In integrated supplier selection and inventory models, there are not many models that specifically consider multiple buyers—except those of Jayaraman et al. (1999), Crama et al. (2004), Keskin et al. (2010a), and Keskin et al. (2010b). However, multi- product models can be formulated and solved as multi-buyer problems. Hence, we prefer to group the research focusing on either multi-product or multi-buyer into this section. Furthermore, we again distinguish between single and multi-sourcing and discuss these models separately in Sections 8.2.2.1 and 8.2.2.2, respectively.
8.2.2.1 Single-Sourcing Models
Similar to the settings in Keskin et al. (2010a) and Keskin et al. (2010b), we consider a manufacturing firm with multiple geographically dispersed buyers. More specifically, we consider a total of M≥ 2 buyers procuring (or producing) a single product to meet buyer-specific stochastic demand. Unit demand arrives at the buyer according to a Poisson distribution with a buyer-specific meani,i =1,. . . ,M. To satisfy the demand at each buyer i, the firm seeks to select a subset of suppliers from a set of potential suppliers that meet initial financial, quality, and delivery criteria.
Each supplier j offers the firm a fixed contractual cost, fj, and a per-unit cost,cj, and has an associated level of quality, qj, and a disruption rate, j. In addition, each supplier j has a specific annual throughput capacity,Wj. Annual throughput capacity is calculated assuming the supplier is never disrupted.
The decision variables in this setting are
Xj 1, if supplier j is selected, 0 otherwise, j =1,. . . ,N
Yi j 1, if supplier j is assigned to buyeri,i =1,. . . ,Mand j =1,. . . ,N Qi Order quantity of buyeri,i =1,. . .,M
Ri Reorder point of buyeri,i =1,. . . ,M
Under a (Q,R) policy, the inventory position (inventory on hand+outstanding backorders) is continuously reviewed, and an order of fixed quantity Qis placed as soon as the inventory position drops to or below a reorder point R(see Federgruen and Zheng 1992). After a supplier–buyer specific lead time,Li j, the orderQiarrives at buyeriprovided that supplier jis not disrupted and has sufficient capacity to do so. In this problem, because the supplier can be disrupted, the lead time, and hence the lead time demand, are affected by the suppliers’ availability. Furthermore, asqj
represents the percentage of products provided by supplier j without any defects, only an order ofqjQi arrives at buyeri.
In terms of inventory system costs at buyers, we assume that each replenishment order incurs a fixed cost ofKiand an inventory carrying cost ofhiper unit, per unit of time. Additionally, stockouts are backlogged at a cost ofsi,i= 1,. . . ,M, per unit, per unit of time. We also explicitly account for the transportation costs between the suppliers and plants. We express the transportation cost associated with each order
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delivery between supplier jand buyeriwith two specific cost parameters: (1) a fixed cost, denoted by pi j, which represents the sum of fixed costs of ordering (setup) and transportation; and (2) a per-mile transportation cost, denoted byri j, which leads to a distance-based transportation cost ofri jdi jper order delivery, wheredi jrepresents the distance between supplier j and buyeri.
Under these assumptions, the multi-supplier, multi-buyer, single-sourcing model, ignoring the impact of disruptions, can be stated as follows:
min M
i=1
N j=1
cjE[Di]Yi j + N
j=1
fiXj+ M
i
N j
(pi j+ri jdi j)E[Di]Yi j
Qi
+ M
i=1
KiE[Di] Qi
+ M
i=1
N j=1
hi
Qi
2 +Ri −E[L T Di j]
Yi j
+ M
i=1
N j=1
sini(Ri)E[Di]Yi j
Qi
(MS-MB-SS)
subject to N
j=1
Yi j =1, ∀i =1,. . . ,M. (8.20)
Yi j ≤Xj, ∀i =1,. . . ,M and ∀j =1,. . .,N. (8.21)
M i=1
E[Di]Yi j ≤WjXj, ∀j =1,. . . ,N. (8.22)
qjXj ≥qiminYi j, ∀i =1,. . . ,M and ∀j =1,. . .,N. (8.23) Qi ≥Qminj Yi j, ∀i =1,. . . ,M and ∀j =1,. . .,N. (8.24) Xj ∈ {0, 1} and Yi j ∈ {0, 1}, ∀i =1,. . . ,M and ∀j =1,. . .,N. (8.25)
Qi ≥0, ∀i =1,. . . ,M.
The objective of this formulation minimizes the expected procurement costs, the fixed contractual costs, transportation costs, inventory ordering and holding costs, and shortage costs. Constraints (8.20) ensure that each buyer should be assigned to a selected supplier. Constraints (8.21) state that buyers can only be assigned to selected
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Analytical Models for Integrating Supplier Selection 145
suppliers. Constraints (8.22) are the capacity restriction of the suppliers; that is, the total average demand assigned to a supplier cannot exceed its annual throughput capacity. Constraints (8.23) state that the selected supplier should meet the assigned buyer’s minimum quality requirement. Constraints (8.24) ensure that each buyer should meet its assigned supplier’s minimum order quantity requirement. Finally, Constraints (8.25) establish the single-sourcing requirements and non-negativity of continuous variables.
This model, even without disruptions, is a stochastic, nonlinear, mixed-integer programming formulation. Unfortunately, it is not possible to solve this problem with traditional optimization techniques. For the deterministic version of this formu- lation, Keskin et al. (2010b) developed a generalized Bender’s decomposition-based special algorithm and discussed the effectiveness. For this formulation, Keskin et al.
(2010a) utilized a scatter-search powered simulation-optimization approach to solve this problem. This approach was not only useful in determining the best suppliers and optimum inventory levels at the buyers, but also in evaluating the impact of qual- ity, disruptions, and delivery performance of the suppliers. Combining the benefits of optimization and simulation, the overall approach provides an increased ability to understand the impact of dynamic events, total system behavior, and (increasingly important today) risk mitigation in the event of expected/unexpected disruptions.
8.2.2.2 Dual- and Multi-Sourcing Models
Before presenting the final analytical model, which focuses on multi-supplier, multi- buyer settings with multi-sourcing, we review the integrated supplier selection and inventory research in this area. Research in this area dates back to the article by Bender et al. (1985) where a computerized mixed-integer programming technique was employed at IBM to improve purchasing contracts. Since then, a number of other researchers contributed to the analytical models in integrated supplier selection and inventory, including Rosenthal et al. (1995), Jayaraman et al. (1999), Dahel (2003), Crama et al. (2004), and Basnet and Leung (2005).
Rosenthal et al. (1995) considered a buyer that should obtain the necessary num- ber of stocking items from various suppliers that charge different prices and have limited capacities and different levels of qualities. A mixed-integer linear program- ming technique is developed to evaluate different bundles offered by the suppliers.
Jayaraman et al. (1999) considered a potential set of suppliers constrained by the quality level of produced and supplied by the suppliers, the lead times to store the products, and storage capacity restrictions imposed by the suppliers. Their model is quite comprehensive except for the fact that they do not only consider the impact of inventory costs. Dahel (2003), on the other hand, presented a multi-objective, mixed-integer programming technique to simultaneously determine the number of suppliers to employ and order quantities to these suppliers in a multi-product, multi- supplier competitive environment. The model takes minimizing procurement cost, maximizing product quality, and maximizing delivery reliability. Crama et al. (2004)
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also described the purchasing decisions faced by a multi-plant company. The sup- pliers of this company offer complex discount schedules based on the total quantity (rather than cost) of ingredients purchased. Linking local and group discounts for purchases, they model the problem as a mixed-integer nonlinear programming for- mulation and solve it using off-the-shelf software. Finally, Basnet and Leung (2005) presented an extension of Tempelmeier’s (2002) work by considering a multi-period inventory lot-sizing problem, where there are multiple products and multiple suppli- ers. The decision maker needs to decide what products to order, in what quantities, with which suppliers, in which periods. An enumerative search algorithm and a heuristic are presented to address the problem.
In this section we present a formulation that generalizes the models of Jayaraman et al. (1999) and Dahel (2003) that considers integrated supplier selection and in- ventory under multi-supplier, multi-buyer, and multi-product under multi-sourcing restrictions. For this purpose, we define the following parameters:
i Index of buyers,i =1,. . .,M j Index of suppliers, j =1,. . .,N k Index of products,k=1,. . .,K ϑk Set of suppliers offering productk Kj Set of products offered by supplier j
Ij Set of buyers that can be supplied by supplier j Ik Set of buyers demanding productk
ik Units of productkdemanded by buyeri
ci j k Unit price of productkquoted by supplier j to buyeri
qi j k Percentage of rejected productkfrom supplier j to buyeri
ti j k Percentage of productkfrom supplier j to buyeri
Sk j Maximum quantity of productkthat may be purchased from supplier j due to capacity restrictions
ujr Upper cutoff point of discount bracketr for supplier j
djr Discount coefficient associated with bracketr of supplier j’s cost function The decision variables for this problem are as follows:
xi j k Units of productkpurchased from supplier j for buyeri
vjr Volume of business awarded to supplier jin bracketr
yjr 1 if the volume of business awarded to supplierjfalls on segmentr, 0 otherwise Then, the mathematical formulation is presented as
min
j∈J
r∈Rj
(1−djr)vjr+
i∈I
j∈Ji
k∈Kj
qi j kxi j k+
i∈I
j∈J
k∈Kj
ti j kxi j k
(MS-MB-MS)
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subject to
j∈Ji
xi j k = Dik, ∀i ∈I, ∀k∈Ki (8.26)
i∈I
xi j k ≤Sk j, ∀k ∈K, ∀j ∈Jk (8.27)
i∈Xj
k∈Ki
ci j kxi j k =
r∈Rj
, ∀jr ∈J (8.28)
vjr ≤ujryjr, ∀j ∈J, ∀r ∈Rj (8.29)
vj,r+1≤ujryj,r+1, ∀j ∈J;r = {1, 2,. . .rj −1} (8.30)
r∈Rj
yjr =1, ∀j ∈J (8.31)
yjr ∈ {0, 1}andvjr ≥0, ∀ ∈J, ∀ ∈Rj (8.32)
xi j k≥0, ∀i ∈I, ∀j ∈Ji, ∀k ∈Kj (8.33)
The objective function of (MS-MB-MS) minimizes the total procurement cost, the cost of defective items, and the cost of items missing their target delivery time.
Constraints (8.26) represent the condition that the total demand of each product at each buyer will be satisfied. Constraints (8.27) ensure that the total number of products procured by each supplier to all plants is within the production and shipping capacity of that supplier. Constraints (8.28) determine the dollar amount of business awarded to supplier j. Constraints (8.29) and (8.30) link the purchase of the product with the business volume discount to the appropriate segment of the discount pricing schedule for each supplier. Constraints (8.31) ensure that only one discount bracket for each supplier’s volume of business will apply. Finally, Constraints (8.32) and (8.33) ensure integrality and non-negativity on the decision variables.
Because the overall formulation is a mixed-integer linear program, this problem can be solved by any off-the-shelf software such as CPLEX. Note that this formula- tion, even though it evaluates the impact of quantity discounts, quality, and delivery performances, does not explicitly account for the inventory decisions. This would be one potential research area for the future.
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