A Mathematical Modeling Approach to the Integrated Supplier Selection and Lot Sizing Problem
Tunç Erel, Peral Toktaş-Palut
Abstract
This study analyzes the integrated supplier selection and lot sizing problem of an international automotive supply industry company owning more than one production facility and supplier. A mathematical model is constructed to solve this problem which is very important for the competitiveness of the company. The aim of this study is to select the most appropriate local or global suppliers and to determine the lot sizes based on the production facility and product. The products are divided into two groups: complex and simple; and the suppliers are divided into three groups: local, strategic, and challenger. Only the simple products can be supplied from local suppliers. Local suppliers offer a quantity discount, whereas strategic and challenger suppliers offer a rebate based on the total value of the order. The objective function of the mathematical model is to minimize the total costs of the supply chain. The mathematical model is solved and the results are discussed.
1. Introduction
Over the past two decades, manufacturers, in the face of increased competition, have been under intense pressure to improve their core operations and are discovering that traditional management tools are not designed for today’s supply chain challenges. To achieve improved performance in managing supply chains, many manufacturers introduced and implemented a wide array of strategies including vertical integration, reduction of suppliers, tight partnerships with suppliers with long term purchasing contracts, supplier managed inventories, and integration of purchasing decisions within the production planning process. In this context, the evaluation and selection of competent suppliers, or vendors, to lower production, distribution, and material costs have become key elements in supply chain management. In addition to total cost, quality and delivery appears critical factors in the supplier selection process. Recent events involving toxic pet food [1], tainted milk [2], tainted cough syrup [3], and lead contaminated toys [4] underscore the importance of supplier selection. Now, manufacturers from all types of industries are being forced to switch to alternate suppliers—despite the higher price—due to quality restrictions. Wal-Mart, accompany whose success is often attributed to its low-cost China sourcing, has introduced strict new supplier control policies [5], including the requirement that suppliers of goods to Wal-Mart stores identify each factory where the products originated. Besides these quality issues, suppliers may experience unplanned disruptions due to natural and financial disasters [6,7] affecting their production which may delay deliveries to the manufacturers. Complex supplier relationships, unreliable sources of material supply, and ever-increasing customer expectations greatly add to the pressure on manufacturers to select the best suppliers and operate their processes more efficiently. Even though the selection of competent suppliers has been traditionally considered as one of the most important functions performed by a purchasing department [8], with these recent developments in business, supplier election has become one of the core focuses of many businesses.
In this study, we concentrate on an integrated supplier selection problem for a manufacturing firm that aims to simultaneously optimize its supplier selection process and inventory replenishment decisions at its plants. More specifically, each plant of the manufacturer faces plant-specific, stationary, deterministic demand. To respond to the demand in a timely manner, each plant holds inventory and operates under a plant-specific policy to replenish its inventory from its assigned supplier. Furthermore, each unmet demand at the plant level is backordered at a cost. Hence, we explicitly account for inventory holding and shortage costs at the plants. We assume that the supplier selection decisions for each plant are conducted centrally at the firm level from a small pool of pre-selected suppliers that offer varying unit costs and reliability (quality). This type of situation is quite common in the automotive industry. In addition to supplier-specific and inventory-specific costs, we also explicitly account for transportation costs and lead time associated with direct shipments between each plant and its assigned supplier. We note that due to the existence of supplier-contract fees and transportation costs, there is a significant trade-off between tactical (inventory replenishment) and strategic (supplier selection) decisions. We take also price discounts and rebates into account. Most companies have several sites located worldwide and are concerned with purchasing raw materials and components for these sites from a global network of suppliers. The sourcing economic development and the need for competitive advantage have increased the trend of sourcing products across the global marketplace. The global purchasing environment has unintended consequences; it exposes organizations to considerable sourcing risks and imposes new challenges that must be considered in the supplier selection process.
In this problem, our goal is to minimize the total cost in the system including the contractual fees of the selected suppliers, the transportation costs between the suppliers and plants, and the inventory holding and backorder costs at the plants by determining (i) the number and selection of suppliers, (ii) the assignment of each selected supplier to a plant, and (iii) the inventory decisions of each plant and beside this maximize the quality and delivery rate. The main restrictions faced in this problem are related to capacity, quality, and late delivery from the suppliers. More specifically, we observe that the capacity limitations of the suppliers impact their selection and supplier–plant assignment since supplier capacity directly relates to delivery performance at the plants. Secondly, the manufacturing firm requires each supplier to provide a specific level of quality to ensure the best performance. The domestic suppliers are relatively reliable but more expensive, while the foreign suppliers offer competitive prices, however material flows from these suppliers are more exposed to unexpected disruptions. Given a set of customer orders for products, the decision maker needs to decide which single supplier or which multiple suppliers, one from each region, to select for purchasing parts required to complete the customer orders and how to schedule the orders over the planning horizon, to mitigate the impact of late delivery.
We emphasize, once again, that our focus is on improving the performance of the supply chain by considering the total operational cost of logistics, rather than just selecting supplier or optimizing the inventory costs at the plants, while adhering to system restrictions. For this purpose, we first analytically formulate the integrated problem, considering supplier selection, supplier–plant assignment, and each plant’s inventory decisions simultaneously with practical constraints. We search answers for the below questions in our study.
a) Correct supplier selection with suitable inventory decisions for all the plants, b) Contribution of single/dual sourcing for disruption of suppliers and cost effect, c) Effect of quantity discounts for integrated models,
The remainder of the study is organized as follows. In the next section, we provide a brief review of supplier selection and lot sizing, and single/multi sourcing literature. In Section 3, we point out the model and constraints. Finally, concluding remarks are given in Section 4.
2. Literature Review
A key component in developing a reliable supply chain is the selection of suppliers. The sourcing of products from across the world has become an increasing trend by companies around the world looking for new sources of competitive advantage. We review the literature in three main parts: (i) supplier selection and lot sizing, (ii) bat algorithm, (iii) single vs multi sourcing.
In the supply chain literature, while lot-sizing is considered as a tactical decision, supplier selection is regarded as a strategic decision. The lot-sizing problem deals with determining order quantity and its timing by striking a tradeoff between ordering and storage costs. The lot sizing problem has also been studied widely in terms of supplier selection during the last several decades. The main goal of the lot sizing problem with supplier selection is to decide which supplier(s) should be selected and how much should be ordered from the selected supplier(s) [9].
Recently, researchers have shown an increased interest in the integrated problem of supplier selection and lot-sizing. Combination of supplier selection and lot-sizing for allocating orders to suppliers over the time planning horizon can significantly reduce costs [10]. Several studies have been carried out in order to combine supplier selection and multi-period order lot-sizing to determine the optimal order quantities in each period. Basnet and Leung [11] presented an uncapacitated mixed integer model for a multi-period inventory lot-sizing scenario where there are multiple products and multiple suppliers. A supplier selection problem integrated with resource-constrained single product, multi-period and inventory lot-sizing problem price discounts schemes was proposed by Hassini [12]. They proposed a single-objective cost-based model. Moghadam et al. [13] proposed an integrated intelligence algorithm for multi-period multi-product lot-sizing with supplier selection considering multiple-echelon inventory system. In their research, a hybrid model of fuzzy neural network was used for demand forecasting. Ustun and Demirtas [14] developed an integrated model of ANP and achievement scalarising functions in order to solve supplier selection problem integrated with multi-period inventory lot-sizing. A multi-period goal programming model including total cost, total defect rate and total value of purchasing objective functions was developed in order to determine the optimum quantities of orders and inventory levels in each period. Ebrahim et al. [15] introduced a multiobjective integer programming model in which qualitative and quantitative factors are considered. They defined defective items, late delivered items and total weighted quantity of purchasing as the objective functions of the multi-objective model and solved the model using a scatter search algorithm (SSA). Keskin et al. [16] proposed a model which was based on non-linear, mixed integer programming formulation (MINLP) and scatter search. They used simulation optimization for an integrated model of lot sizing and supplier selection. The model was designed for single item and period with a single objective model for multiple suppliers. Woarawichaiet et al. [17] proposed a single-objective model for multi-period multi-product lot-sizing with supplier selection under storage space and budget constraints and solved the model using LINGO software. Rezaei and Davoodi [18] proposed two multi-objective mixed integer non-linear models for multi-period multi-product lot-sizing with supplier selection problem and solve the models using GA. Cost, quality and service level were defined as the main objective functions of the models. Hammami et al. [19] proposed a mathematical model in order
to deal with supplier selection problem integrated with multi-period multi-product lot-sizing in an international environment with the objective of minimizing the total cost. Lee et al. [20] constructed a MIP model and genetic algorithm (GA) to solve the lot sizing problem with multiple suppliers. It incorporates the incremental and all-unit quantity discounts and is applicable to determine the replenishment strategy for a manufacturer for multi periods. But it is only suitable for single items. Choudhary and Shankar [21] worked on similar problem with Woarawichai et al. but solved it by using integer linear programming model and goal programming. Ayhan and Kılıç [22] developed two stage algorithm an integrated approach including Fuzzy Analytical Hierarchy Process (F-AHP) and Mixed Integer Linear Programming (MILP) model. Their model based on single period, but multi item, supplier and criteria.
Successful supply chain management necessitates an effective sourcing strategy to combat unreliable supply and stochastic demand. We divide the most frequently used approaches of sourcing into three types: (1) single sourcing, (2) dual sourcing, and (3) multiple sourcing. In the first type it is necessary to mention that single sourcing differs from sole sourcing. Sole sourcing refers to a buyer–supplier relationship where the supply base contains only one supplier, whereas single sourcing is when a buyer chooses a single supplier even though other comparable suppliers exist in the supplier base [23]. The second sourcing strategy, dual sourcing, indicates that a buyer employs two suppliers, one of which may dominate the other in terms of business share, price, reliability, and others. In the last sourcing model, multiple sourcing, a buyer does business with several suppliers and plays one supplier against the other to enjoy the best price advantage. Single sourcing strategy strives for a strategic partnership between a buyer and a supplier to foster a close collaboration and to optimize shared benefits. The tighter coordination between the buyer and supplier is a prerequisite for a successful execution of the just-in-time (JIT) initiative, which encourages the two partners to streamline the supply chain process and encourage the buyer–seller relationship to move toward a single sourcing model. As single sourcing has become a priority for many firms, many researchers have studied the advantages of this strategy. The general benefits of single sourcing, as indicated in a survey study by Larson and Kulchitsky [24], include higher quality at lower total cost to the buyer and that suppliers are linked to higher levels of buyer– supplier cooperation. However, the dependence on a single source also exposes the buying firms to a greater risk of supply chain interruption. For example, Toyota’s brake valve crisis in 1997 exemplifies the possible occurrence of supply chain disruption risks that are resulted from a single sourcing strategy in a JIT system.
With an increasing awareness of the high risks associated with single sourcing and companies’ expanded efforts to mitigate risks or to build in contingency, multiple sourcing is gaining attention again; especially how to determine the optimal supply size has resulted in a great deal of research interest. For example, Berger et al. [25] consider risks associated with a supplier network, which include catastrophic super events that affect all suppliers, as well as unique events that impact only one single supplier, and then present a decision-tree based model to help determine the optimal number of suppliers needed for the buying firm. Following the same decision framework, Ruiz-Torres and Farzad [26] present an extension of a model by considering unequal failure probabilities for all the suppliers. They have also compared their proposed models by conducting a sensitivity analysis in order to better understand the effect of the input parameters on the optimal number of suppliers. In the other effort, Berger and Zeng [27] study the optimal supply size under a number of scenarios that are determined by various financial loss functions, the operating cost functions and the probabilities of all the suppliers being down.
With respect to the selection decision between single and dual sourcing methods, we see a number of important research efforts. A research by Pochard [28] examines how buying firms should prepare for disruptions in their supply chain by using real options theories to compare single sourcing with dual sourcing. Burke et al. [29] indicate that single sourcing is a dominant strategy only when supplier capacities are large relative to the product demand and when the buying firm does not obtain diversification benefits; but in all other cases, dual sourcing is an optimal sourcing strategy. Using the concept of switching costs in a principal-agent frame- work, Wagner and Friedl [30] analyzed whether a firm switches single sourcing to dual sourcing when there is either symmetric or asymmetric information about the alternative supplier’s cost structure. They have found that the sourcing strategies depend on the buying firm’s beliefs in the alternative supplier’s unit costs, switching costs, the price offered by the incumbent supplier, and refinements of the price offered by the incumbent supplier due to competitive reactions and economies of scale.
The other research stream examines how to split orders between different suppliers. In the case of dual sourcing, Lau and Zhao [31] identify the optimal proportion of split between two suppliers by minimizing the sum of annual holding and ordering costs subject to a maximum allowable stockout risk. Through numerical studies, they have found that the optimal proportion of split varies with, among other factors, the difference in the suppliers’ mean lead times. The issue of order split between two suppliers is also studied by Kelle and Miller [32] in the context that the objective of decision is to minimize stockout risk. They not only provide an exact formula for calculating the optimal order split rate, but also show that large lead-time demand and lead- time uncertainty usually favor dual sourcing. In a recent study, Tomlin and Wang [33] examine the flexibility and reliability provided by dual sourcing using the classic newsvendor model. They have identified a number of factors affecting the use of the second supplier, including the resource costs and reliabilities, the firm’s downside risk tolerance, the number of products, the product demand correlations and the spread in product contribution margins In terms of supply chain performance, Bichescu and Fry [34] rely on a numerical analysis approach to examine how decision making rights split between supply chain agents determine order quantity and shipping frequency and then affect the supply chain performance. Among the results presented in the study, they have found that concentrating channel power with the supplier can lead to supply chain profits that are very close to a centralized scenario, but also results in lower customer service levels.
3. Mathematical Model
The objective is to minimize all the cost factors of a manufacturing company, which has multiple sites. The other 2 objectives are to maximize the quality level and minimize the late deliveries. Beside these 3 objectives, there are 11 constrains. The costs are occurred in three main headings: 1) The supplier specific costs: They are variable per unit procurement costs and contract fees. 2) Transportation costs: They are between selected suppliers and plants. There are 2 transportation costs, fix and variable. Variable transportation cost is dependent to distance between plant and supplier. The cost of transportation is given per product.
3) Costs at plant level: They are holding and backorder costs.
The assumptions, which are being done, listed below and the explanations of abbreviates in the objective functions and constraints are being given in 6.3 notation list in appendix:
1) Suppliers are classified as “Strategic”, “Challenger” and “Local”. 2) Products are classified as “Easy” and “Difficult”.
3) Difficult products can only be supplied from strategic and challenger suppliers. 4) Each plant can supply from at most two suppliers.
5) Local suppliers apply all-unit quantity discount and strategic/challenger suppliers apply rebate.
6) Contract fee is only valid for difficult products.
7) Late deliveries are assumed to be received in following period. 8) Demand is deterministic.
Decision variables
n ijkt
Q order quantity of plant i from supplier j for product k in discount interval n in period t
ijkt
Y 1, if plant i gives an order to supplier j for product k in period t
0, otherwise
ikt
I inventory of product k in period t for plant i
n ijkt Y 1, n ijkt Q > 0 0, otherwise Parameters i plant j supplier J j ϵJs, strategic supplier j ϵJc, challenger supplier j ϵJl, local supplier k product k k ϵK , easy product e k ϵK , difficult product d t period n discount interval jk
n index of the last discount interval for product k offered by supplier j J l
jkt
c unit price of product k from supplier j in period t
n jk
b quantity discount percentage of product k from supplier j in discount interval n
n j
r rebate percentage for supplier j J UJ s c in discount interval n
jk
o contract fee for supplier j in period t
ijkt
p fixed dispatch cost per replenishment to plant i from supplier j for product k in period t
ijkt
r unit variable mileage cost per replenishment to plant i from j for product k in period t
ij
d distance between plant i and supplier j jk
late delivery percentage of supplier j for product k ikt
h unit holding cost of product k in period t for plant i ikt
s unit backorder cost of product k in period t for plant i jk
q good quantity product percentage of supplier j for product k
jkt
w production capacity of supplier j for product k in period t
min
jkt
Q minimum order quantity requirement of supplier j for product k in period t ikt
D demand at plant i for product k in period t k
V volume of product k (in m3)
it
f storage capacity of plant i in period t (in m3) n
jk
e lower bound of discount interval n offered by supplier j J for product k l
n jk
u upper bound of discount interval n offered by supplier j J for product k l
n j
e lower bound of discount interval n offered by supplier j J UJ s c
n j
u upper bound of discount interval n offered by supplier j J UJ s c j
n index of the last discount interval offered by supplier j J UJ s c 3.1. Objective Function
First terms is cost of unit procurement, and then minus two terms show all unit discount & rebate from supplier. Forth one is contractual fees and fifth term is transportation cost and last two are holding and back order costs at plant level.
( 1) min . . . . . . ( . .[ .(1 ) ( . )] . . L s c d n n n n n
jkt ijkt jk jkt ijkt j jkt ijkt
i j k n t i j J k n t i j J UJ k n t
n n
jt ijkt ijkt ijkt ij ijkt jk ijk t jk
i j k k t i j k n t
ikt ikt ikt
i k t i k t C c Q b c Q r c Q o Y p r d Q Q h I s I
ikt 3.2. ConstraintsThere are 11 constrains, which are related with suppliers, plants, etc. a) Supplier capacity n ijkt jkt i n Q w
j k t, ,b) Minimum order quantity min. n ijkt jkt ijkt n Q Q Y
i j k t, , ,. n
ijkt ijkt ijkt n
Y
Q M Y i j k t, , , M is very big numberd) Each plant can supply each product in each period from maximum 2 suppliers 2 ijkt j Y
i k t, , e) Inventory constraints max(0, ) ikt ikt I I i k t, , min(0, ) ikt ikt I I i k t, , ( 1) ( 1) [(1 ) (1 )]. . ( 1) n nik t ik t jk jk ijkt jk ijk t ikt ikt ikt
j n j n I I q Q Q D I I
, , i k t f) Inventory constraints-2: Beginning and ending inventories are zero
0 0 0 0 ik ik ikT ikT I I I I ,ik
g) Plant storage capacity .
ikt k it k
I v f
,ikh) All unit discount constraints
. .
n n n n n
jk ijkt ijkt jk ijkt
e Y Q u Y i k n t, , , 1 1 jk n n ijkt n Y
i k t, ,L j J i) Rebate constraints , , , , .max { } . .max { } n n n n nj i k t ijkt jkt ijkt j i k t ijkt
i k t e Y
c Q u Y j J UJs c n 1 1 j n n ijkt n Y
i k t, ,s c j J UJ j) Difficult products can only be supplied from strategic and challenger suppliers 0 d l ijkt k K j J Y
,ik k) Binary variables {0,1} ijkt Y i j k t, , , {0,1} n ijkt Y i j k n t, , , , 4. ConclusionThis paper studies the supplier selection problem integrated with inventory decisions under deterministic demand conditions. In the model, the total cost consists of procurement cost, contract fees, transportation cost and inventory holding and backorder costs. Beside this price discounts and rebates are taken into account. Suppliers were classified as strategic, challenger and local and beside this the products divided into two groups as easy and difficult. Moreover these two points are provided: local suppliers apply all-unit quantity discount and strategic/challenger suppliers apply rebate and difficult products can only be supplied from strategic and challenger suppliers.
After the model is constructed, it is solved using GAMS. As a further study, other objective functions, such as minimization of defective items or maximization of on-time deliveries, can be added to the model and a heuristic approach can be used to solve the problem.
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