An integrated model for lot sizing with supplier selection and quantity discounts

Applied Mathematical Modelling 37 (2013) 4733–4746

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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm

An integrated model for lot sizing with supplier selection and quantity discounts Amy H.I. Lee a, He-Yau Kang b,⇑, Chun-Mei Lai c, Wan-Yu Hong b a

Department of Technology Management, Chung Hua University, Hsinchu 300, Taiwan, ROC Department of Industrial Engineering and Management, National Chin-Yi University of Technology, Taichung 411, Taiwan, ROC c Department of Marketing and Logistics Management, Far East University, Tainan 744, Taiwan, ROC b

a r t i c l e

i n f o

Article history: Received 8 December 2011 Received in revised form 20 August 2012 Accepted 25 September 2012 Available online 5 October 2012 Keywords: Genetic algorithm Lot sizing Supplier selection All-units quantity discount Incremental quantity discount

a b s t r a c t Good inventory management is essential for a firm to be cost competitive and to acquire decent profit in the market, and how to achieve an outstanding inventory management has been a popular topic in both the academic field and in real practice for decades. As the production environment getting increasingly complex, various kinds of mathematical models have been developed, such as linear programming, nonlinear programming, mixed integer programming, geometric programming, gradient-based nonlinear programming and dynamic programming, to name a few. However, when the problem becomes NP-hard, heuristics tools may be necessary to solve the problem. In this paper, a mixed integer programming (MIP) model is constructed first to solve the lot-sizing problem with multiple suppliers, multiple periods and quantity discounts. An efficient Genetic Algorithm (GA) is proposed next to tackle the problem when it becomes too complicated. The objectives are to minimize total costs, where the costs include ordering cost, holding cost, purchase cost and transportation cost, under the requirement that no inventory shortage is allowed in the system, and to determine an appropriate inventory level for each planning period. The results demonstrate that the proposed GA model is an effective and accurate tool for determining the replenishment for a manufacturer for multi-periods. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction Having a good production planning and replenishment control through effective inventory management is important for a firm to keep competitive in the market. The single product, multi-period inventory lot-sizing problem is one of the most common and basic problems, and it has often been tackled in the literature [1]. There are various extensions of the model to consider different issues in real environment. This research considers an environment with multiple periods and multiple suppliers. Different suppliers may have either all-units quantity discounts or incremental quantity discounts. In addition, transportation cost is fixed for each vehicle shipment, and it is different for different suppliers. By adopting the proposed models, the management can decide what quantity to order from which suppliers in which periods. Quantity discount is a common and effective practice for suppliers to promote their products, and buyers can purchase products at a lower unit price when the ordering quantity is over a certain amount. In addition, multi-suppliers give firms a chance to purchase same materials from different sources. Ordering cost for each purchase can be in several different forms, for example, fixed, increasing or decreasing. The assumption of a fixed ordering cost is often seen in works [1–3]. In Rezaei and Davoodi [4], ordering cost consists of a fixed cost and an additional cost. The fixed cost is independent of the lot-size, and ⇑ Corresponding author. Tel.: +886 4 23924505 7624; fax: +886 4 23934620. E-mail address: [email protected] (H.-Y. Kang). 0307-904X/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2012.09.056

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the additional ordering cost depends on the specific lot-size. In Rezaei and Davoodi [5], reduction in ordering cost is positively related to ordering frequency, that is, the higher the ordering frequency from a supplier, the higher the ordering cost reduction from that supplier. In this research, we set the ordering cost to be fixed because this is more suitable for our case study in real practice. In addition, quantity cost is considered in the study so that the more quantity ordered in each purchase, the lower the unit cost is. Even though there have been abundant works which handled lot sizing problems in quantity discount environment and used MIP or other methods to solve the problems, the authors, after reviewing these papers, found out that very few papers have considered an environment with multiple periods, multiple suppliers, and both all-units and incremental quantity discounts simultaneously. With regard to the complexity of the lot-sizing problems, Rezaei and Davoodi [4,5], Florian et al. [6], and Bitran and Yanasse [7], claimed that this kind of problem basically belongs to a class of NP-hard problems. Therefore, to solve such a complicated problem, this paper proposes a MIP model to solve a small-scale problem and to compare the results with the proposed GA model first. The GA model is then used to tackle the problem when it becomes too complicated for the MIP model to solve. The proposed models are then compared with four past works that considered a similar problem environment. The comparisons show that the proposed models have more attributes than the past works. The remaining of this paper is organized as follows. In Section 2, some related methodologies and works are reviewed. In Section 3, the problem under consideration and the assumptions are described. The formulation of the lot-sizing problem by MIP and the construction of the GA model are presented in Section 4. Case studies are carried out in Section 5. Section 6 compares the proposed models with several works with similar environments. Some conclusion remarks are made in the last section. 2. Related methodologies and works 2.1. Inventory management works Intensive research has been carried out to develop models and methods and to find efficient solutions for better inventory management. Recent works of inventory management and lot sizing models are described below. Su and Wong [8] studied a stochastic dynamic lot-sizing problem under the bullwhip effect. A framework of two-stage ant colony optimization (TACO) was proposed, and a mutation operation was added in the second stage to determine the replenishment policy. Kim [9] analyzed the order quantity flexibility, under which a buyer provides to the supplier information about the expected future orders during a predetermined horizon and the supplier provides the buyer with the flexibility to adjust future orders in return. Kogan and Shnaiderman [10] studied a stochastic, optimal control problem characterized by continuous inventory replenishment, and proposed an asymptotically optimal, dynamic, base-stock policies that accounted for continuous inventory costs and continuous replenishment with periodic updates. Koo et al. [11] considered a capacity-constrained multipleproduct system and developed a non-linear optimization model for simultaneous determination of throughput rate and lot size for each product. Since the optimization model could not be solved analytically, a line search algorithm for lot sizing at the capacity constrained workstation was proposed. Lu and Qi [12] studied a multi-product dynamic lot sizing problem, where inventories of all products are replenished jointly with the same quantity whenever a production occurs. A polynomial time algorithm was proposed to solve the problem without the option of lost sales, and two heuristic algorithms were presented to solve the problem with lost sales. Mohammadi and Jafari [13] presented a mathematical model for the problem of integrating lot sizing, loading, and scheduling in a capacitated flexible flow shop with sequence-dependent setups, and three mixed integer programming-based algorithms were proposed to solve the model. Okhrin and Richter [14] analyzed an uncapacitated single-item lot sizing problem where a minimum order quantity restriction guarantees a certain level of production lots. A solution algorithm based on the concept of atomic sub-problems was proposed, and the O(T2) exact algorithm with the worst-case complexity could solve the problem to optimality. Okhrin and Richter [15] further explored a single-item capacitated lot sizing problem with minimum order quantity, and adopted a general dynamic programming technique to develop an O(T3) exact algorithm to optimally solve the problem. Tarim et al. [16] proposed a computational approach to solve the mixed integer programming (MIP) model of a stochastic lot-sizing problem with service level constraints under static-dynamic uncertainty strategy. The approach was based on a relaxation of the MIP model, and an optimal solution to the relaxed model was found by solving an equivalent shortest path problem. Ho et al. [17] investigated the multi-stage logistics and inventory problem in an assembly type supply chain in which a uniform lot size is produced through all stages and unequal sub-batch sizes across stages are allowed. A serial-type supply chain model is constructed first, an assembly-type supply chain model is established next, and finally an optimization algorithm that incorporates the lot size division and recursive tightening methods is developed to solve the problems. 2.2. Quantity discounts Multiple suppliers and quantity discounts are common practices in purchasing activities. When quantity discounts is present and a large order is placed by a buyer, the unit price charged by the supplier is reduced according to a predetermined schedule. There are often a number of price breaks, and the unit discounted price decreases as the quantity level increases. Two major types of quantity discounts are present: all-units quantity discount and incremental quantity discount [18,19].

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Under the all-units quantity discount, if the quantity purchased belongs to a specified quantity level predetermined by the supplier, the discounted price is applied to all units starting from the first unit. Under the incremental quantity discount, the discounted price is applied to the units inside the price break quantity, and therefore, different prices are applied to the units belonging to different price breaks. Quantity discounts has received significant attention in the literature for decades, and several papers have described the quantity discount problems that have been studied [20–22]. Numerous inventory models have considered quantity discounts, such as Chang [23], Chang et al. [24], Li and Liu [25] and Moussourakis and Haksever [26]. Moussourakis and Haksever [26], considering all-units quantity discounts, proposed a zero-one mixed integer programming model for determining order quantities in multi-product multi-constraint inventory systems. Rezaei and Davoodi [27] considered inventory lot sizing and imperfect items in a supply chain and proposed a MIP for supplier selection under an environment with multiple suppliers and multiple products with quantity discounts. Mendoza and Ventura [28] incorporated the discount quantity option in the EOQ and considered a single stage model with both all-units and incremental quantity discount structures. The optimal inventory policies in terms of the reorder interval were devised. Lee and Kang [29] presented a mixed 0–1 integer programming for solving the multi-period inventory problem with the characteristics of price quantity discount, large product size, batch-sized purchase and forbidden shortage. Zhang and Ma [30] presented a mixed integer nonlinear programming approach to simultaneously determine the production level of multiple products, supplier selection, and order quantity allocation for a manufacturer under uncertain demands, where suppliers offer all-unit quantity discount prices. Duan et al. [31] considered a single-vendor, single-buyer supply chain for fixed lifetime product and proposed decision making models to analyze the benefit of coordinating supply chain through quantity discount strategy. Munson and Hu [22] presented methodologies to calculate optimal order quantities and computed total purchasing and inventory costs when products have either all-units or incremental quantity discounts. Shi and Zhang [32] incorporated all unit quantity discounts into the multi-product newsvendor problem with a budget constraint and uncertain demands. The problem was formulated as a mixed integer nonlinear programming (MINLP) model, and a Lagrangian-based solution approach was also developed. Zhang [33] further included all-units quantity discounts to the constrained newsvendor problem, and presented a mixed integer nonlinear programming model and a Lagrangian relaxation approach to solve the problem. Chen and Ho [34] presented an analysis method for the single-period inventory problem with fuzzy demands and incremental quantity discounts, with the objective of minimizing the total cost per unit time. Lin and Ho [35] aimed to find the optimal pricing and ordering strategies for an integrated inventory system when a quantity discount policy is present and market demand is price-sensitive. An algorithm was proposed to determine the optimal retail price, order quantity, and the number of shipment per production run from the vendor to the buyer. Kang and Lee [36] studied the inventory replenishment problem with the consideration of storage space, yield rate, all-units quantity discounts and multiple suppliers, and a fuzzy multiple objective programming, with assigned weights for objectives based on experts’ opinions, is proposed. Krichen et al. [37] constructed a single supplier, multiple cooperative retailers inventory model with quantity discount and permissible delay in payments. A decision rule that helps each retailer to enumerate the preferred coalitions was proposed, and an algorithm that generated coalition structures in the core was presented. Shi et al. [38] aimed to maximize the expected profit of the retailer through jointly determining the ordering quantities and selling prices for the products with the consideration of all-units quantity discounts and newsvendor pricing. The problem was formulated as a generalized disjunctive programming (GDP) model, and a Lagrangian heuristic approach was developed. Taleizadeh et al. [39] studied a multi-product, multi-constraint joint-replenishment inventory control problem in which incremental discount policy was used and transportation, clearance, fixed order, holding and shortage costs were considered. MINLP was applied first, and a hybrid method of harmony search, fuzzy simulation and rough simulation was proposed next to solve the fuzzy integer non-linear problem.

2.3. Genetic Algorithm (GA) GA, a heuristic search process for optimization, was first developed by Holland [40], and it has been widely applied to solve production and operations management problems [41]. GA, based on Darwin’s survival of the fittest principle, mimics the process of natural selection [42]. It is a stochastic solution search procedure, and it is designed to solve combinatorial problems using the concept of evolutionary computation imitating the natural selection and biological reproduction of animal species [43–45]. GA has many advantages in solving the inventory problem, including being easy to implement; having a code which is easy to understand and modify; useful for both complex and loosely defined problems; having inductive nature that it works by its own internal rules; parallel nature of the stochastic search; solving multi-dimensional, nondifferential, non-continuous, and even non-parametrical problems, and dealing easily with many constraints which are neglected by many other methods in spite of their importance in practice [46,47]. The fundamental concept of GA is to code the decision variables of the problem as a finite length array, which is called chromosome, and to calculate the fitness, the objective function, of each string [48]. A chromosome with a high fitness level has a higher probability of survival. Through the crossover and mutation process, the surviving chromosomes reproduce and form the chromosomes for the next generation. Generally, the procedures of GA are as follows: (1) initialize a population of chromosomes with a random method, (2) evaluate each chromosome in the population according to the criteria, (3) create a mating pool by applying selection operator, (4) create the offspring through crossover, mutation, and reordering operators, (5) repeat step 3 and step 4 until the objective function of the problem is optimized or the stop criterion is met [49,50].

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GA is recognized as a powerful tool with an iterative procedure to heuristically search the optimal solution of the problem, and it can generate solutions very close to optimal with less time compared to enumerative tools [44,51,52]. It is suitable for solving a replenishment problem because it can be formulated as a problem with one continuous decision variable (for instance, a basic cycle time) and a set of integer decision variables [46,47]. GA has become a new approach for solving the lot-sizing problem since the last decade, and numerical results from past works show that GA is an effective approach to solve the lot-sizing problem. Maiti et al. [42] adopted GA to solve a multi-item, two-storage, multiple price-break deterministic inventory model with a discount policy. Li et al. [53] applied GA to determine the periods of setups, and a dynamic program was developed to determine the production quantities for solving the capacitated lot-sizing problem. Gupta et al. [54] proposed a GA with ranking selection to solve a finite time horizon inventory problem. Hong and Kim [55] adopted GA to solve the joint replenishment problem based on the exact inventory cost, which is obtained by an unbiased estimator of the exact cost. Widyadana et al. [56] developed a GA and a hybrid of GA and simulated annealing (GASA) to determine the kanban quantity and withdrawal lot sizes in a multi-products supply chain system, and the proposed methods were quite outstanding compared with the MIP. Che [57] proposed a hybrid heuristic algorithm, which combined guided GA and pareto GA, for production planning that integrates supplier selection, assembly sequence planning, and multi-period planning to meet both customer demands and supplier capacities. Ojha et al. [58] considered transportation problems for an item with fixed charge, vehicle cost and quantity discounts, including all-units quantity discount, incremental quantity discount and a combination of both. The problems were formulated using liner programming first, and GA based on roulette wheel selection, arithmetic crossover and uniform mutation was developed and applied next. Jiang et al. [59] studied multiobjective optimization matching for one-shot multi-attribute exchanges with quantity discounts in E-brokerage. A multiobjective nonlinear transportation problem was first developed, and a multi-objective simulated annealing GA was developed next to solve the proposed model. Lee et al. [19] formulated the lot-sizing problem with all-units quantity discount and transportation cost as a MIP model, and a GA model was constructed for solving large-scale lot-sizing problems. Yang et al. [48] suggested replenishment policies that could minimize system-wide cost by considering quantity discounts in the transportation cost structures. The problem was first formulated as an integer programming model, and a GA based approach was proposed to solve the problem. 3. Problem description and assumptions The following assumptions and notations are defined with the modification of those used in the models of Lee et al. [29], Kang and Lee [36] and Kang [60]. The assumptions are summarized as follows:  The demand of each period is independent and known.  At most one order can be placed in each period.  The replenishment lead time is of known duration, and the entire order quantity is delivered at once in the beginning of a period.  The price of each unit is dependent on the order quantity. Both all-units discounts and incremental discounts are considered.  The inventory holding cost for each unit is known and constant, independent of the price of each unit.  The transportation cost is known and constant for each vehicle shipment from a supplier.  Planning horizon is finite and known. In the planning horizon, there are T periods, and the duration of each period is the same.  The initial inventory level (X0) is zero. All the required notations in this paper are defined as below. Notations Indices: t Planning period (t = 1, 2,. . ., T). i Supplier (i = 1, 2,. . ., I). Supplier 1 to supplier i’ provide all-units quantity discounts, and supplier i’ + 1 to I provide incremental quantity discounts. k Price break (k = 1, 2,. . ., K). Parameters: dt Expected demand in period t. h Inventory holding cost, per unit per period. si Transportation cost per vehicle from supplier i. M A large number. oi Ordering cost per replenishment from supplier i. bi The maximum transportation (vehicle) batch size from supplier i.

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pik ^itk p

Unit purchase cost from supplier i with price break k under all-units quantity discounts. Average unit purchase cost from supplier i in period t with purchase quantity under price break k under incremental quantity discounts. qik The upper bound quantity of supplier i with price break k. Decision variables: TC Total cost in a planning horizon. PðQ it Þ Purchase cost for one unit based on the discount schedule of supplier i with order quantity Qit in period t. Qit Purchase quantity from supplier i in period t. Qt Total purchase quantity in period t. dQ it =bi e The smallest integer greater than or equal to Qit/bi. Number of transportations (vehicles) from supplier i in period t. Rit Xt Expected ending inventory level in period t. Fit A binary variable, set equal to 1 if a purchase is made, and 0 if no purchase is made, from supplier i in period t. Ft A binary variable, set equal to 1 if a purchase is made, and 0 if no purchase is made, in period t. Uitk A binary variable, set equal to 1 if a certain quantity is purchased, and 0 if no purchase is made, from supplier i with price break k in period t. P Yt Expected beginning available inventory level in period t, and Y t ¼ X t1 þ Ii¼1 F it  Q it .

Fig. 1 is the graphical representation of a multi-period inventory system for the replenishment problem. The ending inventory level in period t(Xt) is equal to the ending inventory level in period t  1(Xt1) plus the purchase amount in period P t( Ii¼1 F it  Q it ) and minus the expected demand in period t(dt), where Fit represents whether a purchase is made from supplier i in period t (1 if a purchase is made, and 0 if no purchase is made). The beginning available inventory level in period P t(Yt) is equal to the ending inventory level in period t – 1 (Xt1) plus the purchase amount in period t( Ii¼1 F it  Q it ). An instance of inventory level in the planning horizon is also depicted in Fig. 1.

4. Formulation of the lot-sizing problem with multi-suppliers and quantity discounts This paper develops a MIP model and a GA model to solve the lot-sizing problem with multiple suppliers and quantity discounts in determining an appropriate inventory level for each period. The total cost for each period can be calculated by adding up the relevant costs, including ordering cost, holding cost, transportation cost, and purchase cost with quantity discounts. The total cost includes all the costs in a planning horizon. 4.1. Relevant costs Eq. (1) calculates the ordering cost for the system, where oi is the ordering cost per time from supplier i and Fit represents whether a quantity is purchased from supplier i in period t.

Ordering cost ¼ O ¼

T X I X oi  F it :

ð1Þ

t¼1 i¼1

The purchase cost is obtained by Eq. (2), where P(Qit) is the unit purchase cost based on the discount schedule with the order quantity Qit, and Fit represents whether a quantity is purchased from supplier i in period t.

Fig. 1. Graphical representation of the inventory replenishment system.

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Purchase cost ¼ P ¼

T X I X

ðPðQ it Þ  Q it  F it Þ:

ð2Þ

t¼1 i¼1

The transportation cost of the system is calculated using Eq. (3), where si is the transportation cost per vehicle from supplier i in period t, dQ it =bi e is the smallest integer greater than or equal to Qit/bi from supplier i in period t, Rit is number of transportations (vehicles) from supplier i in period t and bi is the maximum transportation batch size from supplier i.

Transportation cost ¼ S ¼

T X I X

si  dQ it =bi e ¼

t¼1 i¼1

T X si  Rit :

ð3Þ

t¼1

The holding cost is defined as follows. The average inventory in period t is one-half of summation of the expected beginning available inventory level in period t(Yt) and the expected ending inventory level in period t(Xt). In addition, the ending inventory in period t(Xt) is equal to the ending inventory level in the period t  1(Xt1) plus the purchase quantity in the period t(Qt) minus the demand in the previous period (dt). Thus, the ending inventory in period t(Xt) is equal to the beginning available inventory level in period t(Yt) minus the demand in period t(dt). The total holding cost for a planning horizon is the summation of the holding cost for each period, as in Eq. (4).

Holding cost ¼ H ¼

T X h t¼1

2

T X h

ðY t þ X t Þ ¼

2

t¼1

ð2Y t  dt Þ:

ð4Þ

4.2. Mixed Integer Programming (MIP) Model In this section, we formulate the lot-sizing problem into a MIP model to solve the multi-period inventory problem and to determine an appropriate replenishment policy for each period. The proposed model can be formulated as follows:

" # T I X X Minimize TC ¼ ðoi  F it þ PðQ it Þ  Q it  F it þ si  dQ it =bi eÞ þ h=2  ð2Y t  dt Þ ; t¼1

ð5Þ

i¼1

subject to X t ¼ Y t  dt ;

t ¼ 1; 2; . . . ; T;

ð6Þ

I X Q it  F it ;

t ¼ 1; 2; . . . ; T;

ð7Þ

Y t ¼ X t1 þ

i¼1

Q it 6 M  F it ; PðQ it Þ ¼

i ¼ 1; 2; . . . ; I

K X pik  U itk ;

and t ¼ 1; 2; . . . ; T;

i ¼ 1; 2; . . . ; i

0

ð8Þ

and t ¼ 1; 2; . . . ; T;

ð9Þ

k¼1

qik1 þ M  ðU itk  1Þ 6 Q it < qik þ M  ð1  U itk Þ; K X U itk ¼ 1;

i ¼ 1; 2; . . . ; i

0

i ¼ 1; 2; . . . ; i

0

and t ¼ 1; 2; . . . ; T;

and t ¼ 1; 2; . . . ; T;

ð10Þ ð11Þ

k¼1

PðQ it Þ ¼

K X ^itk  U itk ; p

0

0

i ¼ i þ 1; i þ 2; . . . ; I

and t ¼ 1; 2; . . . ; T;

ð12Þ

k¼1

aik ¼

k X qik0  ðpik0 1  pik0 Þ;

0

0

i ¼ i þ 1; i þ 2; . . . ; I

and k ¼ 1; 2; . . . ; K;

ð13Þ

0

k ¼1

^itk ¼ pik þ p

aik Q it

F it 2 f0; 1g; U itk 2 f0; 1g;

;

0

0

i ¼ i þ 1; i þ 2; . . . ; I;

i ¼ 1; 2; . . . ; I i ¼ 1; 2; . . . ; I;

and all variables are nonnegative.

t ¼ 1; 2; . . . ; T

and k ¼ 1; 2; . . . ; K;

and t ¼ 1; 2; . . . ; T; t ¼ 1; 2; . . . ; T

and k ¼ 1; 2; . . . ; K

ð14Þ ð15Þ ð16Þ

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The objective (5) is to minimize the total cost, which includes the ordering cost, purchase cost, transportation cost and holding cost in a planning horizon. The operative constraints are as follows. In Eq. (6), the expected ending inventory of period t, Xt, is equal to the expected beginning available inventory level in period t, Yt, minus the demand in period t, dt. In Eq. (7), the beginning available inventory level in period t, Yt, is equal to the ending inventory level in period t  1, Xt1, plus the P purchase quantity from all suppliers in period t, Ii¼1 Q it  F it . Eq. (8) is to set the purchase quantity from supplier i in period t, and Fit is a binary variable. Eqs. (9)–(11) consider the all-units quantity discounts for supplier 1 to i0 , who offer all-units quantity discounts. Eq. (9) determines the purchase cost per unit, P(Qit), under the all-units discount schedule based on the quantity purchased from supplier i in period t. Eq. (10) sets the purchase quantity between a lower bound quantity qik1 and an upper bound quantity qik in a price break k for supplier i. Eq. (11) makes sure that a quantity can only be purchased with one single price break k from supplier i in period t. Eqs. (12)–(14) consider the incremental quantity discounts for supplier i0 +1 to I, who offer incremental quantity discounts. Eq. (12) determines the purchase cost per unit, P(Qit), under the incremental discount schedule based on the quantity purchased from supplier i in period t. Eq. (13) sets the marginal purchase cost (aik) for the quantity (Qit) which is not purchased at pik. Eq. (14) calculates the average unit purchase cost from supplier i with purchase quantity under price break k under incremental quantity discounts. The MIP model contains T variables of Xt, T variables of Yt, IT integer variables of Qit, IT integer variables of Rit, IT variables ^itk , IT 0–1 variables of Fit, and ITK 0–1 variables of Uitk. Furthermore, of P(Qit), (I  i’)K variables of aik, (I  i’)TK variables of p each of Eqs. (6) and (7) contains T equations, each of Eqs. (8)–(11) contains IT equations, Eq. (12) contains (I  i’)T equations, Eq. (13) contains (I  i’)K equations, and Eq. (14) contains (I  i’)TK equations. Thus, the total number of equations is 2T + 4IT + (I  i’)(K + T + KT). 4.3. Genetic Algorithm (GA) model In this research, GA is used to solve the lot-sizing problem with multiple suppliers and quantity discounts so that nearoptimal solutions can be produced in a short period of computational time. The proposed procedures are as follows [19,61,62]: Step 1. Coding scheme. We assume that at most one order can be placed in each period. The replenishment strategy is that a replenishment quantity can serve for an integer number of periods. The chromosome is coded as a string of t binary digits, with the tth gene having a value of 1 if replenishment is made at the beginning of that period and 0 otherwise (see Fig. 2). The replenishment quantity Qit associated with period t is:

Q t ¼ F t dt þ F t ½ð1  F tþ1 Þdtþ1 þ ð1  F tþ1 Þð1  F tþ2 Þdtþ2 þ . . . þ ð1  F tþ1 Þð1  F tþ2 Þ . . . ð1  F T ÞdT ; t ¼ 1; 2; . . . ; T; Qt ¼

I X Q it  F it ;

t ¼ 1; 2; . . . ; T;

ð17Þ ð18Þ

i¼1

Ft ¼

I X

F it ;

t ¼ 1; 2; . . . ; T:

ð19Þ

i¼1

Eq. (17) is to calculate the total purchase quantity in period t, Qt, and Eq. (18) is to allocate Qt to various suppliers in that period. Step 2. Initial population of chromosomes. The initial population is generated randomly. There are two types of chromosomes, and the chromosome type is determined randomly. Step 3. Fitness function. The fitness function for each chromosome is defined as Min TC, where TC is the total cost. Min TC is the minimum cost among all the chromosomes across the population. Step 4. Crossover operation. The standard two-cut-point crossover operator is applied to the selected pair of parent-individuals by recombining their genetic codes and producing two offspring. Step 5. Mutation operator. A mutation operator is used to counteract premature convergence and to maintain enough diversity in the population. It is performed by changing a randomly selected gene in the genetic code (0–1, 1–0). Step 6. Selection of subsequent population. After mutation and crossover operations in a generation, a subsequent population is selected for the next generation. This is also called parent selection. Basically, individuals with higher fitness values are more likely to be selected for the mating pool, and vice versa. Individuals are sorted by their fitness values, and the number of reproductive trials of each individual is then allocated according to its rank. Step 7. Termination. The processes of crossover, selection and replacement are repeated until the objective function of the problem is optimized or the stop criterion is met.

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Fig. 2. Chromosome of coding scheme and replenishment relation.

5. A case study of high technology company in Taiwan 5.1. Basic information input A lot-sizing problem with multi-supplier and quantity discount is studied here. An anonymous manufacturer in Hsinchu, Taiwan purchases electrical parts and assembles them into various products such as LCD monitors, digital audio products, PCs and home appliances. The products are delivered to a distributor in Taoyuan, Taiwan, which re-sells the products to other counties. In order to obtain the optimal replenishment policy, the production management needs to consider multiple suppliers and quantity discounts. The manufacturer currently has one supplier, supplier A, and is considering cooperating with other supplier(s), supplier B, supplier C and supplier D. The objectives of the model are to minimize total cost, and in turn, to determine the optimal purchase amount from each supplier in each period. Based on an interview with the management of the factory, the following assumptions are made. The ordering cost of supplier A(o1), supplier B(o2), supplier C(o3) and supplier D(o4) per replenishment is set to be $250, $220, $165 and $180, respectively. In addition, we set unit holding cost per period (h), which includes the handling cost, storage cost and capital cost, to be $0.11. Table 1 shows the demand in each period for 20 periods. Tables 2-5 show the discount schedules under different purchase quantities for supplier A–D respectively. Note that both suppliers A and B provide all-units quantity discounts while suppliers C and D provide incremental quantity discounts. For instance, if the purchase quantity in a period is between 2001 and 3899 units from supplier A, which offers all-units quantity discounts, the price for each unit, starting from the first unit, is $2.85. If the purchase quantity in a period is 1600 units from supplier C, which provides incremental quantity discounts, the unit cost for the first 900 units is $3.05, and the unit cost for the 901st unit to the 1800th unit is $2.96. Three special cases are examined here, as shown in Table 6. Under case I, the ordering cost per time from supplier A and B is $250 and $220, respectively. The transportation cost per time from supplier A and B is $21 and $22 respectively, and the maximum transportation batch size is 25 units for each supplier. The unit holding cost is $0.11, quantity discounts rule is allunits for both suppliers, and the number of periods is 5. A quantity can be purchased from supplier A and/or B. Under case II, the ordering cost per time is $250, $220, $165 and $180, from supplier A, B, C and D, respectively. The transportation cost per vehicle is $21, $22, $19 and $21.5, from supplier A, B, C and D, respectively. The maximum transportation batch size is 25 units for each supplier. The unit holding cost is $0.11. The quantity discounts rule is all-units for supplier A and B and incremental for supplier C and D. The planning period is 10, and a quantity can be purchased from supplier A, B, C and/or D using

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1 660

2 720

3 510

4 810

5 725

6 465

7 1510

8 2410

9 515

10 1850

11 320

12 540

13 590

14 250

15 780

16 830

17 950

18 650

19 410

20 195

Table 2 Discount schedule for supplier A with all-units quantity discounts. Price break (k)

Purchase quantity (Q)

Price per unit (P(Q))

1 2 3

0–2000 2001–3899 3900 or more

$2.99 $2.85 $2.74

Table 3 Discount schedule for supplier B with all-units quantity discounts. Price break (k)

Purchase quantity (Q)

Price per unit (P(Q))

1 2 3 4

0–1100 1101–2200 2201–3400 3401 or more

$3.00 $2.93 $2.82 $2.75

the discount schedules in Tables 2, 3, 4 and 5, respectively. The data for case III is very similar to that for case II, except that it is a large case with a planning period of 20.

5.2. Experimental results The lot-sizing problem is solved by both the MIP model and the GA model. The MIP model is implemented using the software LINGO [63], and the GA is implemented using the software MATLAB [64]. Case I: The formulation of the MIP for Case I is implemented using the software LINGO [63]. The GA model is implemented by using the software MATLAB [64]. Two-cut-point crossover for crossover operations is applied, and an inversion mutation operator is used to avoid a solution being trapped in a local optimum and to approach the global optimum. The size of the initial population is set as 60. The crossover rate is set as 0.75, meaning that around 75% pairs of individuals take part in the production of offspring. The mutation rate is set as 0.009, meaning that each gene of a newly created solution is mutated with the probability 0.009. In addition, the algorithm is terminated after 200 generations have elapsed. The best generation occurs at the 42nd generations, as shown in Fig. 3. The solutions of Case I obtained by the MIP model and by the GA algorithm are the same and are summarized in Table 7. The purchases are made once from supplier B in period 1 for 3425 units, and the total cost is $15,143. Case II: A larger case with the consideration of both types of quantity discounts and more suppliers is constructed here to examine the MIP and GA models. Again, the MIP model is implemented using the software LINGO [63], and the GA is implemented using the software MATLAB [64].

Table 4 Discount schedule for supplier C with incremental quantity discounts. Price break (k)

Purchase quantity (Q)

Price per unit (P(Q))

1 2 3

0–900 901–1800 1801 or more

$3.05 $2.96 $2.83

Table 5 Discount schedule for supplier D with incremental quantity discounts. Price break (k)

Purchase quantity (Q)

Price per unit (P(Q))

1 2 3 4

0–999 1000–2599 2600–4099 4100 or more

$2.98 $2.82 $2.79 $2.76

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Fig. 3. The convergence of GA for Case I.

Table 6 Data for the three cases. Case

o (o1, o2, o3, o4)

h

s (s1, s2, s3, s4)

Quantity discounts

Period (t)

Supplier (i)

I II III

$250, $220,———,——— $250,$220,$165,$180 $250,$220,$165,$180

$0.11 $0.11 $0.11

$21, $22,———,——— $21,$22, $19, $21.5 $21,$22, $19, $21.5

All-units All-units/incremental All-units/incremental

1–5 1–10 1–20

A, B A, B, C, D A, B, C, D

Table 7 Results from the MIP and the GA model for Case I.

Yt Xt Q1t Q2t P(Q1t) P(Q2t)

t=1

t=2

t=3

t=4

t=5

TC

3425 2765

2765 2045

2045 1535

1535 725

725 0

$15,143

3425 $2.75

Table 8 Results from the MIP model for Case II. t

1

2

3

4

5

6

7

8

9

10

TC

Yt Xt Q1t Q2t Q3t Q4t P(Q1t) P(Q2t) P(Q3t) P(Q4t)

1900 1240

1240 520

520 10

3510 2700

2700 1975

1975 1510

1510 0

4775 2365 4775

2365 1850

1850 0

$39,495

3500 1900 $2.74 $2.75 $2.90

The solution obtained by the MIP model is shown in Table 8. The replenishment policy is to purchase 1900 units from supplier D in period 1, 3500 units from supplier B in period 4, and 4775 units from supplier A in period 8, respectively. The total cost is $39,495.

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Fig. 4. The convergence of GA for Case II.

Table 9 Results from the GA for Case II. t

1

2

3

4

5

6

7

8

9

10

TC

Yt Xt Q1t Q2t Q3t Q4t P(Q1t) P(Q2t) P(Q3t) P(Q4t)

3890 3230

3230 2510

2510 2000

2000 1190

1190 465

465 0

3920 2410 3920

2410 0

2365 1850

1850 0

$39,678

3890

2365

$2.74 $2.75

$2.82

Case II is also solved by the GA model. Fig. 4 shows that a solution is obtained in the 98th convergence under a fast searching process. Table 9 shows that the replenishment policy is to purchase 3890 units from supplier B in period 1, 3920 units from supplier A in period 7 and 2365 units from supplier B in period 9, respectively. The total cost is $39,678. The percentage error, calculated by (TC from GA–TC from MIP) / TC from MIP, is 0.4647%. Since the percentage error is rather low, the GA model can provide a satisfactory solution with a very short computational time. Case III: An even larger case with 20 periods is constructed here to examine the MIP and GA models. The MIP model cannot be solved using the LINGO within a reasonable computational time (48 h). For the GA, we increase the termination of generation to 600 to ensure that a near optimal solution is obtained. The best generation occurs at the 171th generations, as shown in Fig. 5. Table 10 shows that the replenishment policy is to purchase 3890 units from supplier B in period 1, 3920 units from supplier A in period 7, 4065 units from supplier A in period 9, 1610 units from supplier D in period 15, and 2205 units from supplier B in period 17, respectively. The total cost is $61,447. The computational time is 16 s.

6. Comparison of the proposed models with four previous works In this section, four previous works are compared with the proposed models in this research. According to a comprehensive review of supplier selection and order lot sizing modeling by Aissaoui et al. [65], single item models with discounts were proposed by Chaudhry [66] and Tempelmeier [2]. We searched the works after the publication of Aissaoui et al. [65] and found two other related works. Chaudhry [66] used linear and mixed binary integer programming to solve the supplier

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Fig. 5. The convergence of GA for Case III.

Table 10 Results from the GA for Case III. t

1

2

3

4

5

6

7

Yt Xt Q1t Q2t Q3t Q4t P(Q1t) P(Q2t) P(Q3t) P(Q4t)

3890 3230

3230 2510 2000 1190 465 3920 2510 2000 1190 465 0 2410 3920

8

9

2410 4065 0 3550 4065

10

11

12

13

14

15

3550 1700 1380 840 250 1610 1700 1380 840 250 0 830

16

17

18

830 2205 0 1255

3890

19

20

TC

1255 605 195 $61,447 605 195 0

2205 1610 $2.74

$2.74

$2.75

$2.82 $2.92

Table 11 Comparisons among similar approaches. Compared items

Chaudhry et al. [66]

Tempelmeier [2]

Basnet and Leung [1]

Moghadam et al. [3]

Proposed GA

Proposed MIP

Single-objective Algorithm Accuracy

Yes Exact Optimal

Yes Heuristic Near optimal

Yes Heuristic Near optimal

Yes Heuristic Near optimal

Yes Exact Optimal

Multi-periods All-units quantity discount Incremental quantity discount Batch size Ordering cost Holding cost Transportation cost Purchase cost Solve linear behavior Low CPU time

No Yes Yes

Yes Yes Yes

Yes No No

Yes No No

Yes Heuristic Near optimal Yes Yes Yes

No No No No Yes Yes No

No Yes Yes No Yes No Yes

No Yes Yes No Yes Yes Yes

No Yes Yes No Yes Yes Yes

Yes Yes Yes Yes Yes Yes Yes

Yes Yes Yes Yes Yes Yes No

Yes Yes Yes

selection problem with price breaks. Tempelmeier [2] considered the problem of supplier selection and purchase order sizing for a single item under dynamic demand conditions, and a heuristic procedure was proposed. Basnet and Leung [1] used an enumerative search algorithm and a heuristic to solve the problem. Moghadam et al. [3] developed a hybrid GA model for the replenishment problem.

A.H.I. Lee et al. / Applied Mathematical Modelling 37 (2013) 4733–4746

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These four works are compared with the models proposed in this research. As shown in Table 11, among the compared items, Chaudhry [66], Basnet and Leung [1] and Moghadam et al. [3] considered seven of them, and Tempelmeier [2] performed well in eight items. On the other hand, the proposed GA performed well in eleven items, while the proposed MIP considered twelve of them. Both the GA and MIP models are outstanding overall; however, the MIP model cannot solve a problem when it becomes too complicated. This is due to the fact that an increase in the number of periods (t), suppliers (i), or price breaks (k) will increase the problem size, and the problem can become NP-hard and computationally prohibitive. On the other hand, the proposed GA model can solve a NP-hard problem by providing a satisfactory outcome in a short computational time. In summary, this research successfully formulated the lot-sizing with supplier selection and price breaks problem. Practical case studies have illustrated the effectiveness of the proposed models. 7. Conclusions The purpose of this paper is to construct a lot-sizing model with multi-suppliers and quantity discounts to minimize total cost over the planning horizon. The contribution can be summarized as follows. First, a general formulation of the lot-sizing problem by MIP is proposed. The model can consider costs such as ordering cost, holding cost, purchase cost and transportation cost, and inventory level for each planning period can be obtained to minimize the total cost in the system without any inventory shortages. Second, a Genetic Algorithm (GA) model is constructed to solve the problem when it becomes too complicated. Third, a small-scale case with two suppliers which both provide all-units quantity discounts for a horizon of five periods is studied using both the MIP and GA models. Replenishment level and system cost can be determined after calculating ordering cost, holding cost, purchase cost and transportation cost. Both models can obtain the optimal solution. Fourth, two more complicated cases are studied. When the case becomes too complicated or NP-hard, the GA model can find solutions very close to the optimal ones. Thus, the GA model can be very effective in searching for solutions, and it can be very useful for managers in real practice. The results of the method can satisfy decision makers’ desirable service level of replenishment in a production environment. Moreover, it is readily available for real applications. For future studies, a more complete case for supply chain management in manufacturing can be considered. A model that takes into account variable lead time, probability demand, different priority of orders, and safety stock can be established. The ordering cost for each purchase can be variable based on the environment of the new case. 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