MIP formulation and heuristics for multi-stage capacitated lot-sizing and scheduling problem with availability constraints

Journal of Manufacturing Systems 32 (2013) 392–401

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Journal of Manufacturing Systems journal homepage: www.elsevier.com/locate/jmansys

Technical paper

MIP formulation and heuristics for multi-stage capacitated lot-sizing and scheduling problem with availability constraints Reza Ramezanian a,∗ , Mohammad Saidi-Mehrabad a , Parviz Fattahi b a b

Department of Industrial Engineering, Iran University of Science and Technology, Tehran, Iran Department of Industrial Engineering, Bu-Ali Sina University, Hamedan, Iran

a r t i c l e

i n f o

Article history: Received 10 August 2012 Received in revised form 8 November 2012 Accepted 16 January 2013 Available online 28 February 2013 Keywords: Dynamic lot-sizing and scheduling Flow shop Availability constraint Sequence-dependent setup Mixed-integer programming MIP-based heuristic

a b s t r a c t In this paper, the problem of lot-sizing and scheduling of multiple product types in a capacitated flow shop with availability constraints for multi-period planning horizon is considered. In many real production systems, machines may be unavailable due to breakdowns or preventive maintenance activities, thus integrating lot-sizing and scheduling with maintenance planning is necessary to model real manufacturing conditions. Two variants are considered to deal with the maintenance activities. In the first, the starting times of maintenance tasks are fixed, whereas in the second one, maintenance must be carried out in a given time window. A new mixed-integer programming (MIP) model is proposed to formulate the problem with sequence-dependent setups and availability constraints. The objective is to find a production and preventive maintenance schedule that minimizes production, holding and setup costs. Three MIP-based heuristics with rolling horizon framework are developed to generate the integrated plan. Computational experiments are performed on randomly generated instances to show the efficiency of the heuristics. To evaluate the validity of the solution methods, problems with different scales have been studied and the results are compared with the lower bound. Computational experiments demonstrate that the performed methods have good-quality results for the test problems. © 2013 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.

1. Introduction Lot-sizing and scheduling are the most important tasks carried out in manufacturing systems with goals such as optimal lot-sizes, optimal sequence of products on machines, minimum total costs, balanced machine utilization rate, and short average customer waiting time. These issues have a considerable impact on the productivity of production systems. Due to interrelationships that exist between various levels of the planning problem, there are many trade-offs between the decisions made at different planning modules. To obtain globally optimal solutions, the inter-dependencies between different planning functions should be considered and the planning decisions should be taken simultaneously [1]. Lot-sizing, scheduling and maintenance planning have separately received considerable attention in operations research literature. In these domains, issues of lot-sizing and scheduling modeling and maintenance modeling are studied both from theoretical and practical views but, integrating of lot-sizing and scheduling with availability constraints has received much less

∗ Corresponding author. Tel.: +98 9125397764. E-mail addresses: [email protected], [email protected] (R. Ramezanian), [email protected] (M. Saidi-Mehrabad), [email protected] (P. Fattahi).

attention. In this area it is usually assumed that machines are available and that the system will function at its maximum performance during the whole planning horizon. However, in many real situations, machines may be unavailable due to breakdowns or preventive maintenance (PM) activities [2]. Moreover, in maintenance planning models the effects of maintenance activities on production capacity are often disregarded [3]. The primary goal of preventive maintenance activities is to prevent the failure of equipment before it actually occurs. It can retain or restore a system to an acceptable operating condition. Although PM activities take time that could otherwise be used for production, delaying them may increase the probability of machine failure [4], therefore integrating lot-sizing and scheduling with preventive maintenance is necessary to model real manufacturing conditions. This problem has many applications in assembly workshops such as automobile assembly lines [5], automotive parts manufacturing, assembly lines for household electric appliances, assembly workshop in an auto body plant, printed circuit board manufacturing system [6] and others. For example, automobile assembly lines consist of determining lot-sizes and sequences of automobiles (such as buses, trucks and cars) to be assembled. Ghodratnama et al. [7] considered the maintenance activities, deterioration of jobs and the learning effect in a single-machine scheduling problem. Chen [8] studied the integrated problem of production and preventive maintenance for an imperfect production process. They tried to determine the optimal

0278-6125/$ – see front matter © 2013 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jmsy.2013.01.002

R. Ramezanian et al. / Journal of Manufacturing Systems 32 (2013) 392–401

inspection interval and frequency and production quantity for an imperfect production system. The term “Lot-sizing” is used in discrete manufacturing environments to define the process of determining the production quantity of each product over a finite multi-period planning horizon. A lot is the quantity of a given product manufactured on a machine continuously without interruption after its setup. The goal of lotsizing is to determine the optimal production amount by balancing the tradeoffs between production, inventory, backorder, and setup costs. Since Wagner and Whithin [9] published their seminal paper on lot-sizing and scheduling problems in 1958, it has remained a topic of interest for researchers and practitioners. For a more detailed review of the relevant work done in this area, interested readers are referred to Drexl and Kimms [10], Karimi et al. [11] and Buschkühl et al. [12]. Models proposed for dynamic lot-sizing and scheduling may be classified into small bucket and big bucket problems. “Big Bucket” problems have long time periods in which several products can be manufactured while “Small Bucket” problems divide the planning horizon into small time periods which limits the number of items produced in a single period. These models enable simultaneous lot-sizing and scheduling. The capacitated lot-sizing problem (CLSP) is a large bucket problem in which several products may be produced in a given macro-period. Small bucket problems are classified into the discrete lot-sizing and scheduling problem (DLSP), the continuous setup lot-sizing problem (CSLP), the proportional lot-sizing and scheduling problem (PLSP) and the general lot-sizing and scheduling problem (GLSP). In DLSP at most one product can be produced per micro-period. In CSLP, similar to the DLSP, only one item may be produced per period and production does not use the full capacity of that period (the ‘all-or nothing’ assumption is removed). PLSP modeling allows up to two products in single period [11], GLSP proposed by Drexl and Kimms [10] and Fleischmann and Meyr [13] integrates lot sizing and scheduling of several products in macro-periods by subdividing the macro-periods into a predefined number of non-overlapping micro-periods. The computational complexity of models is increased using this procedure. Lin et al. [14] studied scheduling in multiple flow line manufacturing system. They integrated the lot sizing and scheduling decisions with a simulation module to obtain more accurate schedules for printed circuit board production systems using a utility function to determine lot size. To find a suitable rule for the scheduling, four sequencing rules are evaluated by the authors. Buschkühl et al. [12] present a review of four decades of research on dynamic lot-sizing with capacity constraints. With a focus on the multi-level capacitated lot-sizing problem (MLCLSP). MLCLSP is a big bucket model in which several products can be processed in a given macro-period, but it cannot determine lot-sizes and schedules simultaneously. Multi-level production planning models of the literature in small bucket time are the multilevel discrete lot-sizing and scheduling problem (MLDLSP) [15], the multi-level proportional lot-sizing and scheduling problem (MLPLSP) [16] and the multi-level general lot-sizing and scheduling problem (MLGLSP) [17]. MLDLSP and MLPLSP models enable simultaneous lot-sizing and scheduling, but limit the number of products to be produced per period. The MLGLSP model proposed by Fandel and Stammen-Hegene [17] attempts to unite the advantages of the MLPLSP and MLCLSP based on subdividing the macro-period into a fixed number micro-period. It integrates lot sizing and the scheduling of several products in each period. Due to, the high level of complexity of their proposed model, caused by the great number of variables, only problems with a low number of products, machines and periods can be optimally solved. Mohammadi et al. [18] proposed a mathematical formulation model for lot-sizing and scheduling in a flow shop environment with sequence-dependent

393

setups. The artificial setup concept used to formulate this problem similar is to Clark and Clark [19] which assumes that during every planning period, N (number of products) setups occur. This presumption increases the computational complexity of the problem, therefore they developed some MIP-based heuristics for solving the problem. Later, Mohammadi and Fatemi Ghomi [20] developed a genetic algorithm-based heuristic to solve the problem. Ramezanian et al. [21] studied a multi-product multi-period lot-sizing and scheduling problem in capacitated permutation flow shop with sequence-dependent setups and proposed a more efficient mathematical model for the problem and used two MIP-based heuristics for solving related problem. The two-machine flow shop scheduling problem with availability constraint is first studied by Lee [22]. He proved that this problem, with an availability constraint only on one machine is NP-hard. Ma et al. [23] provided a thorough survey of deterministic scheduling problems with machine availability constraints. Hadda [24] considered two machine flow shop scheduling problem with several availability constraints where only the first machine is unavailable. He presented a polynomial-time approximation scheme for the problem, under the resumable scenario, to minimize the makespan. Aggoune et al. [25] proposed a genetic algorithm approach to solve flow shop scheduling problem with availability constraints. They considered the makespan and the total weighted tardiness as performance measures. Aggoune [2] considered a flow shop scheduling problem with availability constraints and proposed a heuristic based on a genetic algorithm and a tabu search to solve the makespan minimization of the problem. Later, Aggoune and Portmann [26] developed a temporized geometric approach to solve the problem with two jobs. Besbes et al. [27] presented an approximate approach based on genetic algorithm for hybrid flow shop scheduling problems under availability constraints with the makespan minimization as the objective function. Zhao and Tang [28] studied two-machine no-wait flow shop scheduling with deteriorating jobs and machine availability constraints in terms of makespan minimization. They assumed that there are unavailability intervals on only one machine. As mentioned before, a primitive form of big bucket time models, which is known as CLSP, cannot determine the scheduling decisions [10,11]. Recent studies on big time buckets try to determine at least the first and the last lots by linking adjacent periods. In this paper, a new big bucket time model approach is proposed that is able to determine the lot sizing and scheduling problems together. This new approach considerably reduces the complexity of the model. This modeling approach can use in other environments such as single stage system with single or multi machine workstation. In this paper, a multi-product and multi-period capacitated production system with sequence-dependent setups and availability constraints that has been organized as a flow shop is considered. The main contribution of this paper is providing an efficient MIP model for modeling the integrated lot-sizing and scheduling problem with preventive maintenance, which can simultaneously achieve a production plan and schedule. This paper is the first attempt to successfully formulate integrated lot-sizing and scheduling with availability constraints for the flow shop problem. Three MIP-based heuristics are developed to solve proposed model. Also, to evaluate the performance of heuristics two lower bounds are presented. This paper has the following structure. In Section 2, mathematical model for formulation of multi-stage lot-sizing and scheduling problem with availability constraints is completely described. Section 3 provides heuristic approaches to solve the considered problem, computational experiments are described in Section 4, and the conclusions and future researches are presented in Section 5.

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2. Problem formulation This section provides an MIP model to formulate the problem of integrated lot sizing and scheduling in capacitated flow shop environment with sequence-dependent setups and availability constraints. 2.1. Assumptions The main assumptions for the problem of multi-level capacitated lot-sizing and scheduling are as follows: • Simultaneous lot-sizing and scheduling are considered. • Several products can be manufactured in each period. • Each product requires multiple operations that are in a designated series of work centers in a capacitated flow shop system. • External demands for the finished products are known and deterministic. • Machines are not available at all times in planning horizon. Breakdowns and preventive maintenance are allowed. • Maintenance tasks are flexibly organized. In other words, completion of preventive maintenance occurs in predefined time window. • Setting up for a certain item incurs a setup time and relevant setup cost, and is sequence-dependent. • The setting up of a machine must be completed in a period. • Shortages are not permitted. • A component cannot be produced in a period until the production of its required components is finished. In other words, production at a given level can only be started if a sufficient amount of the required items from the previous level is available. This is called vertical interaction. • At the beginning of the planning horizon, all machines are set up for predefined products. • The triangle inequality holds for setups, i.e. it is never faster to change over from one product to another by means of a third product. • At any time, each machine can process at most one job or PM task. • At any time, each job can be processed on at most one machine.

Notation

Definition

Wijm

Sequence-dependent setup cost of switching from product i to j on machine m. We also assume that the relation between setup costs and times can be considered as: wijm = fw × Sijm where fw denotes the opportunity cost per time unit The early completion time of the lth preventive maintenance task on machine m in period t The late completion time of the lth preventive maintenance task on machine m in period t Inventory of product j in the beginning of planning horizon A large real number (bigM → ∞)

EMlmt LMlmt IjM 0 bigM

Decision variables Inventory of finished product j at level m in period t Ijmt Quantity of product j produced on machine min period t Xjmt Starting time of product j on machine m in period t SOjmt Completion time for product j on machine m in period t COjmt Completion time of the lth preventive maintenance task on FMmlt machine m in period t A binary variable that is equal to 1 if job j is processed immediately Yijmt after job i on machine m in period t, 0 otherwise A binary variable that is equal to 1 if job j is processed before PMjlmt maintenance task l when processing on machine m in period t, 0 otherwise A binary variable that is equal to 1 if Xjmt > 0, 0 otherwise Zjmt

The proposed MIP model is as follows: Min

 

pjmt · Xjmt +

j∈J t ∈T m∈M

+

   i∈J

j∈J

Wijm · Yijmt

s.t. IjM(t−1) + XjMt = IjMt + djt

Xjmt ≤ bigM · Zjmt

∀j ∈ N, t ∈ T, m = 1, . . . , M − 1

∀j ∈ N, m ∈ M, t ∈ T

∀j ∈ N, m ∈ M, t ∈ T

(4) (5)

∀j ∈ N, m ∈ M, t ∈ T

(6)

∀j ∈ N, t ∈ T, m = 2, . . . , M

(7)

SOjmt ≥ COj(m−1)t

/ j, SOjmt ≥ COimt + Sijm · Yijmt − bigM · (1 − Yijmt ) ∀i, j ∈ N, i = m ∈ M, t ∈ T

Parameters Number of different products (set of product types) N Number of machines (production levels) (set of machine types) M Planning horizon (in terms of number of planning periods) (set of T periods) Number of different preventive maintenance tasks on machine m Lm (set of maintenance tasks) Production coefficient. Required capacity of machine m to produce bjm one unit of product j (in time units) Duration of the lth preventive maintenance task on machine m tlm Processing time (capacity) available for machine m in period t (in Cmt time units) External demand for the jth product at the end of period t djt Production cost per unit of finished product j on machine m in pjmt period t Holding cost per unit of finished product j on machine m hjm Sequence-dependent setup time of switching from product i to j Sijm on machine m

(2)

(3)

In order to formulate the model, the following notations and data are used:

Indices of product types Index of maintenance tasks Machine index Planning period index

∀j ∈ N, t ∈ T

Ijm(t−1) + Xjmt = Ijmt + Xj(m+1)t

COjmt = SOjmt + bjm · Xjmt

Indices i, j l m t

(1)

m∈M t ∈T

2.2. Mathematical model

Definition

hjm · Ijmt

j∈J m∈M t ∈T

i= / j

COjmt ≤ Cmt

Notation



 

Yijmt ≥

i ∈ J,i = / j j∈J



(8)



Zjmt − 1 ∀m ∈ M, t ∈ T

(9)

j∈J

Yijmt ≤ Zjmt

∀j ∈ N, m ∈ M, t ∈ T

(10)

Yjimt ≤ Zjmt

∀j ∈ N, m ∈ M, t ∈ T

(11)

i ∈ J,i = / j



i ∈ J,i = / j

Xjmt , Ijmt , COjmt , SOjmt ≥ 0; IjM0 = 0 ∀j ∈ N, m ∈ M, t ∈ T

(12)

Yijmt , Zjmt = {0, 1} ∀i, j ∈ N, m ∈ M, t ∈ T

(13)

The objective function (1) considers the minimization of the sum of production cost, holding cost, and sequence-dependent setup costs. Constraints (2) and (3) determine the relationship among demand, inventory levels, and planned quantities. Constraints (2) ensure the demand of products supply in each period. Constraints given in Eq. (3) determine that in a network, total of in-flows to each

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node should be equal to out-flows from that node. Constraints (4) ensure that whenever Xjmt > 0, the indicator variable Zjmt is automatically set to 1. Constraints (5) are capacity constraints for each machine in each period. Constraints (6) correspond to the computation of the completion time for planned production. Constraints (7) force to start the processing of planned product only when it has been completed on the precedent machine (vertical interaction). Constraints (8) force to start the processing of each product only when its precedent product has been completed on the same machine in addition to its setup time. Constraints (9)–(11) determine the sequence of products. According to constraints (10) and (11), job sequence is included product j if it produces in period t. Relations (12) and (13) represent the type of variables. 2.3. Flexible availability constrains in flow shop environment One of the most common assumptions in the production planning literature is that machines are continuously available for their use throughout the production planning horizon. However, this assumption ignores real industrial conditions, since a machine may be stopped for several reasons, such as failures and maintenances. Preventive maintenance is a set of preplanned actions performed to prevent the potential failure of equipment before it actually occurs [29]. It can retain or restore a system to an acceptable operating condition. Taking into account that machines and processors are a fundamental part of production system and maintenance costs are a great percentage of the total operation cost [30], it is necessary to have coordination and integration between production planning and maintenance planning. There are two strategies to plan joint production scheduling and preventive maintenance in the literature [31]. The first strategy consists of two steps: First scheduling the production jobs, then inserting the maintenance tasks. The second one consists of simultaneously scheduling both maintenance and production activities. In this paper, the second approach is adapted to schedule products and maintenance tasks on machines. Flexible availability constraints are considered for all machines, i.e., maintenance tasks on machines are completed in theirs predefined time window. SOjmt − FMlmt + bigM · PMjlmt ≥ 0 ∀j ∈ N, l ∈ Lm , m ∈ M, t ∈ T (14)

FMlmt − tlm − COjmt + bigM · (1 − PMjlmt ) ≥ 0 ∀j ∈ N, l ∈ Lm , m ∈ M, t ∈ T

395

environment has been formulated successfully. However, due to the inherent complexity of the problem, only small-size problems can be solved exactly using optimization approaches such as “branch and bound”, “branch and cut” and so on. The proposed mathematical model is not a practical approach for solving large instances of the problem. Therefore, it is necessary to provide a computable lower bound close to the optimal solution to test the accuracy of the heuristics used in this paper (which are described in the next section). We note that the solution obtained by heuristics is an upper bound on the optimal solution. In this subsection, two lower bounds are developed for the problem. The first lower bound is obtained by relaxing all the binary variables of the problem. After relaxing the model’s binary variables, with regards to the non-integer values of Yijmt , the right hand side of inequality set (8) will have negative values. Thus, this set of inequality will have no significant effect on the solution. Since most of the binary variables of the problem are associated with the sequencing variables Yijmt and maintenance variables PMjlmt (N2 .M.T and N.M.L.T binary variables, respectively), in the second lower bound; only these two variables are relaxed. In order to evaluate the performance of two lower bounds against the exact solution, a problem sizes with the range of (N.M.L.T) = (3 × 3 × 2 × 3) have been considered. Table 1 lists the results obtained for five instances of this problem sizes. The evaluating approach is used for accuracy of lower bound, which is similar to that of Gupta and Magnusson [32] and Mohammadi et al. [18]. For given problem size, five instances are randomly generated. The required parameters are extracted from the following uniform distributions: bjm ≈ U(1.5,2), djt ≈ U(0,150), hjm ≈ U(0.2,0.4), pjmt ≈ U(1.5,2), wijm ≈ U(35,70), Sijm ≈ U(35,70), Cmt ≈ U(a1 ,b1 ),EM ≈ U(a2 ,b2 ) and LM ≈ U(a3 ,b3 ); a1 = 300N + 200(m − 1), b1 = 300N + 300(m − 1), a2 = (300N + 200(m − 1)) × (l/(L + 1)), b2 = (300N + 200(m − 1)) × (l/(L + 1)) + 50, a3 = (300N + 200(m − 1)) × (l/(L + 1)) + 50 and b3 = (300N + 200(m − 1)) × (l/(L + 1)) + 100. The proposed model and lower bounds are coded as GAMS models using GAMS IDE (ver 23.6.5) and solved using GLPKGAMS module. All tests were conducted on a PC with Intel Core Duo CPU 2 GHzand1 GB of RAM. Table 1 and Fig. 1 confirm the advantages of the second lower bound in comparison to another lower bound. Thus, the second lower bound is used to compare heuristic algorithms.

(15) 3. Heuristic algorithms based on rolling horizon

∀l ∈ Lm , m ∈ M, t ∈ T

(16)

3.1. Idea and description of heuristics

FMlmt ≥ 0 ∀l ∈ Lm , m ∈ M, t ∈ T

(17)

PMjlmt = {0, 1} ∀j ∈ N, l ∈ Lm , m ∈ M, t ∈ T

(18)

Since manufacturing environments are dynamic, the production quantities should be continually updated in response to disturbances such as delay in the delivery of raw materials, equipment failure, demand changes, etc. Rolling horizon heuristic methods are usually used in dynamic environments and especially in dynamic lot-sizing and scheduling problems. In dynamic production planning, demand gradually emerges in the planning horizon. While the estimation of the demands of the early periods is more precise, the demands of the late periods are approximate. If, in the integrated lot-sizing and scheduling problem, there is a subset of early periods that include detailed scheduling decisions, the late planning periods only include high-level decisions (i.e. production quantities), or relaxation formulation is used for these periods, the computational complexity will decrease.

EMlmt ≤ FMlmt ≤ LMlmt

For each machine, if a product is processed before/after a maintenance task then the finish/start time of that product must be less/greater than the maintenance start/finish time which are mentioned in the constraint sets (14) and (15). The constraint set (16) ensures that a maintenance task is performed in the corresponding time window. 2.4. Development of lower bounds In the previous sections, the problem of integrated lot-sizing and scheduling with availability constraints in a flow shop

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R. Ramezanian et al. / Journal of Manufacturing Systems 32 (2013) 392–401

Table 1 Comparison between the lower bounds and optimal solutions in problem size N = 3, M = 3, L = 2 and T = 3. No.

Exact solution

The first lower bound

The second lower bound

O.V.

CPU time

O.V.

CPU time

Gap

O.V.

CPU time

Gap

1 2 3 4 5

4532.6 4800.5 4417.4 4147.8 4608.9

67.58 54.38 51.39 60.86 85.20

3767.3 3996.6 3713.1 3454.8 3814.1

1.49 1.57 1.20 1.37 1.22

16.89% 16.75% 15.94% 16.71% 17.24%

4290.2 4529.6 4159.3 3899.6 4432.6

13.91 14.20 13.05 12.25 14.09

5.35% 5.64% 5.84% 5.98% 3.82%

Average

4501.4

63.88

3749.2

1.37

16.71%

4262.3

13.50

5.33%

The values in CPU time’s column are the elapsed computational time in seconds and the values in Gap’s column are the difference between the objective values (O.V.) of the lower bound against the exact solution. Gap = 100 × (O.V.Exact − O.V.Lower bound )/O.V.Exact .

Using rolling horizon heuristics for large-sized MIP problems greatly reduces the computational complexity by replacing binary variables with continuous variables for late periods. This approach is useful, even when all the parameters are completely known. This approach first decomposes the model into a set of smaller MIP models, each of which has a small number of binary variables, and then the decision variables are then iteratively determined [18,19,33]. According to the framework of Merece and Fonton [33], at each step of the iterative approach, the planning horizon is divided into three sections (initial, central, and final). For every specified iteration k:

At the end of each iteration k, one period is rolled for entering all sections of the algorithm into the new iteration. The mentioned procedure is terminated when iteration T is carried out. The last iteration of the algorithm determines all decision variables for the overall planning horizon. Fig. 2 illustrates the iterative procedure based on rolling horizon for two successive iterations.

• The initial section consists of (k − 1) periods. In this section, according to the previous iterations of the algorithm, values are partially or completely assigned to the decision variables, according to the selected freezing strategy. • The central section only includes the kth planning period. For this period, the problem is considered as a whole. In this section, all binary variables that are associated with this planning period are accounted in the model as binary forms. • The final section includes the remaining periods, from period (k + 1) to period T, which is simplified according to the chosen simplification strategy.

3.2.1. Heuristic algorithm 1 (HA1) Beginning section: Only binary variables are frozen over the beginning section. Thus, production quantities are free and can be determined through the algorithm. Central section: Consists of one period, the whole problem is considered. Ending section: Binary variables and also constraint set (8) are relaxed for the ending section. Using a simplified representation for the ending section in the rolling horizon makes the model less difficult to solve therefore permits the solution of larger problems. We should note that the beginning section does not exist for the first iteration.

3.2. Heuristic algorithms In this paper, three heuristics based on the previously mentioned iterative framework are used to solve the related problem.

3.2.2. Heuristic algorithm 2 (HA2) The only different between heuristics HA1 and HA2 is in the freezing policy in the beginning section. In heuristic algorithm 2, all decision variables related to the beginning section are completely frozen. Both of the central and ending sections for this heuristic are similar to the heuristic algorithm 1.

Fig. 1. Comparative results for the problem size N = 3, M = 3, L = 2 and T = 3.

3.2.3. Heuristic algorithm 3 (HA3) As mentioned above, the whole problem is considered for the central section. Thus, each iteration of previous heuristics (HA1 and HA2) has M × (N2 + N + 1) binary variables in this section. When the number of binary variables increases in a MIP model, the time needed to solve the problem with the exact procedure

Beginning section

Centeral section

Ending section

Freezing strategy

Whole model

Simplification strategy

Iteration k Period 1

Period 2

Period k-1

Period k

Period k+1

Period T

t

Period 1

Period 2

Period k-1

Period k

Period k+1

Period T

t

Iteration k+1

Fig. 2. Illustration of the iterative procedure.

R. Ramezanian et al. / Journal of Manufacturing Systems 32 (2013) 392–401

397

explode exponentially. Therefore, a faster solution approach that has smaller MIPs is needed to solve larger problems. In heuristic algorithm 3 (HA3), the search space is limited to the permutation flow shop problem. Furthermore, the sequence vector of products for all machines is similar and the lot sizes of each product are assumed to be constant on all machines. Therefore, Xjmt , Ijmt , Zjmt and Yijmt variables are reduced to Xjt , Ijt , Zjt and Yijt , respectively. Other variables and parameters are similar to Section 2. This simplified version is still practical in production systems. In each iteration of HA3 there exist only (N2 + N + 1) binary variables. The modified model is as follows: Min

 

pjmt · Xjt +

j∈J t ∈T m∈M

+

   i∈J

j∈J



hjm · Ijt

j∈J m∈M t ∈T

wijm · Yijt

Fig. 3. Comparative results obtained by the heuristics for the problem size N = 3, M = 3, L = 2 and T = 3.

(19)

m∈M t ∈T

i= / j s.t. Ij(t−1) + Xjt = Ijt + djt Xjt ≤ bigM · Zjt COjmt ≤ Cmt

∀j ∈ N, t ∈ T

∀j ∈ N, t ∈ T

∀j ∈ N, m ∈ M, t ∈ T

COjmt = SOjmt + bjm · Xjt SOjmt ≥ COj(m−1)t

∀j ∈ N, m ∈ M, t ∈ T

∀j ∈ N, t ∈ T, m = 2, . . . , M

(20) (21) (22)

 

Yijt ≥

(24)

i ∈ J,i = / j j∈J



(25)



Zjt − 1 ∀t ∈ T

(26)

j∈J

Yijt ≤ Zjt

∀j ∈ N, t ∈ T

(27)

Yjit ≤ Zjt

∀j ∈ N, t ∈ T

(28)

i ∈ J,i = / j



i ∈ J,i = / j

SOjmt − FMlmt + bigM · PMjlmt ≥ 0 ∀j ∈ N, l ∈ Lm , m ∈ M, t ∈ T (29)

FMlmt − tlm − COjmt + bigM · (1 − PMjlmt ) ≥ 0 ∀j ∈ N, l ∈ Lm , m ∈ M, t ∈ T

EMlmt ≤ FMlmt ≤ LMlmt

(30)

∀l ∈ Lm , m ∈ M, t ∈ T

(31)

Xjt , Ijt , COjmt , SOjmt , FMlmt ≥ 0; Ij0 = 0 ∀j ∈ N, l ∈ Lm , m ∈ M, t ∈ T

Yijt , Zjt , PMjlmt = {0, 1} ∀i, j ∈ N, l ∈ Lm , m ∈ M, t ∈ T

4. Computational results

(23)

SOjmt ≥ COimt + Sijm · Yijt − bigM · (1 − Yijt ) ∀i, j ∈ N, i = / j, m ∈ M, t ∈ T

the ending section (simplification strategy). Thus, the simplification strategy chosen for the ending section ignores setup mechanisms from period k + 1 to period T. In each iteration, the only binary variables involved in the model are related to the central section, thus the computational effort required for solving the problem is considerably reduced.

(32)

(33)

The procedure used in heuristic algorithm 3 (HA3) is similar to HA1. In each iteration k the production planning and scheduling variables are determined from period 1 to period k (periods covering the beginning and central section). It should be noted that the decision variables related to period 1 up to period k − 1 have been set by previous iterations. In all heuristics, binary variables are relaxed for

In this section, the computational experiments are used to evaluate the performance of the heuristics in finding good quality solutions are described. For this purpose, the algorithm is tested versus the selected lower bound presented in Section 2.4. To evaluate and compare the performance of the developed heuristics, 20 problems with different sizes were selected. For each problem size, five instances were randomly obtained using the uniform distributions described in Section 2.4, and the average results were considered as a measure for the lower bound and the heuristic methods. These test problems are classified into three classes: small size integrated lot-sizing and scheduling problems (SILS1:5), medium size integrated lot-sizing and scheduling problems (MILS6:15) and large size integrated lot-sizing and scheduling problems (LILS16:20). Problem hardness is dependent on the number of products, machines, maintenance tasks and periods. To solve mentioned problems, we coded the mathematical model, the lower bounds (presented in Section 2) and the heuristic algorithms (described in Section 3) with GAMS IDE (ver 23.6.5) software and solved using GLPK GAMS solver. These programs were run on a PC that has Intel Core Duo CPU 2 GHz, with 1 GB RAM. The required parameters for all computational experiments are similar to those used in Section 2.4. To evaluate the performance of the heuristics against the selected lower bound, a specific problem size with (N.M.L.T) = (3 × 3 × 2 × 3) is studied in more detail. Table 2 and Fig. 3 show the results obtained for this problem size. Table 2 and Fig. 3 show that the percentage difference of the objective value obtained from the heuristic HA1 against the lower bound is less than those from the other heuristics HA2 and HA3. In order to evaluate the effect of flexible availability constraints against constant availability constraints on objective functions, five instances of problem size (N.M.L.T) = (3 × 3 × 2 × 3) are examined in more detail. Two objective functions consisting of total flow time and total cost are studied and the obtained results are shown in Table 3 and Table 4, respectively. As mentioned in constraint set (25), the completion time of maintenance tasks at each machine is planned for a predefined time window. For constant condition, two cases are considered. The completion time of maintenance task on each machine is equal to the pre-specified early completion time (FMlmt = EMlmt ) or late completion time (FMlmt = LMlmt ).

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Table 2 Comparison between the lower bound and heuristics’ results in problem size N = 3, M = 3, L = 2 and T = 3. No.

The second lower bound

Heuristic algorithm 1

O.V.

CPU time

O.V.

CPU time

Gap

Heuristic algorithm 2 O.V.

CPU time

Gap

Heuristic algorithm 3 O.V.

CPU time

Gap

1 2 3 4 5

4290.2 4529.6 4159.3 3899.6 4432.6

13.91 14.20 13.05 12.25 14.09

4568.3 4879.6 4429.1 4173.1 4781.0

13.20 12.08 12.08 12.36 11.80

6.48% 7.73% 6.49% 7.01% 7.86%

4658.3 4999.6 4536.1 4319.917 4854.626

13.03 13.19 12.67 13.05 12.55

8.58% 10.38% 9.06% 10.78% 9.52%

4957.3 4931.6 4569.1 4423.2 5040.7

0.79 0.80 0.77 1.30 0.83

15.55% 8.87% 9.85% 13.43% 13.72%

Average

4262.3

13.50

4566.2

12.30

7.11%

4673.7

12.90

9.66%

4784.4

0.89

12.28%

Table 3 Obtained results for flexible or constant availability constraints in problem size N = 3, M = 3, L = 2 and T = 3 considering total flow time as an objective function. No.

Flexible maintenance

The early maintenance, FMlmt = EMlmt

The late maintenance, FMlmt = LMlmt

Total flow time

CPU time (s)

Total flow time

CPU time (s)

Gap

Total flow time

CPU time (s)

Gap

1 2 3 4 5

4404.5 4908.9 4819.5 4537.0 5141.0

31.09 57.27 36.20 25.22 237.81

4644.5 5327.1 4852.7 4611.0 5290.5

140.02 118.98 17.84 21.28 132.19

5.45% 8.52% 0.69% 1.63% 2.91%

4638.71 5043.9 4831.8 4557.0 5284.6

39.28 29.19 19.78 22.38 189.88

5.32% 2.75% 0.26% 0.44% 2.79%

Average

4762.2

77.52

4945.2

86.06

3.84%

4871.2

60.10

2.31%

Table 4 Obtained results for flexible or constant availability constraints in problem size N = 3, M = 3, L = 2 and T = 3 considering total cost as an objective function. No.

Flexible maintenance

The early maintenance, FMlmt = EMlmt

The late maintenance, FMlmt = LMlmt

Total cost

CPU time (s)

Total cost

CPU time (s)

Gap

Total cost

CPU time (s)

Gap

1 2 3 4 5

4464.7 4390.9 4204.8 4059.2 4549.1

16.34 16.03 14.30 19.53 17.03

4535.1 4392.3 4239.0 4153.1 4617.4

17.66 15.63 23.11 15.97 17.73

1.58% 0.03% 0.81% 2.31% 1.50%

4552.10 4413.0 4428.7 4135.1 4598.1

16.39 19.16 17.53 16.27 58.66

1.96% 0.50% 5.33% 1.87% 1.08%

Average

4333.7

16.65

4387.4

18.02

1.25%

4425.4

25.60

2.15%

TFT 5500

5000

4500

4000

Flexible maintenance The early maintenance The late maintenance

3500 1

2

3

4

5

Problem No.

Fig. 4. Comparative total flow time in flexible or constant maintenance conditions for the problem size N = 3, M = 3, L = 2 and T = 3.

Total cost 5000

4500

4000

Flexible maintenance The early maintenance The late maintenance

3500 1

2

3

4

5

Problem No.

Fig. 5. Comparative total cost in flexible or constant maintenance conditions for the problem size N = 3, M = 3, L = 2 and T = 3.

Tables 3 and 4 and Figs. 4 and 5 show that when the flexible availability constraint was considered in the model better results were obtained for both objective functions. The average results obtained from three trials of each of the 20 mentioned problems have been listed in Table 5. Table 5 compares the objective values and the run times elapsed to solve the problems using heuristics and also the lower bound. The “Gap” column indicates the difference of the objective values resulting from the specified heuristic method relative to the selected lower bound. A lower percentage shows better performance for the solution methods. As can be seen from Table 5 and Fig. 6, only solutions for the small-sized problem class can be obtained using the three heuristics. The objective value for the heuristic HA1 is less than that obtained for the other heuristics HA2 and HA3. Feasible solutions were not found after 3600 s of computing time for medium-sized and large-sized problems with HA1 and HA2. Heuristic algorithm 3 (HA3) is able to solve these problem sizes in a reasonable time. Thus, the heuristic method HA1 has a better performance for small-sized problems and the heuristic method HA3 can be used for large-sized problems. The obtained production plan that does not consider machine availability constraints is most likely unfeasible or sub-optimal. The following instance is used to clarify the impact of preventive maintenance on lot-sizing. An instance of the capacitated lot sizing and scheduling problem in a flow shop with three stages is considered to satisfy the customer demands for five products in two periods. The required data for the related problem are given in Table 6. A capacity of 1000 units of time is considered for each machine in each period.

R. Ramezanian et al. / Journal of Manufacturing Systems 32 (2013) 392–401

399

Table 5 Comparative results obtained by the second lower bound and the heuristics. Problem No.

Size (N.M.L.T)

SILS1 SILS2 SILS3 SILS4 SILS5 MILS6 MILS7 MILS8 MILS9 MILS10 MILS11 MILS12 MILS13 MILS14 MILS15 LILS16 LILS17 LILS18 LILS19 LILS20

3.3.2.3 5.3.2.3 3.5.2.3 3.3.2.5 5.5.2.5 7.5.2.5 5.7.2.5 5.5.2.7 7.7.2.7 10.5.2.5 5.10.2.5 5.5.2.10 10.7.2.7 7.10.2.7 7.7.2.10 10.10.2.10 15.10.2.10 10.15.2.10 10.10.2.15 15.15.2.15

The second lower bound

Heuristic algorithm 1

Heuristic algorithm 2

Heuristic algorithm 3

CPU time (s)

Gap

CPU time (s)

Gap

CPU time (s)

Gap

CPU time (s)

13.50 14.06 14.28 15.72 20.77 26.97 41.14 311.47 >3600a 58.72 1461.88 >3600a >3600a >3600a >3600a >3600a >3600a >3600a >3600a >3600a

5.99% 7.38% 8.07% 8.26% 6.96% 10.58% – 5.84% – – – – – – – – – – – –

12.30 12.77 12.22 12.97 16.83 1291.27 – 197.17 – – – – – – – – – – – –

9.66% 13.01% 14.32% 14.61% 15.39% 16.15% – 11.46% – – – – – – – – – – – –

12.90 18.94 14.13 13.22 71.14 157.97 – 139.53 – – – – – – – – – – – –

12.28% 13.91% 14.29% 19.78% 14.88% 21.12% 18.24% 18.95% 20.79% 20.29% 21.12% 19.60% 22.19% 20.11% 20.77% 18.84% 25.28% 25.73% 23.19% 28.02%

0.89 2.02 1.78 1.39 15.22 27.88 14.97 15.19 29.42 35.92 21.66 19.92 117.41 60.94 36.70 822.49 953.44 1724.67 1624.33 1807.50

–, means that a feasible solution has not been found after 3600 s of elapsed time. a Means that finding the optimum value for the third lower bound requires more than 3600 s and the objective value at this time is recorded. Table 6 Example of five products, three machines and two periods problem. Job

1 2 3 4 5

Product demand (djt ) Period

Process time (bmj ) Machine

1

2

1

2

3

14 53 123 0 77

118 36 0 83 55

1.6 1.8 1.8 1.9 1.5

1.9 2 1.7 1.9 1.6

1.5 1.7 1.8 1.8 1.6

First, this instance is optimally solved to determine the optimal lot size of products and their sequence without consideration of availability constraints. Second, we explore the impact of machine failure and maintenance activities on the planed lot sizes. The optimal lot size and schedule of products without considering machine availability constraints are depicted in Fig. 7. A general review of this figure shows that: (a) the obtained production plan is

feasible, (b) in each period, maximum completion time of products at each machine is smaller than available capacity for that machine and (c) the minimum objective function (total costs) is equal to 3304.56. To examine the effect of machine unavailability and maintenance activities, we set machine 1 in period 1 and machine 2 in period 2 to be unavailable for PM activities with 100 and 50 units of time after processing product 1. Also, after 200 units of time, machine 1 in period 2 is unavailable to restore the machine to an acceptable operating condition for 100 unite of time. The impact of these issues on the mentioned problem is illustrated in Fig. 8. As can be observed from Fig. 8, the obtained production plan in the previous step is unfeasible considering machine availability constraints. Overtime is needed to satisfy customer demand and to produce products 4 and 5. Obviously overtime, production costs are higher than those in regular time, therefore, considering the availability constraint in the related problem is important for both the feasible and the optimal production plan. Also, this issue has considerable impact on the productivity of the system and in manufacturing qualified products.

Fig. 6. Comparative results obtained by the heuristics for the test problems.

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Fig. 7. Lot-sizing and scheduling for the problem size (5 × 3 × 2) without availability constraints (Total cost = 3304.56).

Fig. 8. Effect of maintenance activities on lot-sizing and scheduling: (a) for period 1 and (b) for period 2.

5. Conclusions and future work Multi-product and multi-period integrated lot-sizing and scheduling for capacitated flow shop environment with availability constraints has been discussed in this paper. One of the most important and practical conditions in real manufacturing system is machine unavailability due to breakdowns or preventive maintenance tasks. An efficient mathematical model is proposed to formulate this problem. Assumptions such as capacity constraint and sequence-dependent setup costs and times have been considered in the problem. Moreover, three MIP-based heuristics based on iterative procedures are also used to solve problem instances. The first two heuristics are based on the original model and the last heuristic is based on the permutation flow shop problem which considers a similar sequence vector of products on all machines. For small-sized problems, solutions can be acquired with the three heuristics but the heuristic algorithm 1 (HA1) results in better objective values. For non-small size problems, the heuristic algorithm 3 (HA3) is able to solve these problem sizes in reasonable computing time. Thus, the heuristic method HA1 has a better performance for small-sized problem and the heuristic method HA3 can be used for large-sized problems. Also, results show that the objective value can be improved by considering flexible availability constraints. One area for future research is the development of the proposed model to include real conditions of manufacturing environments such as limited intermediate buffer space, lot transportation constraints, etc. Meta-heuristic approaches for solving combinatorial optimization problems, and the multiobjective optimization approach could be other subjects for future research.

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