shows the statistically signicant fac-tors and interactions associated with the four responses in the experiment based on analysis of variance(ANOVA, Montgomery, 2001). The signicance level for all analyses is 1%. Signicant factors are indicatedby a check mark (

p). Some results in Table 5 are based on transformations that were needed to maintain the

ANOVA assumptions. When ANOVA results indicate that a factor is insignicant while an interaction involv-ing the same factor is signicant, it is customary to assume that the factor is still signicant, but that its sig-nicance is masked by the interaction. Such a situation is indicated with an asterisk in Table 5. To obtain an

440 L. Gaafar / Computers & Industrial Engineering 51 (2006) 433444estimate of the experimentation error for ANOVA, the 200 instances investigated in each run were randomlydivided into two sets of 100 instances, and the averages of these instances were used to provide two replicatesof the responses in each run.

Results in Table 5 conrm the conclusions of the visual analysis based on Fig. 3. They further point out thesignicance of several interactions. A signicant interaction indicates that the way a certain factor aects aresponse depends on the level of another factor. Fig. 4 depicts the eects of interaction AE on the two respons-es dSM and dGA. It is well documented that the SM is sensitive to demand variation (Gaafar & Choueiki,

Factor A (demand pattern) Factor B (batch size)

Factor C (ratio of ordering cost to carrying cost) Factor D (ratio of carrying cost to backorder cost)

Factor E (planning horizon)

Key:

Fig. 3. Factor eects on the deviation responses.

L. Gaafar / Computers & Industrial Engineering 51 (2006) 433444 441Table 5Summary of ANOVA results showing the signicant factors/interactions (0.01 signicance level)

Factor/interaction Responses

% Deviation from optimum % Optimum solutions

dSM dGA nSM nGA

Ap p p

B *p

*C

p pD * * *E

p p p pAC

p pAE

p p pBD

p pCE

p p

Key:p

signicant factor/interaction.* Factor masked by interaction.2000; Lee, Kramer, & Hwang, 1991; Zhiwei et al., 1994). This is clearly the case in Fig. 4. Also, a long planninghorizon leads to an increased decision frequency and an accumulation of deviations; hence, the MSM perfor-mance is expected to degrade with longer planning horizons. The interaction aspect here is that the demandpattern (Factor A) has an eect at the low level of Factor E only. All combinations of AE, except a constantpattern and a short horizon, lead to a bad performance as measured by the dSM response. For the dGAresponse, only the combination seasonal demand pattern and long planning horizon leads to a badperformance.

In addition to interactionAE, interactions AC, BD, andCE signicantly aect dSM. The signicance of theseinteractions may be explained based on the nature of the problem. The signicance of interaction AC indicatesthat the eect of seasonality on deteriorating the performance of the MSM is higher when Factor C is at its highlevel. As mentioned before, the low level of Factor C usually leads to JIT orders that are easy to obtain usingMSM. In this case, the ease of obtaining the optimum solution when Factor C is low dampens the seasonalityeect. The signicance of interactionBD is also expected as the batch size (B) is themain driver of backorder deci-sions. Accordingly, the higher the value of Factor B, the more signicant the eect of Factor D will be. The sig-nicance of the CE interaction is a result of the fact that the eect of a long planning horizon on degrading theMSM performance is more pronounced when the demand is seasonal than when it is constant.

Finally, results in Table 5 show that Factor D is signicant, but that its signicance is masked by interac-tions. Considering the two levels of Factor D and the fact that the high level implies that the carrying andbackorder cost are equal. It is expected that optimization would be harder at the high level of Factor D asthe results indicate.

A Effect of interaction AE on dSM

A Effect of interaction AE on dGA

Fig. 4. Interaction plots for the deviation response.

It is clear that the GA has a more robust performance than the MSM as it is aected by fewer factor andinteraction eects.

Comparisons between the two approaches have been so far based on averages. They indicate that GAobtains the optimum solution 70.7% of the time, while MSM obtains it 30.7% of the time. More signicantly,the average percentage deviation of the MSM is more than ten times that of the GA. Nevertheless, the averagepercentage deviation of the MSM is still less than 2%, and considering its simplicity and fast execution time,one may conclude that it is a better alternative. However, a more detailed picture is provided by the distribu-tion of the 3200 observations on the percentage deviation from the optimum solution over the three rangesshown in Table 6. The majority of the deviations of the GA are less than 1%, and very few deviations arein the range of 1% to 4%. On the other hand, for the MSM, a signicant percentage of the deviations arein the range of 14%, and more than 10% of the deviations exceed 4%. Deviations as high as 19.3% wereobserved for the MSM, while the maximum observed deviation for the GA did not exceed 3.7%.

Fig. 5 provides a further detailed comparison between the two approaches. The curve in Fig. 5 shows thevarious percentiles of the percentage deviations of the solutions obtained using the MSM from those obtainedusing the GA. For all 3200 instances a value was calculated based on the dierence between the two costsdivided by the minimum of the two. These values were sorted and percentiles were calculated. Fig. 5 showsthat the MSM provides a better solution than the GA in less than 5% of the instances by a maximum improve-ment of about 3.5%. On the other hand, the GA provides a better solution in more than 60% of the cases by asmush as 19.3%. In the remaining instances, the two algorithms obtained the same cost. Results in Table 6 andFig. 5 show clearly that using the GA is signicantly more advantageous in many cases.

Finally, a detailed investigation of Table 4 indicates that the performance of the MSM at the low level ofFactor C is relatively reasonable compared to that of the GA. In the eight runs (1600 instances) satisfying thiscondition, the MSM deviates from the optimum solution by an average of less than 0.8%. For the same runs,the GA deviates from the optimum solution by an average of about 0.13%. Considering the simplicity and fast

442 L. Gaafar / Computers & Industrial Engineering 51 (2006) 433444execution time of the MSM, the use of the MSM may be recommended in this case. On the other hand, TheGA performs at its best when Factor F (planning horizon) is low regardless of the levels of all other factors.This is expected as a smaller planning horizon leads to a smaller solution space. In the 8 runs (1600 instances)

Table 6Distribution of the % deviation from the optimum solution over three ranges

Algorithm Range of the % deviation from the optimum solution

D 6 1% 1% < D 6 4% D > 4%MSM 56.6% 32.5% 10.9%GA 96.0% 4.0% Fig. 5. Percentiles of the percentage deviations of the MSM solutions from the GA solutions.

L. Gaafar / Computers & Industrial Engineering 51 (2006) 433444 443satisfying this condition, the GA deviated from the optimum solution by about 0.007%. For the same 8 runs,the MSM deviated from the optimum by an average of about 1.23%.

The results obtained in this research are, in general, much better than those obtained by Shittu (2003).However, a formal comparison was not conducted because Shittus results used a dierent reference and someslightly dierent factors. Also, Shittus approach of using post processing to determine the order quantitiesdoes not accurately test the performance of the GA.

5. Conclusions

This paper investigated the applicability of GAs to the dynamic lot sizing problem with batch ordering andbackorders. A special GA was developed with a new 012 coding approach that explicitly models the batchordering decisions. The performance of the GA was compared to that of a modied version of the Silver-Mealheuristic. Comparisons were based on the average percentage deviations from the optimum cost and the fre-quency of obtaining the optimum cost. The GA had an impressive performance with an average deviation ofabout 0.15% from the optimum over 3200 instances. The deviation was never larger than 3.7% in any of the3200 instances. On the other hand, the MSM had an average deviation of about 1.5% from the optimum, withdeviations as large as 19.3%.

Within the investigated ranges, Factors A (demand pattern) and E (planning horizon) have the most inu-ential eects on the responses. Their eects on dSM and nSM are more pronounced. Factor C (the ratio of theordering cost to the carrying cost) also has a signicant eect on the MSM responses. Factors B (batch size)and D (the ratio of the carrying cost to the backorder cost) were involved through interactions in aectingsome responses. The GA demonstrated a more robust performance than the MSM as it was aected by fewerfactors and interactions.

In general, the MSM performs at its best when the ratio of the ordering cost to the carrying cost is small.On the other hand, the GA performs at its best when the planning horizon is short (small solution space). Thismay indicate that a better GA performance may be obtained by using an adaptive approach that increases thenumber of objective function evaluations as the solution space gets larger.

Many assumptions were imposed in this paper to facilitate obtaining the optimum solution to evaluate theGAs performance. The excellent performance of the GA in the evaluated instances may now motivate futureresearch to investigate the performance of the GA on other batch ordering problems for which the optimumsolution does not exist or is hard to obtain. Future plans are to extend the developed GA to the multi-level andthe rolling horizon cases.

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Applying genetic algorithms to dynamic lot sizing with batch orderingIntroductionApproach: The genetic algorithmThe evaluation experimentResultsConclusionsReferences