International Journal of Management Science and Engineering Management
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A bi-objective multi-item capacitated lotsizing model: Two Pareto-based meta-heuristic algorithms Esmaeil Mehdizadeh, Vahid Hajipour & Mohammad Reza Mohammadizadeh To cite this article: Esmaeil Mehdizadeh, Vahid Hajipour & Mohammad Reza Mohammadizadeh (2015): A bi-objective multi-item capacitated lot-sizing model: Two Pareto-based meta-heuristic algorithms, International Journal of Management Science and Engineering Management, DOI: 10.1080/17509653.2015.1086965 To link to this article: http://dx.doi.org/10.1080/17509653.2015.1086965
Published online: 01 Nov 2015.
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Date: 07 November 2015, At: 22:16
International Journal of Management Science and Engineering Management, 2015 http://dx.doi.org/10.1080/17509653.2015.1086965
A bi-objective multi-item capacitated lot-sizing model: Two Pareto-based meta-heuristic algorithms Esmaeil Mehdizadeha, Vahid Hajipourb and Mohammad Reza Mohammadizadeha a
Faculty of Industrial and Mechanical Engineering, Qazvin Branch, Islamic Azad University, Qazvin, Iran; bYoung Researchers and Elite Club, Qazvin Branch, Islamic Azad University, Qazvin, Iran
ARTICLE HISTORY
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ABSTRACT
Lot-sizing problems (LSP) form a class of production planning problems in which available quantities are always considered as decision variables in the production plan. The goal of this paper is to present a multi-item capacitated lot-sizing problem (MICLSP) with setup times, safety stock deficit costs, demand shortage costs – both backorder and lost sale states – and different manners of production. Although a considerable amount of research has concentrated on model development and solution procedures in the terms of single-objective problems in the past decade, to make the model more realistic this paper develops a bi-objective mathematical programming model with two conflicting objectives including: (1) minimizing the total cost considered by the production plans including production costs with different manners of production, inventory costs, safety stock deficit costs, shortage costs and setup costs; (2) minimizing the required storage space. Considering that the proposed model is NP-hard, we propose two novel Pareto-based multi-objective meta-heuristic algorithms called multi-objective vibration damping optimization (MOVDO) and the non-dominated ranking genetic algorithm (NRGA) for the literature on LSP. In order to validate the performance of the proposed MOVDO and NRGA, a non-dominated sorting genetic algorithm (NSGA-II), one of the most common multi-objective metaheuristic algorithms, is applied. The optimal solutions are also reported to justify the results. Finally, we calibrate both algorithms by robust response surface methodology (RSM); then, the results are analysed on some test problems, both graphically and statistically.
1. Introduction Production planning problems consist in deciding how to transform raw material into final goods in such a way as to satisfy demand at minimum cost. The lot-sizing problem (LSP) is a crucial step and well-known optimization problem in production planning involving time-varying demand for a set of N items over T periods. In industrial applications, considering different factors may make the best decisions more complicated. For instance, considering multi-items can lead to it’s being impossible to satisfy demand. Moreover, safety stock is also a complicating constraint as a target to aim at rather that an industrial constraint to satisfy (Tempelmeier & Derstroff, 1996). The capacitated lot-sizing problem (CLSP) is a typical example of a big bucket problem, where many different items can be produced with the same resource in one time period. In fact, CLSP is an important challenge for industrial companies because it has a strong impact on their performance in terms of customer service quality and operating costs. However, production planning often proves itself to be a very complex task. The classical CLSP consists in determining the amount and timing of the production of products in the planning horizon: the outcome is a production plan giving for each planning period the quantity (lot size) of each item that should be produced. However, detailed warehouse space decisions are not
CONTACT Esmaeil Mehdizadeh
[email protected]
© 2015 International Society of Management Science and Engineering Management
Received 22 October 2014 Accepted 17 August 2015 KEYWORDS
Lot-sizing problem; multiobjective; MOVDO; NRGA; NSGA-II; RSM JEL CLASSIFICATION
C44; C52; C60; C61; C63
integrated in the CLSP. The usual approach is therefore to solve the CLSP first and to solve an inventory control problem for each period separately afterwards. Production planning typically includes three time scopes for decision making: long-term, medium-term and short-term. In long-term planning, the concentration mostly involves making strategic decisions on product, equipment, facility location, and resource planning. Medium-term planning often involves making decisions on material requirements planning, determining production quantities, and lot-sizing decisions during the planning period. In short-term planning, decisions usually involve daily scheduling of operations such as job sequencing or control in a workshop (Karimi, FatemiGhomi, & Wilsonb, 2003). This paper concentrates on medium-term production planning and lot-sizing decisions. In the former, Wagner and Whitin (1958) and Manne (1958) introduced various types of lot-sizing problem in terms of model development and solving methodologies. Following these, the single-item problem has attracted special interest for its relative simplicity and for its importance as a sub-problem of some more complicated lot-sizing problems (Kazan, Nagi, & Rump, 2000). In the literature, production planning models involving multiple items, restrictive capacities, and significant setup times, which occurred frequently in industrial situations, are formulated to determine optimal outputs. Loparic, Pochet, and Wolsey (2001) developed valid inequalities for the sin-
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gle-item uncapacitated lot-sizing problem with sales instead of fixed demands and lower bounds on stock variables. Aksen, Altinkemer, and Chand (2003) introduced a profit maximization version of the well-known Wagner–Whitin model for the deterministic uncapacitated single-item lot-sizing problem with lost sales. Absi and Kedad-Sidhoum (2007) proposed a multi-item capacitated lot-sizing problem (MICLSP) with setup times and safety stock in which demand can be totally or partially lost. They also presented mixed integer programming heuristics based on a planning horizon decomposition strategy to find a feasible solution. Following this, Absi and Kedad-Sidhoum (2008) developed the above model with consideration of shortage costs. Moreover, they presented a fast combinatorial separation algorithm within a branch-and-cut framework to solve the proposed model. Absi and Kedad-Sidhoum (2009) proposed an MICLSP with setup times, safety stock deficit costs and demand shortage costs. To solve their model, they first proposed a Lagrangian relaxation of the resource capacity constraints; then, a dynamic programming algorithm was developed to solve the induced sub-problem. Mehdizadeh and Kivi (2014a) proposed a mixed integer programming model for the single-item capacitated lot-sizing problem with setup times, safety stock, demand shortages, outsourcing and inventory capacity. Three meta-heuristic algorithms, called simulated annealing (SA), vibration damping optimization (VDO) and harmony search (HS), have been used to solve the proposed model. Also, Mehdizadeh and Kivi (2014b) proposed a new mixed integer programming model for the multi-item capacitated lot-sizing problem with setup times, safety stock and demand shortages in closed-loop supply chains with returned products consideration. With regard to expanding the applicability of these problems in industrial operations, LSPs represent challenges to be solved owing to their combinatorial nature. Chen and Thizy (1990) proved that the MICLSP with setup times is strongly NP-hard. Many researchers have attempted to solve MICLSP to very close to optimality (Sural, Denizel, & Van Wassenhove, 2009; Tempelmeier & Derstroff, 1996). Thus, they were not quite successful in solving large-scale problems because they could not anticipate the number of cutting planes that needed to be generated, or the number of iterations required in a branch-and-bound approach. One group of researchers developed heuristics to solve large-scale problems (Berretta & Rodrigues, 2004; Han, Tang, Kaku, & Mu, 2009; Tang, 2004; Xiao, Kaku, Zhao, & Zhang, 2011). Today, many realistic problems involve simultaneous optimization of several objectives (Coello, Lamont, & Van Veldhuizen, 2007). Regarding the aforementioned work on multiple-objective LSPs, a vast variety of solution methodologies including exact and approximation techniques has been utilized to find Pareto solution sets of different multi-criteria lot-sizing models. Among multi-objective algorithms, the non-dominated sorting genetic algorithm (NSGA-II) is a commonly used Pareto-based algorithm (Deb, Pratap, Agarwal, & Meyarivan, 2002). Furthermore, the non-dominated rank genetic algorithm (NRGA) is another multi-objective meta-heuristic introduced by Al Jadaan, Rao, and Rajamani (2008) that can be used to find Pareto front solutions. Rezaei and Davoodi (2011) developed two multi-objective mixed integer nonlinear models for multi-period LSPs involving multiple products and multiple suppliers. The first model represents this problem in situations
where shortage is not allowed, while in the second one, all the demand during the stock-out period is backordered. The three conflicting objectives are cost, quality and service level. To solve the multi-objective models, they applied NSGA-II to find the best Pareto fronts. Karimi-Nasab and Aryanezhad (2011) proposed a novel multi-objective model for the production smoothing problem in a single-stage facility for which some of the operating times could be determined in a time interval. Karimi-Nasab and Konstantaras (2012) presented a new multi-objective production planning model and a random search heuristic to explore the feasible solution space with the hope of finding the best solution in a reasonable time. Kian, Gürler, and Berk (2014) formulated the problem as a mixed integer, nonlinear programming problem and obtained structural results used to construct a forward dynamic-programming algorithm that obtains the optimal solution in polynomial time. Almeder, Klabjan, Traxler, and Almada-Lobo (2015) concentrated on the CLSP by explicitly modeling these two aspects and the synchronization of batches of products in the multi-level lot-sizing and scheduling formulation. Park and Klabjan (2015) considered the single-item lot-sizing problem with minimum order quantity where each period has an additional constraint on the minimum production quantity. Mehdizadeh and Tavakkoli-Moghaddam (2009) proposed a new meta-heuristic optimization algorithm, namely vibration damping optimization (VDO), which is based on the concept of vibration damping in mechanical vibration. They first utilize the VDO algorithm to solve the parallel machine scheduling problem. This algorithm simulates the vibration phenomenon. For more explanation, please see Aliabadi, Jolai, Mehdizadeh, and Jenabi (2011), Mehdizadeh and Nezhad-Dadgar (2014), Mehdizadeh, Tavakkoli-Moghaddam, and Yazdani (in press), Mehdizadeh, Tavarroth, and Hajipour (2011), Mehdizadeh, Tavarroth, and Mousavi (2010) and Mousavi, Niaki, Mehdizadeh, and Tavarroth (2013). Toledo, Ribeiro de Oliveira, and Morelato Franca (2013) proposed a new hybrid multi-population genetic algorithm (HMPGA) as an approach to solve the multi-level capacitated lot-sizing problem with backlogging. This method combines a multi-population based meta-heuristic using a fix-and-optimize heuristic and mathematical programming technique. Xiao, Zhang, Zhao, Kaku, and Xu (2014) improved the variable neighborhood search (VNS) algorithm for solving uncapacitated multilevel lot-sizing (MLLS) problems. In this paper, we propose in the following a new MICLSP with setup times, safety stock deficit costs, demand shortage costs both backorder and lost sale states, and different manners of production. As the main contribution in the model formulation area, this paper develops a bi-objective mathematical programming model with two conflicting objectives to make the model closer to reality. The objectives include (I) minimizing the total cost considered by the production plans including production costs with different production manners, inventory costs, safety stock deficit costs, shortage costs and setup costs; and (II) minimizing the required storage space. With regard to the fact that the proposed model is NP-hard, we propose two Pareto-based multi-objective meta-heuristic algorithms, called MOVDO and NRGA, that are novel in the literature on production planning. In order to validate the performance of the proposed MOVDO and NRGA algorithms, a well-developed evolutionary algorithm called NSGA-II is employed. To evaluate efficiency of the
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Figure 1. Scheme of the single-item lot-sizing problem.
algorithms, we first calibrate both algorithms using RSM; then, the results are analysed on some test problems, both graphically and statistically. The rest of the paper is organized as follows: Section 2 provides the proposed mixed integer programming formulation. Section 3 illustrates three Pareto-based multi-objective meta-heuristic algorithms in detail. Section 4 describes the experimental design and tuning of the parameters of all algorithms. Section 5 provides the results of all solving methodologies, both statistically and graphically. Finally, Section 6 gives conclusions and implications for future work.
2.1. Assumptions • The demand is deterministic. • Shortage comprises both backorders and lost sales, proportionally. • Shortage and inventory costs must be taken into consideration at the end. • Storage capacity limitations are considered. • Raw material resources are capacitated. • The quantities of inventory and shortage at the beginning of the planning horizon are zero. • The quantity of shortage at the end of the planning horizon is zero.
2. Problem formulation In this section, first the problem, assumptions, parameters and decision variables are thoroughly discussed, and then the proposed bi-objective linear programming model is formulated. Today, in most production centers, the need to answer the question of appointing a mixture of commodities production is felt more than ever before. In order to close the gap between the conditions of the problem and real-world conditions in this research, the multi-item lot-sizing problem has been studied considering production line equilibrium limitation and capacity limitation. Not only has there been consideration of different manners of product production, but also the model has been designed with the conditions of having safety stock and allowed shortage. Figure 1 represents the single-item lot-sizing problem schematically. To the best of our knowledge, two objective functions are simultaneously considered to make the model near to reality. Based on viewpoint of the owner of warehouse in presence of expenses like taxes, utilities, building and inventory insurance, wages and warehouse space, two conflicting objective functions are defined. Therefore, assuming the holding cost is independent of the storage space, the second objective can be formulated to minimize the total required storage space. The main goal is to present a bi-objective mathematical model to optimize production, inventory and shortage quantities as well as to determine the best production manner in which the totals of production, setup, inventory and shortage costs, as well as total storage cost, are minimized. In order to formulate the mathematical model of the problem, the assumptions, parameters, decision variables and mathematical formulation are provided in the following subsections.
2.2. Parameters T N J dit φit ξ πit yit− Lit δit αijt βijt yit+ Ct vi M wi at fij
Number of periods in the planning horizon t = 1, …, T Number of items i = 1, …, N Number of production manners j = 1, …, J Demand (forecast) for item i in period t Unitary shortage cost of item i in period t Probability of backorder shortage Unitary lost sale shortage cost of item i in period t Unitary safety stock deficit cost of item i in period t Safety stock value of item i in period t Safety stock variation between two consecutive periods Unitary production cost of item I by production manner j in period t Setup cost of item i by production manner j in period t Unit holding cost of item i in period t Amount of resource available in period t Unit amount of resource necessary to produce item i A large number Required space for unitary produce item i Unitary cost of storage space in period t Lost resource amount when product i produced by manner j
2.3. Decision variables xijt yijt rit sit+ sit− Ft
Quantity of item i produced by production manner j in period t Binary variable equal to one if item i is produced by production manner j in period t Shortage of item i in period t Overstock deficit variables of item i in period t Safety stock deficit variables of item i in period t Required storage capacity in period t
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2.4. The mathematical model min Z1 =
n T ∑ ∑
{[
] } J ( ) ∑ − + + − 𝛼ijt ⋅ xijt + 𝛽ijt ⋅ yijt + 𝜉 ⋅ 𝜑it ⋅ rit + (1 − 𝜉) ⋅ 𝜋it ⋅ rit + yit ⋅ sit + yit ⋅ sit
i=1 t=1
j=1
min Z2 =
T ∑
(2)
Ft ⋅ a t
t=1
s.t.
si,+t−1 − si,−t−1 − 𝛼 ⋅ ri, t−1 + rit +
J ∑
xijt =
j=1
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dit + 𝛿it +
sit+
−
sit−
si,+t−1
−
si,−t−1
− 𝛼 ⋅ ri, t−1 +
J ∑
xijt =
3. Multi-objective Pareto-based meta-heuristic algorithms (4)
dit + 𝛿it + sit+ − sit− ∀ i = 1, 2,..., n; t = T
(vi ⋅ xijt + fij ⋅ yijt ) ≤ Ct ∀t = 1, 2,..., T
(5)
i=1 j=1
xijt ≤ M ⋅ yijt ∀i = 1, 2,..., n; j = 1, 2,..., J; t = 1, 2,..., T (6) rit ≤ dit ∀i = 1, 2,..., n; t = 1, 2,..., T
(7)
sit− ≤ Lit ∀i = 1, 2,..., n; t = 1, 2,..., T
(8)
J n ∑ ∑
wi ⋅ xijt ≤ Ft ; ∀t = 1, 2,..., T
(9)
i=1 j=1
xijt , rit , sit+ , sit− , Ft
≥ 0; yijt ∈ {0, 1} ∀ i =
1, 2,..., n; j = 1, 2,..., J; t = 1, 2,..., T .
(10)
• The objective function (1) minimizes the total cost including unit production costs with different manners of production, inventory costs, overtime costs, shortage costs and setup costs. • The objective function (2) minimizes the total storage cost. • Constraints (3) are the inventory balance through the planning horizon. • Constraints (4) balance the inventories at the end of the period because the shortage is not permitted at the end period. • Constraints (5) are the capacity constraints; the overall consumption must remain lower than or equal to the available capacity. • Constraints (6) impose that the quantity produced must not exceed a maximum production level Mit, which is set to the minimum of the total demand requirement for item i on section [t, T] of the horizon and the highest quantity of item i that could be produced regarding the capacity constraints. Mit is then equal to } { T ∑ / � Min di ⋅ t , (ct − fit ) vit . t � =t
• Constraints (7) and (8) define upper bounds on, respectively, the demand shortage and the safety stock deficit for item i in period t. • Constraints (9) states storage space limitation. • Constraints (10) characterize the variable’s domains: xijt, rit, sit+ and sit− are non-negative and γijt is a binary variable.
(3)
∀ i = 1, 2,..., n; t = 1, 2,..., T − 1
j=1
J n ∑ ∑
(1)
In this section, three Pareto-based meta-heuristic algorithms called MOVDO, NRGA and NSGA-II are proposed for solving the bi-objective mathematical formulation at hand. However, some required multi-objective backgrounds are first defined in the following subsection. 3.1. Fundamental concept of multi-objective algorithms Consider a multi-objective[ model with ] a set of conf (⃗x ) = f1 (⃗x ), ..., fm (⃗x ) flict objectives subject to gi (⃗x ) ≤ 0 , i = 1, 2, ..., c, x⃗ ∈ X , where x⃗ denotes n-dimensional vectors that can get real, integer or even Boolean values and X is the feasible region. Then, for a minimization model, we say solution a⃗ dominates solution b⃗ (⃗a, b⃗ ∈ X) if
⃗ ∀i = 1, 2, ..., m and (1) fi (⃗a) ≤ fi (b), ⃗ ∃ i ∈ {1, 2, ..., m}: fi (⃗a) < fi (b) (2) Furthermore, a set of solutions that cannot dominate each other is called a Pareto solutions set or Pareto front. A good Pareto front has two features: (1) good convergence; and (2) good diversity of the solutions. Accordingly, Pareto-based algorithms aim to achieve the Pareto optimal front during the evolution process. The Pareto optimal front is called to the front of the last iteration of the algorithms. This front is expected to have the most convergence and the highest diversity (Deb et al., 2002). 3.2. Multi-objective vibration damping optimization (MOVDO) algorithm VDO is a meta-heuristic algorithm that works on the concept of vibration damping in mechanical vibration improvisation of musicians (Mehdizadeh & Tavakkoli-Moghaddam, 2009). In this paper, the multi-objective version of the VDO algorithm, introduced by Hajipour, Khodakarami, and Tavana (2014) and Hajipour, Mehdizadeh, and Tavakkoli-Moghaddam (2014), has been proposed for the area of production planning problems. The details of this algorithm are explained in the following subsections. 3.2.1. Solution representation In this paper, the illustration of the solution structure is of the single-strand (A) type, the length of which is equal to the total number of periods. Each part of the string itself contains a strand (Bt) the length of which equals the total number
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Figure 2. Solution representation.
of different manners of production. On the other hand, the components of the (Bt) string can define the production program, depending on the type of decision variable (TavakkoliMoghaddam, Rahimi-Vahed, Ghodratnama, & Siadat, 2009). To clarify that, the schematic of the solution group of a specific problem with three products and three different manners of production for each of the products in five periods is shown in Figure 2. However, since some constraints are likely to be violated, they are penalized using the method given in Yeniay and Ankare (2005). In other words, infeasible solutions are fined using Equation (11):
[( P(x) = M ×
g(x) b
)
] −1≥0 ,
(11)
where M, g(x), P(x) and f(x) represent a large number, the constraint under consideration, the penalty value, and the objective function value of chromosome x, respectively. This equation that is designed for a constraint like g(x) ≤ b, more violations receive bigger penalties. Moreover, penalty values are considered for all of the three objective functions through an additive function given in Equation (12):
{ F(x) =
f (x) ; f (x) + P(x);
x ∈ feasible region x ∉ feasible region .
(12)
3.2.2. MOVDO main loop In order to explain the proposed algorithm, we apply two main concepts of multi-objective meta-heuristics, namely fast non-dominated sorting (FNDS) and crowding distance (CD), to compare the solutions. In FNDS, R initial populations are compared and sorted. In order to do this, all chromosomes in the first non-dominated front are first found. Since both objective functions in the mathematical model are to be minimized, the chromosomes are chosen using the concept of domination. In this case, we say { that}xi is the non-dominated solution within the solution set xi , xj . Otherwise, it is not. Then, in order to find the chromosomes in the next non-dominated front, the solutions of the previous fronts are disregarded temporarily. This procedure is repeated until all solutions are set into fronts. After sorting the populations, a CD measure is defined to evaluate solution fronts of populations in terms of the relative density of individual solutions (Deb et al., 2002). To do this, consider Z and fk ; k = 1, 2, ..., M as the number of non-dom-
inated solutions in a particular front (F) and the objective functions, respectively. Additionally, let di and dj be the value of the CD on the solutions i and j, respectively. Then, the CD is obtained using the following steps. (I) Set di = 0 for i = 1, 2, ..., Z . (II) Sort all objective functions fk ; k = 1, 2, ..., M in ascending order. (III) The CD for the end solutions in each front (d1and dZ) are d1 =dZ → ∞ . (IV) The crowding distance for dj , j = 2, 3, … , (Z − 1), are dj =dj +(fkj+1 − fkj−1 ) . To select individuals of the next generation, the crowded tournament selection operator ‘f ’ is applied (Coello et al., 2007). In order to do that, the following steps are required to be carried out. Step 1: Choose n individuals in the population randomly. Step 2: The non-dominated rank of each individual is obtained and the CD of the solutions having equal non-dominated rank calculated. Step 3: The solutions with the least rank are selected. Moreover, if more than one individual shares the least rank, the individual with the highest CD should be selected. In other words, the comparison criterion of MOVDO algorithm’s solutions can be written as follows. If rx dy) then xfy, where rx and ry are the ranks, and dx and dy are the CDs. In this paper, a polynomial neighborhood structure for the selected chromosome is performed. After operating the aforementioned concepts and operators, the parents and offspring population should be combined to ensure the elitism. Since the combined population size is naturally greater than the original population size N, once more, non-domination sorting is performed. In fact, chromosomes with higher ranks are selected and added to the population until the population size becomes N. The last front is also consisted of the population based on the crowding distance. The algorithm stops when a predetermined number of iterations (or any stopping criteria) are reached. 3.2.3. Evolution process of MOVDO The process is started by initializing the initial population of the solution vectors Pj. Then, the new operators are implemented on Pj to create a new population Qj. The combination of Pj and Qj creates Rj for keeping elitism in the algorithm. In this step, vectors of Rj are sorted into several fronts based on FNDS and CD. Using the proposed selection method, a population
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of the next iteration Pj+1 is chosen to have a predetermined size. Figure 3 illustrates the evolution process of the proposed MOVDO, schematically. It is valuable to mention that, using Pareto dominance solutions, it is a computationally efficient algorithm implementing the idea of a selection method based on classes of dominance of all the solutions.
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3.2.4. The pseudo code of the MOVDO Figure 4 illustrates the pseudo code of the MOVDO algorithm based on the basic operators of a VDO algorithm and the described multi-objective operators. The main multi-objective parts are shown in the shaded boxes.
Figure 3. Evolution process of the proposed MOVDO.
Figure 4. Pseudo code of MOVDO.
3.3. Proposed non-dominated ranking genetic algorithm (NRGA) In addition to MOVDO, we also propose another multi-objective evolutionary algorithm called NRGA in the production planning area. The main difference of the NRGA from the MOVDO algorithm is the evolution process of the algorithm fromPt to Qt. While for the MOVDO algorithm the evolution is depicted in Figure 3, in NRGA the evolution process of a genetic algorithm (GA) is used. Accordingly, after generating or modifying populations by means of single-objective operators of the algorithms (a GA or VDO), the population is dealt with in a multi-objective way in a similar fashion in all algorithms. In contrast with NSGA-II, which utilizes binary tournament
International Journal of Management Science and Engineering Management
selection, the NRGA uses a roulette wheel selection strategy in which each individual is assigned a fitness value equal to its rank in the population. Additionally, to minimize the impact of using different operators on the performance comparison process of the algorithms, operators are designed identically. To do so, the neighborhood operator of the MOVDO algorithm is designed to be similar to the mutation operator of the NRGA. Moreover, in the NRGA, the crossover operator is designed using a uniform crossover operator (Pasandideh, Niaki, & Hajipour, 2011).
tionary algorithm called NSGA-II is applied. As mentioned, the main difference of the NSGA-II and NRGA from MOVDO is the evolution process of the algorithm. Moreover, in NSGA-II the binary tournament selection strategy is applied. The mutation and crossover operators are also designed to be similar to NRGA using swap mutation and uniform crossover operators as depicted in Figures 5 and 6, respectively. To clarify the trend of both the NSGA-II and NRGA frameworks, a flowchart that simultaneously contains both NSGA-II and NRGA algorithms is given in Figure 7.
3.4. NSGA-II
4. Analysis and comparison of results
To demonstrate the performance of the proposed MOVDO and NRGA, a well-developed Pareto-based multi-objective evolu-
This section provides application of the proposed methodology and performance comparison of the three meta-heu-
Parent
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7
689819
561940
452794
152930
195140
Offspring 152930
561940
452794
689819
195140
Figure 5. An example of a swap mutation operator.
Parent 1
152930
561940
452794
689819
195140
Offspring 1 367433 838563 503744 261565 255366
763117
Offspring 2 291019 649619 474762 660608 702891
Beta 0.391639 0.240675 0.301271 0.063854 0.893964 Parent 2
505522
926242
525712
232354
Figure 6. An example of a continuous uniform crossover operator.
Initialization
Chromosome Evaluation
Calculating FNDS and CD of the individuals
Front Determination
Stop Criterion
Pareto Optimal Front
NO
Algorithm Determination
NSGA-II
NRGA Roulette Wheel Selection
Crossover & Mutation
Chromosome Evaluation
Elitism
Calculating FNDS & CD
Sort Population & Choose N Individuals
Figure 7. The NRGA and NSGA-II framework.
Binary Tournament Selection
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Table 1. Search range of algorithm parameters. Solving methodologies NSGA-II
Parameter nPopNSGA Pc Pm nItNSGA
Range 25–200 0.6–0.99 0.01–0.4 100–500
Low (−1) 25 0.6 0.01 100
Medium (0) 100 0.8 0.2 300
High (+1) 200 0.99 0.4 500
NRGA
nPopNRGA Pc Pm nItNRGA
25–200 0.6–0.99 0.01–0.4 100–500
25 0.6 0.01 100
100 0.8 0.2 300
200 0.99 0.4 500
MOVDO
A0 𝜎 𝛾 L nPopMOVDO
6–10 1–2 0.005–0.5 50–100 4–12
6 1 0.005 50 4
8 1.5 0.05 75 8
10 2 0.5 100 12
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Table 2. Computational results obtained for the tuning parameters of NSGA-II. NSGA-II parameters Run order 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Pm −1 1 1 0 0 1 1 −1 0 −1 −1 0 1 −1 0 0 0 0 0 0
Pc 1 1 −1 0 0 1 −1 −1 0 1 −1 0 0 0 1 −1 0 0 0 0
nItNSGA −1 −1 −1 0 0 1 1 1 0 1 −1 0 0 0 0 0 1 −1 0 0
nPopNSGA 1 −1 1 0 0 1 −1 1 0 −1 −1 0 0 0 0 0 0 0 1 −1
NSGA-I implementing diversity 336,015,476.1 1,093,243,726 3,641,817,551 68,430,748.01 51,107,522.37 2,758,889.604 67,459,424.23 1,226,269,621 11,920,777.93 0 839.8104 11,019,339.86 27,593,327.35 152,812,369.7 60,468,941.25 62,333,783.02 13,636,116.26 783,465,256.6 32,669,796.26 419,798,229.7
Table 3. Computational results obtained for the tuning parameters of NRGA. NRGA parameters Run order 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Pm −1 1 1 0 0 1 1 −1 0 −1 −1 0 1 −1 0 0 0 0 0 0
Pc 1 1 −1 0 0 1 −1 −1 0 1 −1 0 0 0 1 −1 0 0 0 0
nItNRGA −1 −1 −1 0 0 1 1 1 0 1 −1 0 0 0 0 0 1 −1 0 0
ristic algorithms using a parameter tuning procedure. In order to calibrate the algorithms, the design of experiments (DOE) approach is first applied to investigate the effect of the parameters. Then, the response function is estimated and optimized using response surface methodology (RSM). The developed algorithms are coded in a MATLAB® software (Ver-
nPopNRGA 1 −1 1 0 0 1 −1 1 0 −1 −1 0 0 0 0 0 0 0 1 −1
NRGA implementing diversity 1,679,770,337 1,109,467,107 999,494,565.4 8,744,941.954 91,631,314.87 1,637,731.626 7,264,677.735 61,208,646.71 72,920,660.03 0 1,934,429.212 117,831,668.7 5,118,834.598 5,187,106.521 26,797,745.7 63,178,324.31 11,076,795.79 282,398,389.9 32,713,302.15 36,413,038.52
sion 7.10.0.499, R2010a) environment and the experiments are performed on a 2 GHz laptop with 4 GB of RAM to estimate the response functions. Although the responses may have curvatures over the search ranges of the factors, the central composite design of a fractional factorial with four central points is chosen to run
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the experiments (Montgomery, 2004), wherein there are k = 4 factors for NSGA-II and NRGA and k = 5 for MOVDO, each factor having three levels, i.e. low, medium and high, signed by (−1), (0), and (+1), respectively, and p = 1 . Table 1 reports search ranges of parameters for all algorithms in three levels. The design points along with the results of the experiments are represented in Tables 2–4 for NSGA-II, NRGA and MOVDO, respectively. Furthermore, the analysis of variance results indicate that both regression functions are appropriate and can be used in RSM, which is reported in Tables 5, 6 and 7 for NRGA, NSGA-II and MOVDO, respectively. Then, the models in (13), (14) and (15) are solved by LINGO™ 11 software within the range of the parameters, and the optimum combinations of the parameters are shown in Table 8 for each algorithm.
9
To evaluate and compare the performance of the solution methodologies under different environments, the experiments are implemented on 15 problems, which are reported in Table 9. Then, these problem instances are solved by three algorithms. Furthermore, to eliminate uncertainties in the solutions obtained, each problem is used three times under different random environments. Then, the averages of these three runs are treated as the ultimate responses. In order to evaluate the performance of the three multi-objective meta-heuristic algorithms, the following four metrics are used (Zitzler & Thiele, 1998). (I) Number of Pareto solutions (NOS) (II) Spacing (III) Diversity (IV) Computational Time
RNRGA = − 7,703,886 + 168,459,228 × Pc + 37,488,240 × Pm + 161,974,533 × nPopNRGA − 399,187,698 × nItNRGA 2 2 + 106,349,276 × Pc2 + 66,514,212 × Pm2 + 95,924,412 × nPopNRGA + 208,098,834 × nItNRGA
(13)
− 189,035,208 × Pc × Pm − 231,830, 505 × Pm × nPopNRGA − 59,945,393 × Pm × nItNRGA RNSGA = 13,948,356 − 350, 539,419 × Pc + 311,777,461 × Pm + 365,902,911 × nPopNSGA − 454,441,880 × nItNSGA + 61,900, 500 × Pc2 + 90, 701,986 × Pm2 (14)
2 2 + 399,049,824 × nItNSGA − 215,377,422 × Pc × Pm + 226,733,151 × nPopNSGA
+115,198,629 × Pm × nPopNSGA − 694,387,034 × Pm × nItNSGA RMOVDO = 10,491,601,614 − 2,737,627,775 × A0 + 994,920,614 × 𝜎 + 353,059,683 × 𝛾 + 1,672,356,871 × L + 42,143,648 × nPOP + 6,794,873,703 × A20 − 687,213,642 × 𝜎 2 + 1,210,353,336 × 𝛾 2 − 10,087,200,000 × L2 + 478,367,448 × nPOP 2 − 3,742,427,026 × A0 × 𝜎 − 3,742,616,868 × A0 × 𝛾 − 117,658,235 × A0 × L + 664,914,221 × A0 × nPOP + 1,576,911,535 × 𝜎 × 𝛾 + 768,185,078 × 𝜎 × L − 276,908,770 × 𝜎 × nPOP − 1,615,567,798 × 𝛾 × L −2,966,151,380 × 𝛾 × nPOP + 59,893,830 × L × nPOP .
(15)
Table 4. Computational results obtained for the tuning parameters of MOVDO. MOVDO parameters Run order 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
A0 1 0 1 0 −1 1 1 −1 0 −1 1 −1 −1 1 −1 1 −1 −1 0 1 1 −1 0 0 0 0 0 0 0 0
𝜎 −1 0 1 0 1 −1 1 1 0 1 −1 −1 −1 −1 −1 1 −1 1 0 1 0 0 1 −1 0 0 0 0 0 0
𝛾 −1 0 −1 0 −1 −1 1 −1 0 1 1 −1 1 1 −1 1 1 1 0 −1 0 0 0 0 1 −1 0 0 0 0
L −1 0 −1 0 −1 1 1 1 0 −1 1 −1 −1 −1 1 −1 1 1 0 1 0 0 0 0 0 0 1 −1 0 0
nPopMOVDO −1 0 1 0 −1 1 1 1 0 1 −1 1 −1 1 −1 −1 1 −1 0 −1 0 0 0 0 0 0 0 0 1 −1
MOVDO implementing diversity 1,725,003,528 5,354,787,511 643,863,392.6 2,156,528,713 1,948,922,239 17,234,668,768 135,886,910.7 17,234,668,768 276,634,244.1 19,945,780,036 3,461,670,796 3,714,855,440 14,273,828,331 2,211,039,679 2,693,442,050 987,993,149.2 9,267,592,917 28,507,076,371 7,964,684,657 672,535,573.9 31,181,298,932 9,945,094,535 14,288,082,452 11,874,136,326 1,694,759,298 28,262,593,435 1,854,191,153 5,508,023,838 6,567,041,373 21,926,339,584
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Table 5. Analysis of variance results for the NRGA. Source Regression Linear Square Interaction Residual error Lack-of-fit Pure error Total
DF 11 4 4 3 8 5 3 19
Seq-SS 3.717,03E+18 2.153,70E+18 8.187,45E+17 7.445,85E+17 3.881,97E+17 3.817,12E+17 6.485,53E+15 4.105,23E+18
Adj-SS 3.717,03E+18 2.153,70E+18 8.187,45E+17 7.445,85E+17 3.881,97E+17 3.817,12E+17 6.485,53E+15
Seq-SS 1.220,26E+19 5.604,85E+18 2.263,07E+18 4.334,65E+18 1.456,71E+18 1.454,22E+18 2.483,25E+15 1.365,93E+19
Adj-SS 1.220,26E+19 5.604,85E+18 2.263,07E+18 4.334,65E+18 1.456,71E+18 1.454,22E+18 2.483,25E+15
Seq-SS 1.221,35E+21 2.053,38E+20 3.274,66E+20 6.885,45E+20 1.298,23E+21 1.263,43E+21 3.480,07E+19 2.519,57E+21
Adj-SS 1.221,35E+21 2.053,38E+20 3.274,66E+20 6.885,45E+20 1.298,23E+21 1.263,43E+21 3.480,07E+19
Adj-MS 3.379,12E+17 5.384,26E+17 2.046,86E+17 2.481,95E+17 4.852,46E+16 7.634,23E+16 2.161,84E+15
F-test 6.96 11.10 4.22 5.11
P-value 0.005 0.002 0.040 0.029
35.31
0.007
F-test 6.09 7.70 3.11 7.94
P-value 0.008 0.008 0.081 0.009
351.37
0.000
F-test 0.42 0.28 0.45 0.48
P-value 0.947 0.910 0.801 0.868
18.15
0.019
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Table 6. Analysis of variance results for NSGA-II. Source Regression Linear Square Interaction Residual error Lack-of-fit Pure error Total
DF 11 4 4 3 8 5 3 19
Adj-MS 1.109,32E+18 1.401,21E+18 5.657,67E+17 1.444,88E+18 1.820,88E+17 2.908,45E+17 8.277,51E+14
Table 7. Analysis of variance results for MOVDO. Source Regression Linear Square Interaction Residual error Lack of fit Pure error Total
DF 20 5 5 10 9 6 3 29
Table 8. Optimal values of algorithm’s parameters.
Table 9. Input parameters of the model for the generated test problems.
Solving methodologies NSGA-II
Parameter nPopNSGA Pc Pm nItNSGA
NRGA
nPopNRGA Pc Pm nItNRGA
25 0.6 0.4 100
A0 𝜎 𝛾 L nPopMOVDO
6 1 0.5 100 4
MOVDO
Adj-MS 6.106,74E+19 4.106,76E+19 6.549,32E+19 6.885,45E+19 6.885,45E+19 2.105,71E+20 1.160,02E+19
Optimum amount 25 0.99 0.4 100
Test problemnumber 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
N 2 2 3 3 5 5 5 5 10 10 10 12 15 15 20
J 2 2 2 3 2 3 2 3 2 3 3 2 2 3 3
T 3 5 5 5 6 6 12 12 12 5 12 12 5 12 12
Table 10. Computational results of multi-objective metrics comparisons for NSGA- II, NRGA and MOVDO. Test problem number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
NSGA-II
Proposed NRGA
Proposed MOVDO
NOS
Spacing
Diversity
Time
NOS
Spacing
Diversity
Time
NOS
Spacing
Diversity
Time
12 8 10 4 5 3 11 6 5 8 6 4 6 10 4
6,011,94.09 130,122,720 127,655,720 45,231,121 11,742,479 25,840,836 2,620,129 9,970,298 148,436,567 82,236,657 11,162,182 35,537,360 16,091,661 64,969,726 110,561,941
3,917,051.4 576,382,174 66,5617,767 123,882,851 63,777,190 86,817,677 187,725,518 324,469,478 601,126,197 367,942,362 272,714,941 386,958,527 104,946,666 1.333E+09 417,644,791
28.747,798 31.759,981 31.214,237 37.141,955 32.630,218 42.343,969 32.522,637 36.766,985 30.593,923 30.946,569 30.127,575 30.196,617 30.856,064 31.538,927 31.571,051
10 3 6 5 9 5 8 3 6 2 6 8 6 8 4
19,795,855 4,047,977.9 15,180,598 21,008,200 31,704,332 81,633,421 33,411,245 2,684,243 209,620,349 0 9,5782,531 281,296,977 435,930,223 376,387,259 324,746,244
105,288,932 16,736,906 344,437,663 648,217,331 394,913,513 410,083,441 157,667,871 34,666,706 1.28E+09 14,860,905 69,4004,647 1.658E+09 2.154E+09 3.079E+09 1.351E+09
26.159,775 25.932,745 28.478,866 30.879,915 27.084,275 35.596,236 28.068,086 30.219,928 26.788,964 27.981,75 28.984,739 28.690,177 27.445,478 28.491,154 28.311,697
12 11 11 9 10 6 15 16 13 8 9 10 8 10 7
127,171,514 54,859,074 161,350,941 70,325,037 326,980,456 57,875,769 84,698,663 99,468,331 159,127,388 44,963,597 325,626,389 388,465,663 135,631,921 130,953,577 305,198,280
5.784E+09 456,303,724 959,036,787 89,4165,024 2.554E+09 543,075,485 2.585E+09 7.61E+09 4.096E+09 1.771E+09 2.466E+09 4.2E+09 2.153E+09 2.342E+09 1.432E+09
26.302,011 28.280,835 30.207,168 33.698,049 28.814,995 30.113,291 29.382,029 28.078,475 29.258,903 29.382,952 30.340,892 30.155,596 32.964,616 29.691,251 30.088,913
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Table 11. Analysis of variance results for the NOS metric. Source Algorithm Error Total
DF 2 42 44
SS 162.98 300.67 463.64
MS 81.49 7.16
F 11.38
P 0.00
Table 12. Analysis of variance results for the Spacing metric. Source Algorithm Error Total
DF 2 42 44
SS 9.436,32E+16 5.509,15E+17 6.452,78E+17
MS 4.718,16E+16 1.311,70E+16
F 3.60
P 0.036
Source Algorithm Error Total
DF 2 42 44
SS 126.08 297.07 423.14
MS 63.04 7.07
F 8.91
P 0.001
Table 14. Analysis of variance results for the Diversity metric. Source Algorithm Error Total
DF 2 42 44
SS 4.403,27E+19 6.990,75E+19 1.139,40E+20
MS 2.201,64E+19 1.664,46E+18
P 0.00
500000000
7000000000
400000000
6000000000
Spacing
5000000000 Diversity
F 13.23
Boxplot of Spacing by Alg.
Boxplot of Diversity by Alg. 8000000000
4000000000 3000000000 2000000000
300000000
200000000
100000000
1000000000 0
0 MOVDO
NRGA
MOVDO
NSGA
NRGA
NSGA
Alg.
Alg.
Figure 8. Boxplot of the Diversity metric for comparison of the algorithms.
Figure 10. Boxplot of the Spacing metric for comparison of the algorithms.
Boxplot of NOS by Alg.
Boxplot of Time by Alg.
16
42.5
14
40.0
12
37.5 Time
10 NOS
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Table 13. Analysis of variance results for the Time metric.
8 6
35.0 32.5 30.0
4
27.5
2 0
25.0 MOVDO
NRGA
NSGA
Alg.
Figure 9. Boxplot of the NOS metric for comparison of the algorithms.
MOVDO
NRGA
NSGA
Alg.
Figure 11. Boxplot of the Time metric for comparison of the algorithms.
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Figure 12. Graphical comparison of the algorithms in terms of the metrics.
Table 15. Analysis of variance results for the Time metric. Comparison between the NRGA and MOVDO algorithms. Source Algorithm Error Total
DF 1 28 29
SS 10.38 122.78 133.16
MS 10.38 4.39
F 2.37
P 0.135
Boxplot of TIme NRGA&MOVDO by Alg.2 35.0
TIme NRGA&MOVDO
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12
32.5
30.0
27.5
25.0 MOVDO
NRGA Alg.2
Figure 13. Boxplot of the Time metric. Comparison between the NRGA and MOVDO algorithms.
Table 16. Summarized comparison of the algorithms. Comparison Metrics NOS Spacing Diversity Time
Statistically MOVDO NSGA-II MOVDO NRGA and MOVDO
Graphically MOVDO NSGA-II MOVDO NRGA
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13
Table 17. Initial parameters of numerical example. Period:
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Production manner: αijt
1
2
Item 1 Item 2
I 82 70
II 84 76
I 67 85
II 84 85
βijt
Item 1 Item 2
25,743 24,753
22,912 25,757
24,801 23,934
20,851 20,214
φit
Item 1 Item 2
202 214
212 159
πit
Item 1 Item 2
110 150
160 135
yit+
Item 1 Item 2
71 50
58 51
yit−
Item 1 Item 2
123 88
146 63
dit
Item 1 Item 2
3000 3000
1000 2000
Lit
Item 1 Item 2
30 20 20
20 30 30
at
Table 18. Secondary parameters of numerical example. Parameter wi vi fij
Item 1 Item 2
Value 1 2
Item 1 Item 2
0.7 0.75
Item 1
Production manner 1 Production manner 2
5 3
Item 1
Production manner 1 Production manner 2
2 2
ξ Ct
0.8 3000
Table 19. Optimal solution of the MICLSP example by object-tive integration. Period: Production manner: xijt
1
2
Item 1 Item 2
I 0 3004
II 1060 0
I 2572 0
II 0 0
yijt
Item 1 Item 2
0 1
1 0
1 0
0 0
rit
Item 1 Item 2
1940 0
0 1996
sit+
Item 1 Item 2
0 0
0 0
sit−
Item 1 Item 2
30 16
0 30
7068
2572
Ft Objective function value
The results comparisons in terms of all multi-objective metrics for all algorithms are reported in Table 10. Moreover, the algorithms are compared based on the properties of their obtained solutions by the analysis of variance method, statistically. The analyses of variance of these tests on each
1,433,585
metric are represented in Tables 11–14. To make the results of the tests more visible, boxplots are shown in Figures 8–11 for the cases where a significant difference is obtained. For these cases, all metrics are also plotted and graphically compared in Figure 12.
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To clarify the trend of comparisons, the analysis of variance for the Time metric between the NRGA and MOVDO algorithms is carried out in Table 15 and Figure 13. We note that while in terms of the diversity and NOS metrics, bigger values are desired, for spacing, MID and CPU time, smaller values are better. Thus, in general, it is clear that MOVDO shows better performances in terms of diversity, Time, and NOS. However, with statistical comparisons of the metrics, Figs. 8-11 and 13 shows that in terms of NOS and diversity the algorithms have significant differences. This conclusion is also confirmed at 95% confidence level. Furthermore, Fig.12 supports this conclusion as well. However, to clarify the algorithms comparison, Table 16 summarizes this fact both graphically and statistically. In order to justify our results, consider an MLCLSP with two products, two periods and two production manners. The parameters corresponding to this problem are reported in Tables 17–18. Table 19 presents optimal solutions of the problem using LINGO™ software. To do so, we integrated objectives by the weighted-sum approach with the same weights.
5. Conclusion In this paper, we developed an MICLSP with setup times, safety stock deficit costs, demand shortage costs both backorder and lost sale states, and different production manners as a bi-objective problem. The conflicting objectives are: minimizing the total cost considered by the production plans including production costs with different production manners, inventory costs, safety stock deficit costs, shortage costs, and setup costs and minimizing the required storage space. Since the problem is NP-hard, two Pareto-based multi-objective meta-heuristic algorithms called MOVDO and NRGA are proposed and introduced into the literature on LSP. To justify the performance of the proposed MOVDO and NRGA, one of the best-developed multi-objective evolutionary algorithms called NSGA-II is implemented. RSM is also executed to tune the parameters of the three algorithms. The statistical and graphical comparisons demonstrate the robustness of the proposed MOVDO in terms of the diversity and NOS metrics, and both the MOVDO and NRGA algorithms in terms of the Time versus in NSGA-II. However, NSGA-II works better in terms of the Spacing metric. For future research, the demand can be considered as a random variable or a fuzzy number.
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