A Math-heuristic for the Multi-Level

Capacitated Lot Sizing Problem with Carryover

No Author Given

No Institute Given

Abstract. We present a math-heuristic algorithm for the lot sizingproblem with carryover. The proposed algorithm uses mathematical pro-gramming techniques in a metaheuristic fashion to iteratively solve smallerportions of the original problem. More specifically, we draw ideas fromthe corridor method to design and impose exogenous constraints onthe original problem and, subsequently, we solve to optimality the con-strained problem using a MIP solver. The algorithm iteratively buildsnew corridors around the best solution found within each corridor and,therefore, explores adjacent portions of the search space. In the attemptof fostering diversification while exploring the original search space, wegenerate a pool of incumbent solutions for the corridor method and,therefore, we reapply the corridor method using different starting points.The algorithm has been tested on instances of a standard benchmark li-brary and the reported results show the robustness and effectiveness ofthe proposed scheme.

1 Introduction

The Multi-Level Capacitated Lot Sizing Problem (MLCLSP) finds applicationin production systems where setup times are significant (see, e.g., [2, 14]). Anexample of such a production system is the job-shop system, i.e., a productionsystem where operations require specialized manufacturing processes such ascostumer adapted orders or small batch jobs. The MLCLSP is an extension ofthe Capacitated Lot Sizing Problem (CLSP) [9, 13]. In the problem, multipleitems (products) must be produced following a known Bill of Material (BOM).The objective is to find an optimal production plan that minimizes production,setup, and inventory costs, and delivers optimal lot sizes and production periodsfor each product. In the problem, the external demands (volume) are given forpredefined periods. When an item is produced, machine (resource) capacity isconsumed, which is limited. When there is a change of production from one itemto another, it is also necessary to account for setup costs as well as setup times.Incurring in setup times means reducing the available machine capacity. Lastly,whenever there is an excess of production over the current demand in a givenperiod for a specific item, inventory is built up and, consequently, inventoryholding costs must be paid.

This work deals with the MLCLSP with setup carry-over (MLCLSP-CO),also called MLCLSP with linked lot sizes [2, 14, 10, 3]. The MLCLSP-CO intro-

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duces some scheduling information into the classical lot sizing problem. When-ever an item is produced over two consecutive periods on the same machine,its setup in the second period can be discounted, under the assumption thatits setup state is “carried over” the two periods. This implies that a partialscheduling is introduced, i.e., which item is produced as the last one in the firstperiod and the first one in the second period is established [9]. Under the clas-sical “small bucket vs. big bucket” classification, MLCLSP-CO is a big-bucketmodel [10, 12], as it is an extension of the CLSP. Thus, the MLCLSP-CO, beingan extension of the standard CLSP, is NP-hard [8].

In [14] a lagrangean heuristic is presented. First, inventory balance con-straints, capacity constraints, and carryover constraints are relaxed in a la-grangean fashion. A Wagner-Within algorithm is used to compute a lower bound.Next, a pool of repair schemes is used to find a feasible solution.

Similarly, in [3] a corridor method algorithm paired with dynamic program-ming and a lagrangian based heuristic is used to generate a (possible infeasible)solution. Subsequently, three different repair mechanisms are implemented toreach a feasible solution.

In this paper we present a math-heuristic algorithm for the MLCLSP-CO.The proposed algorithm hybridizes mathematical programming techniques witha metaheuristic to iteratively solve smaller portions of the original problem.More specifically, we utilize the corridor method together with a mathematicalprogramming formulation which is solved with a MIP solver.

The paper is structured as follows. First, in the next section, we providea mathematical programming formulaton of the problem. Then, the basic ideaof the algorithm is outlined in Section 3. A simple scheme for the generationof feasible incumbent solutions is presented in Section 4. Section 5 presentscomputational results on a set of benchmark instances and, finally, Section 6offers a few final remarks.

2 A Formal Model for the MLCLSP-CO

In this section, we present a mixed integer formulation for the problem. TheMLCLSP-CO model presented in this work is based on some models from theliterature, particularly [2], [14], [10], [3], and [12].The following notation is usedto formulate the MLCLSP-CO.

Indices and index sets:j is the items index, with j = 1, 2, ..., J ;m is the machine index, with m = 1, 2, ..., M ;t is the period index, with t = 1, 2, ..., T ;Γ (m) is the set of items produced on machine m;Λ(j ) is the set of predecessors of item j;Σ(j ) is the set of successors of item j.

Parameters:

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fjt is the setup cost of item j in period t;cjt is the unitary production cost of item j in period t;hjt is the unitary holding cost of item j in period t;tpjt is the unitary resource usage of item j in period t;tsjt is the setup time of item j in period t;bmt is the capacity of resource m in period t;djt is the external demand of item j in period t;γij is the number of units of item i required to produce one unit of item j;Z big number;sj is the physical inventory of item j at the beginning of the planning horizon.

Decision variables:xjt is the binary setup variable for item j in period t;ωjt is the binary setup carry-over variable for item j at the beginning of periodt;yjt is the production quantity (lot size) of item j in period t;sjt is the inventory of item j at the end of period t.

A MIP formulation for the problem is given below (we assume a lead time of1 period). It captures the following features: Constraints (1)–(3) are balanceconstraints; Constraint (4) is a capacity constraint, taking into account bothvariable production times and setup times; Constraint (5) establishes the relationbetween production and setup variables; Constraint (6) ensures that only onecarryover per machine is fixed; Constraints (7) and (8) account for the relationbetween setup and carryover variables; Constraint (9) ensures that consecutivecarryover for the same item are allowed only when no other items are setup onthat machine; and Constraint (10) makes sure that no carryover is establishedin the first period.

3 General Idea and Algorithm

The general idea of the proposed algorithm relies on the combination of mathe-matical programming techniques with metaheuristic features. More precisely, weuse mathematical programming techniques in a metaheuristic fashion, followingthe paradigm of “math-heuristic” algorithms.

Small-size instances of the MLCLSP-CO can be solved to optimality usingmodern MIP solvers. However, when the size of the instances grows, such solversfail to deliver optimal solutions in a timely fashion. Nevertheless, in many cases,the same MIP solvers are able to quickly provide good quality solutions in a shortcomputational time when used on constrained versions of the original problem.The key idea of the proposed approach is to iteratively define smaller versionsof the MLCLSP-CO problem by imposing exogenous constraints on the originalproblem. Thus, the MIP solver is used to solve, perhaps to optimality, suchsmaller problems. The mechanism guiding the generation of subproblems is, inturn, guided by a metaheuristic.

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minT

X

t=1

JX

j=1

[cjtyjt + fjt(xjt − ωjt) + sjthjt]

s.t.

sjt−1 + yjt = sjt + djt +X

k∈Σ(j)

γjkxkt+1,j = 1, . . . , J,t = 1, . . . , T − 1,

(1)

sjT−1 + yjT = sjT + djT , j = 1, . . . , J, (2)

sj0 = sj −X

k∈Σ(j)

yk1, j = 1, . . . , J, (3)

X

j∈Γ (m)

tpjtyjt + tsjt(xjt − ωjt) ≤ bmt,m = 1, . . . , M,t = 1, . . . , T,

(4)

yjt ≤ Zxjt,j = 1, . . . , J,t = 1, . . . , T,

(5)

X

j∈Γ (m)

ωjt ≤ 1,m = 1, . . . , M,t = 2, . . . , T,

(6)

ωjt ≤ xjt,j = 1, . . . , J,t = 2, . . . , T,

(7)

ωjt ≤ xjt−1,j = 1, . . . , J,t = 2, . . . , T,

(8)

Z (2− ωjt − ωjt+1) + 1 ≥X

i∈Γ (m)

xit,t = 2, . . . , T,m = 1, . . . , M,j ∈ Γ (m),

(9)

ωj1 = 0, j = 1, . . . , J, (10)

yjt, sjt ≥ 0,j = 1, . . . , J,t = 1, . . . , T,

(11)

xjt, ωjt ∈ 0, 1 ,j = 1, . . . , J,t = 1, . . . , T.

(12)

The corridor method (CM) is a hybrid metaheuristic proposed by [11]. Whilesome successful applications of the method have already been reported (e.g., [5,6]), its full exploitation to solve complex combinatorial optimization problemsstill needs to be advanced. The central idea of the CM is the use of a (possiblyexact) method to solve smaller “versions” of the original optimization problem,i.e., to find the optimal solution on a limited portion of the solution space.

Given a problem P belonging to the class of NP-hard problems, a very largesolution space X , and an exact method M that could be used to solve problem P

to optimality if X were not too large, the method receives as input an incumbentsolution x ∈ X . The CM defines a “corridor” around the incumbent solution byimposing exogenous constraints on P and, therefore, by cutting out portions ofthe solution space X . The nature of the exogenous constraints is problem and

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method specific; however, ideally, the resulting neighborhood built around x,i.e., N (x), should be such that it could be explored in (pseudo) polynomial timeusing method M.

Let us now consider how the CM can be applied to the MLCLSP-CO. Inthe following, let us indicate with X the feasible space of the problem , whilewe indicate with M a branch and bound algorithm implemented within a MIPsolver. Let us suppose that we are given an incumbent solution (x,y, s, ω)I . Webuild a neighborhood around the incumbent solution by imposing the followingconstraint:

J∑

j=1

T∑

t=1

xIjtxjt ≥ δni (13)

where δ ∈ [0, 1] is a parameter used to define the corridor width, and ni =∑J

j=1

∑Tt=1 xI

jt accounts for the number of setups in the incumbent solution.Therefore, the corridor constraint establishes a lower bound in the number ofsetups of a possible solution. It is worth noting, though, that such constraint is“flexibly” fixing some of the setup variables to 1. Constraint (13) cuts out of thesolution space X all those (originally feasible) solutions that do not fix to 1 atleast δ% of the setup variables currently set to 1 in the incumbent solution. Thedifference between arbitrarily fixing to 1 δ% of the variables in MLCLSP-CO andthe introduction of the corridor constraint lies in the fact that, with the corridor

constraint, we let the MIP solver the flexibility to choose which variables haveto be fixed to 1.

It is easy to see that, by increasing (decreasing) the value of δ, we reduce(enhance) the corridor width and, consequently, the size of the search space.Constraint (13) can also be seen as a type of distance, or diversity, measure and,therefore, the addition of such constraint to MLCLSP-CO generates a neighbor-hood around the incumbent solution (x,y, s, ω)

I, defined as:

Nδ(xI) =

(x,y, s, ω) ∈ X :

J∑

j=1

T∑

t=1

xIjtxjt ≥ δni

(14)

In Figure 1 we present an outline of the overall algorithm. The algorithm isiteratively repeated using different incumbent solutions for every iteration. Wecreate and store each incumbent solution in Ω. Such set is used to check that noincumbent solution is fed to the corridor method more than once.

Each iteration of the corridor method (step S2) stops when one of the twostopping criteria is met, i.e., either a maximum running time for the explorationof the neighborhood is reached, or a maximum number of new feasible solutionshave been collected. It is worth noting that the incumbent solution is used asstarting point by the MIP solver and its objective function value as a cut. There-fore, every new feasible solution found by the MIP solver within the corridor hasan objective function value better than the incumbent itself.

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S1. Initialization

– Generation of the incumbent solution. A simple scheme (Sec-tion 4) is used to generate a new feasible incumbent (x,y, s, ω)I ,i.e., a feasible solution such that (x,y, s, ω)I /∈ Ω.

– Add (x,y, s, ω)I to Ω.

S2. Corridor Method“

(x,y, s, ω)I , δ, δm”

– Add the corridor constraint (13) and solve the resulting MIPproblem.

– Stopping criteria of the MIP solver:• maximum running time• maximum number feasible solutions

– (x,y, s, ω)c is the best solution found in Nδ(xI)

S3. Update“

(x,y, s, ω)I , δ, δm”

– If (x,y, s, ω)c is better than (x,y, s, ω)I , then set (x,y, s, ω)c asincumbent and go to S2.

– If no solution better than the incumbent has been found:• If 0.9δ ≥ δm, set δ ← 0.9δ (enhance the corridor) and go to

S2.• If a minimum δ value has been reached (δm), stop the corridor

method and go to S1.

Fig. 1. Outline of the proposed algorithm.

The overall algorithm terminates once a predefined stopping criterion isreached, i.e., (i) a maximum running time; or (ii) a maximum number of in-cumbent solutions in Ω.

4 Incumbent Solution Generation: A Metaheuristic

Scheme

As presented in Section 3, one of the main ingredients of the proposed algorithmis the generation of an incumbent solution, around which a corridor is going tobe build. In this section, we illustrate how a metaheuristic method can be usedto generate a set of incumbent solutions for the corridor method.

The cross entropy method (CE) has been presented by [7] and its applicationto the CLSP has been proposed by [4]. The CE can be used to generate apopulation of N binary matrices X1, . . . , XN , to identify the periods in whicheach item is setup. Given a binary matrix Xi, we fix the corresponding xjt

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variables accordingly, hence fixing periods in which a setup is scheduled. Theproblem obtained after fixing the setup variables is composed of continuousvariables (s and y) and binary variables (ω) strongly connected to the valueof the setup variables, as established by constraints (6)– (8). Such restrictedproblem can quickly be solved to (near) optimality using a standard MIP solver.If the restricted problem obtained after fixing the setup variables has a feasiblesolution, we associate the objective function value of the best solution foundto the corresponding stochastic matrix Xi. Otherwise, if the derived problem isinfeasible, we simply discard the stochastic matrix Xi and we generate a newstochastic matrix.

Let us define a probability matrix P = pjt, where pjt is the probabilitythat a setup is scheduled for item j in period t. In order to exploit the stochasticnature of the proposed approach, we generate a population of Xi of size N , alldrawn under the same initial probability matrix P . Inspired by the spirit of CE,we then use the “Maximum Likelihood Estimator” method to generate a newprobability matrix P 1 that better describes the best individuals of the currentpopulation. Therefore, we update the set of probabilities pjt in order to reflecthow likely it is that, in a high-quality solution, a setup of item j in period t isscheduled. Once a new probability matrix P 1 is obtained, a new population ofsize N can be drawn under such matrix. Hopefully, such matrix better describeshigh quality solutions obtained in the previous generation and, therefore, thechances of obtaining high quality individuals under the new matrix are higher.

This process of “probability matrix update” and “population generation” canbe iterated until either the P matrix converges to a binary matrix (therefore,the process converges to a unique solution) or a pre-specified maximum numberof iterations has been reached.

The “Maximum Likelihood Estimator” method is used to modify probabili-ties pjt in such a way that the new stochastic matrix better reflects the chancesof obtaining high quality solutions. Let us assume that, based upon the currentstochastic matrix P k, we have generated a population of size N , i.e., X1, . . . , XN .Let us now find, within the current population, the objective function value ofthe (1 − ρ)% quantile, i.e., the value γ for which ρ% of the population havea better objective function value and (1 − ρ)% of the population have a worseobjective function value.

We modify the probability matrix using the following updating rule:

pjt =

N∑

i=1

xijt × If(Xi)≤γ

ρN(15)

where xijt indicates component (j, t) of matrix Xi, f(Xi) is the objective function

value of the ith solution (Xi, ωi,yi, si), and If(Xi)≤γ is the indicator function,whose value is 1 if f(Xi) ≤ γ and 0 otherwise.

Remark. As pointed out by [7], in order to prevent the CE from convergingtoo fast to a suboptimal solution, a smoothing factor α (typically 0.7 ≤ α ≤ 0.9)

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could be used in the updating rule. Therefore, to foster a more thorough explo-ration of the solution space, at each iteration k we use the following updatingrule:

pk+1jt = αpjt + (1 − α)pk

jt. (16)

5 Computational Results

In order to evaluate the quality of the proposed approach, we ran the algorithmon benchmark instances from the literature. More specifically, we solved someof the instances of Tempelmeier et al. [14]. In this section, we present results onthe first and fourth classes of such instances. The two classes had been chosenfirstly, to see for a simpler class to which extent the approach could providemeaningful results, and secondly, to see whether those results are also confirmedwhen considering the hardest instances among those literature data. Class onecovers one general and one assembly product structure, 10 products, 3 resources,and 4 periods. A total of 480 instances are contained in this class. Class four ischaracterized by 20 products, 6 resources, and 16 periods and composed of 240instances. These instances have been downloaded from the author’s web page,and recreated following the instructions provided by the authors themselves, asindicated in [14].

All computational experiments have been carried out on a 2.0 GHz Pentium4 Workstation with 2Gb of RAM running Linux. The algorithm has been codedin C++ and compiled with the GNU g++4.3 compiler.

The values of the algorithmic parameters N , ρ, and α have been determinedusing the response surface methodology [1], as presented in [4]. The MIP solverused is Cplex 11. As reported by [14], many of these instances could not besolved to optimality using a MIP solver within a time limit of 3600 seconds.

The stopping criteria for both the overall algorithm and the corridor methodare (These values have been chosen such as to find a balance between solutionquality and CPU time.):

– Maximum running time of the overall algorithm: 300 seconds.– Maximum number of incumbent solutions in Ω: 10.– Maximum running time of the constrained MIP (corridor method): 10 sec-

onds.– Maximum number of new solutions visited within a corridor: 10.– Initial value of δ : 0.1.– Maximum value of δ : 0.2.

In Table 1 we summarize the computational results on these 720 benchmarkinstances. The first three columns describe the instance, in terms of number ofitems, number of periods and number of resources (e.g., machines), respectively.Column four reports the results of the algorithm presented in [14]. The computa-tional time of these authors is provided as an average over all the instances, andreported as being 0.15 seconds. Since many of the benchmark instances could

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not be solved to optimality within the established time limit, we measure thesolution quality computing the dual gap, as in [14]. Dual gap and computationaltime are used to measure the algorithmic performance of the proposed algorithm.The lower bound values used to compute the distance from optimality have beenobtained solving the MLCLSP-CO with Cplex 11 for 3600 seconds. Therefore,for each instance of the class, we computed the optimality gap as

γ =zH − lb

lb× 100,

where zH is the objective function value of the heuristic solution found by theproposed algorithm, and lb is the best lower bound obtained by Cplex.

Instance Details Tempelmeier et al. Proposed Algorithm

No. Items No. Periods No. Machines Gap Gap Time

10 4 3 1.39 0.10 2.720 6 16 21.90 8.56 59.3

Table 1. Results on first and fourth classes of benchmark instances. Time measuredin CPU seconds and averaged over each instance class.

From the table we can confirm the robustness and the effectiveness of theproposed scheme in dealing with these classes of instances of the MLCLSP-CO.If we compare the running time of the proposed algorithm with that of [14], wecan observe that the average time of the proposed algorithm is of 31 seconds asopposed to 0.15 seconds of [14]. However, such an increase in computational timeallows to sensibly reduce the gap and, therefore, to find better quality solutions.

6 Conclusions

We have presented a math-heuristic algorithm for the multi-item multi-periodcapacitated lot sizing problem with carryover, which is a variation of the standardlot sizing problem where some scheduling information is included in the finalsolution. More specifically, given any two consecutive periods, it is possible tointroduce information about the last item scheduled in one period and the firstitem scheduled in the subsequent period. Therefore, by extending a setup of anitem over two consecutive periods, a setup in the second period can be saved,hence leading to more realistic solutions.

We have proposed a three-step algorithm, where (i) an incumbent solution isgenerated through the use of the cross entropy method; (ii) an iterative corridormethod algorithm is applied starting from the incumbent solution and movingin the direction dictated by the best solution found in each corridor; and (iii)an updating phase, in which either the corridor is enhanced or the method isreapplied starting from a different portion of the original search space.

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The method has been tested on a limited set of benchmark instances, moreprecisely, on “extreme” classes one and four of a larger set of benchmark instancesfrom [14]. Our results show that the algorithm is effective, both from the pointof view of the solution quality as well as from the perspective of computationaltime.

References

1. Box, G. and Wilson, K. On the Experimental Attainment of Optimum Conditions.Journal of the Royal Statistical Society, Series B - 13:1–45 (1951)

2. Buschkuhl, L., Sahling, F., Helber, S., Tempelmeier, H. Dynamic capacitated lot-sizing problems: a clasification and review of solution approaches. OR Spectrum.(2008) doi: 10.1007/s00291-008-0150-7

3. Caserta, M., Ramirez, A., Voß, S., Moreno, R. A hybrid algorithm for the multilevel capacitated lot sizing problem with setup carryover. In S. Voß, J. Pahl andS. Schwarze (eds.), Logistik Management. Physica, Heidelberg, 123-138 (2009)

4. Caserta, M. and Quinonez, E. A Cross Entropy-Lagrangean Hybrid Algorithmfor the Multi-Item Capacitated Lot-Sizing Problem with Setups. Computers and

Operations Research. 36(2), 530–548 (2009)5. Caserta, M., Voß, S., Sniedovich, M. Applying the corridor method to a block

relocation problem OR Spectrum. (2009) doi: 10.1007/s00291-009-0176-5.6. Caserta, M., Voß, S. A corridor method-based algorithm for the pre-marshalling

problem Applications of Evolutionary Computing, Lecture Notes Computer Sci-

ence 5484, 788-797, M. Giacobini et al. (eds.), (2009)7. De Boer P. and Kroese, D. P. and Mannor, S. and Rubinstein, R. Y. A tutorial on

the cross-entropy method Annals of Operations Research. 134(1), 19–67 (2005)8. Maes, J., McClain, J., Van Wassenhove, L.. Multilevel capacitated lotsizing com-

plexity and LP-based heuristics. European Journal of Operational Research. 53,131–148 (1991)

9. Quadt, D., Kuhn, H. Capacitated lot-sizing with extensions: a review. 4OR: A

Quarterly Journal of Operations Research. 6(1), 61–83 (2008)10. Sahling, F., Buschkuhl, L., Tempelmeier, H., Helber, S. Solving a multi-level

capacitated lot sizing problem with multi-period setup carry-over via a fix-and-optimize heuristic. Computers and Operations Research . 36(9), 2546–2553 (2009)

11. Sniedovich, M. and Voß, S. The Corridor Method: a Dynamic Programming In-spired Metaheuristic. Control and Cybernetics. 35(3), 551–578 (2006)

12. Suerie, C., Stadtler, H. The capacitated lot-sizing problem with linked lot sizes.Management Science. 49(8), 1039–1054 (2003)

13. Tempelmeier, H., Destroff, M. A lagrangian-based heuristic for dynamic multilevelmultiitem constrained lotsizing with setup times. Management Science. 42(5),738–757 (1996)

14. Tempelmeier, H., Buschkuhl, L. A heuristic for the dynamic multi-level capac-itated lotsizing problem with linked lotsizes for general product structures. OR

Spectrum. 31, 385–404 (2009)