Multi-supplier, single-buyer models are the most common problem setting addressed in the supplier selection literature. This is mainly due to having decentralized pur- chasing operations within several companies of the same firm. Our integrated sup- plier selection and inventory model focuses on a two-stage supply chain consisting of Nsuppliers (j =1,. . .,N) and one buying firm. All Nsuppliers are assumed to have been prescreened by the firm and are thus included in the supplier base.
Within this setting, one of the concerns while determining which supplier(s) to
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136 Burcu B. Keskin
work with is the type of sourcing, that is, single sourcing, dual sourcing, or multi- sourcing.
Single sourcing may be justified when (1) lower total cost results from a much higher volume (economies of scale); (2) the buying firm has more influence with the supplier; (3) lower costs are incurred to source, process, expedite, and inspect; (4) sig- nificantly lower freight costs may be obtained; and (5) more reliable, shorter lead times are required (Benton 2007). When single sourcing is enforced, the underlying integrated supplier selection model boils down to the single-supplier, single-buyer model, which is the most stylistic model for the supply chain coordination models.
In Section 8.2.1.1, we summarize some of those integrated inventory/transportation models.
On the other hand, dual or multiple sourcing would be more appropriate when (1) the buyer requires some protection from shortages, strikes, and other disruptions of the supplier; (2) the buyer wants to maintain competition and has a backup source;
(3) the buyer is constrained by a supplier’s capacity issues; (4) local content is required for, especially for more global companies, international manufacturing locations.
Dual and multi-sourcing bring up the issues of order splitting, order assignments, inventory, and transportation considerations. We highlight some analytical models that address these problems in Section 8.2.1.2.
8.2.1.1 Single-Sourcing Models
Single-sourcing models, despite increasing attention given to JIT systems, are still rare in the literature. The methods utilized for selecting suppliers under single-sourcing restrictions include (Ghodsypour and O’Brien, 1998)
Linear weight point methods that depend on human judgment
The cost ratio, where cost of each criterion for each supplier is calculated with respect to total cost
The analytical hierarchy process that is a pairwise comparison of each criteria Unfortunately, there are not many mathematical models dedicated to supplier selection with single sourcing except those of Benton (1991) and Akinc (1993).
Benton (1991) developed a nonlinear programming formulation using the eco- nomic ordering quantity (EOQ) model to represent the inventory decisions. In that article, the buyer’s objective was to minimize the sum of purchasing cost, inventory cost, and ordering costs subject to an aggregate inventory investment constraint and an aggregate storage limitation constraint by selecting a single supplier for multiple items from a set of multiple suppliers offering all-unit quantity discounts. A heuristic procedure using Lagrangian relaxation for supplier selection, lot sizing resource lim- itations, and all-unit quantity discounts was developed. Akinc (1993), on the other hand, developed a decision support approach for selecting suppliers under the con- flicting criteria of minimizing the annual procurement costs, reducing the number of suppliers, and maximizing suppliers’ delivery and quality performances. Because
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it is challenging to quantify the delivery and quality performances, rather than trying to estimate a utility function for these criteria, the author focused on building three simple models and assessed the different trade-offs within these models.
Generalizing the results of these two aforementioned papers, we now describe a simple mathematical model that can be utilized for an integrated supplier selection and an inventory model under single sourcing. For this purpose, we first define the following notation:
D Annual demand of the buyer K Inventory ordering cost of the buyer
h Annual inventory holding cost rate of the buyer
cj Unit procurement cost offered by supplier j, j =1,. . .,N fj Annual contractual cost required by supplier j, j =1,. . .,N tj Per-mile transportation cost from supplier jto the buyer dj Distance between supplier jand the buyer
uj Per-unit transportation cost from supplier j to the buyer Wj Annual throughput capacity of supplier j
The decision variables are
Qj Order quantity of supplier j, j =1,. . .,N,Qj ≥0 Xj 1 if supplier j is used at all, 0 otherwise
Then, the mathematical model is formulated as min
N j=1
(cj +uj)D Xj+ N
j=1
fjXj+ N
j=1
K D Qj + 1
2h Qj
Xj
+ N
j=1
tjdjD
Qj Xj (MS-SB-SS) subject to
j∈J
Xj =1 (8.1)
D≤WjXj, ∀j =1,. . .,N (8.2)
Xj = {0, 1}andQj ≥0, ∀j =1,. . .,N (8.3) The objective function minimizes the total cost of selecting a supplier including
Annual procurement and unit-based transportation costs (first term)
Annual fixed contractual costs (second term)
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138 Burcu B. Keskin
Annual inventory ordering and holding (third term)
Annual distance-based (and trip-based) transportation costs (final term) Constraints (8.1) and (8.3) establish the single sourcing requirements, whereas Constraints (8.2) establish that the selected supplier should have enough capacity to serve the demand of the buyer.
Note that given Xj = 1 for a potential supplier j, j = 1,. . . ,N, the re- mainder of the problem becomes a generalized EOQ formulation with generalized transportation costs. The order quantity for supplier j is given as
Qj =
2[K +tjdj]D hcj
(8.4)
Then, the total cost of selecting supplier j would be Cj =(cj+uj)D+ fj+
2[K +tjdj]Dhcj (8.5) Finally, the supplier that has enough capacity should be selected if j∗ = arg minNj=1{Cj : D≤Wj}(i.e., Xj∗ =1), granted that the capacity requirements are satisfied.
With this transformation all of the generalization models developed for EOQ, including quantity discounts, storage and space constraints, and transportation and cargo restrictions, are applicable and usable for the integrated supplier selection and inventory problem with single-sourcing restrictions.
8.2.1.2 Dual- and Multi-Sourcing Models
The buying firm, before implementing a multi-sourcing policy, must specify the number of suppliers to employ. This is equivalent to selecting a subset of suppliers from a predetermined set that conforms the buyer’s quality, delivery, and financial requirements. The buyer then needs to determine the distribution of order quan- tities among selected suppliers. Some authorities have suggested that two suppliers should be used for a material category that has an annual dollar volume higher than a given predetermined level, which is also known as dual sourcing. Some others require multiple suppliers even though managing more than one source is obviously more cumbersome than dealing with a single source. Web-based SCM applications enable closer management of diverse suppliers, streamline supply chain processes, and drive down procurement costs. Furthermore, multiple suppliers provide more flexibility under dire disruption conditions. For this purpose, multi-sourcing models have received more attention than their single-sourcing counterparts.
Due to its ability to optimize the explicitly stated objective subject to a multi- tude of constraints, mathematical programming is the most appropriate technique that allows the decision maker to formulate such multi-sourcing integrated supplier selection and inventory problems (Aissaoui et al. 2007). It considers internal policy
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Analytical Models for Integrating Supplier Selection 139
constraints and externally imposed system constraints placed on the buying process in order to determine an optimal ordering and inventory policy simultaneously while selecting the best combination of suppliers. Due to these advantages, substantial work has been done in this area and a great number of studies is conducted that considered different aspects and instances of the problem since the 1990s. Among all the research in this area, four articles deserve special credit, including the articles by Pan (1989), Chaudhry et al. (1993), Ghodsypour and O’Brien (2001), and Tempelmeier (2002), due to their integrated supplier selection and inventory consideration. Furthermore, the two analytical models we present in this section are based on these articles. We discuss the contributions of these works in detail next.
Pan (1989) presented a linear programming model that can be used in the si- multaneous determination of the number of suppliers to utilize and the purchase quantity allocations among suppliers in a multiple sourcing system a single objective linear programming model to choose the best suppliers, in which three criteria are considered—price, quality, and service. The total cost is taken into account as an objective function, and quality and service are considered as constraints. Chaudhry et al. (1993) considered the price, delivery, and quality objectives of the buyer, as well as the production or rationing constraints of the suppliers. In that work, suppliers were assumed to offer price breaks that depend on the sizes of the order quantities.
They presented linear and mixed binary integer programming models that provide unifying frameworks for models of supplier performance measures. In a similar con- text, Ghodsypour and O’Brien (2001) developed a single-objective supplier selection model to minimize the total cost of logistics, including aggregate price, ordering, and inventory costs, subject to suppliers’ capacity constraints and the buyers’ limi- tations on budget. Our first analytical model is based on these articles, considering the EOQ-based inventory model together with multiple-supplier selection.
The final article by Tempelmeier (2002) considered the problem of supplier se- lection and purchase order sizing for a single item under dynamic demand conditions.
Suppliers offer all-units and/or incremental quantity discounts that may vary over time. Time-varying deterministic prices were dictated by time-varying (all-units or incremental) quantity discount structures with several discount levels. This includes rising or falling prices, starting with a specific period, as well as special prices for limited time intervals. The model formulation is based on the well-known analogy between the plant location problem and the dynamic lot sizing problem. Our second analytical model is based on Tempelmeier (2002).
8.2.1.2.1 Multi-Supplier, Single-Buyer with Multi-Sourcing Model 1 In addition to the notation introduced in the previous section, we define the following decision variables:
Q Total order quantity ordered from all the selected suppliers Qj Order quantity ordered from supplier j, j =1,. . . ,N
Yj Percentage ofQ
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140 Burcu B. Keskin
The revised mathematical model is then given as
min N
j=1
(cj +uj)DYj + N
j=1
fjXj+ N
j=1
K D Xj
Q +
N j=1
1
2hcjQYj
+ N
j=1
tjdjD Qj
Xj (MS-SB-MS-1)
subject to
j∈J
Xj =1 (8.6)
D≤WjXj, ∀j =1,. . .,N (8.7)
Xj = {0, 1}andQj ≥0, ∀j =1,. . .,N (8.8) Note that the total order quantity Q only appears in the objective function.
Furthermore, given Xj andYj, the objective function is convex inQ. Hence, the optimal total order quantity is given as
Q= 2D
Nj=1K Xj +N
j=1tjdjXj
N
j=1hcjYj
(8.9)
Replacing Equation (8.9) in the objective function, we obtain N
j=1
(cj+uj)DYj+ N
j=1
fjXj
+ 2D
N
j=1
K Xj+ N
j=1
tjdjXj
N
j=1
hcjYj (8.10)
Then, the overall formulation is a mixed-integer nonlinear programming formu- lation. It can be solved with any general-purpose nonlinear programming package (Ghodsypour and O’Brien 2001).
8.2.1.2.2 Multi-Supplier, Single-Buyer with Multi-Sourcing Model 2 In this model, instead of an average annual cost model, we consider the integrated supplier selection and inventory model over a finite horizon. For this model, we define the following parameters:
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Analytical Models for Integrating Supplier Selection 141
T Number of periods in the planning horizon
Dt Demand requirements of the buyer in periodt,t =1,. . .,T Kj t Fixed ordering cost for supplier j in periodt
h Holding cost percentage per period
Rj t Number of discount levels for supplier j in periodt
pjr t Unit price in discount levelr for supplier jin periodt,r =1, 2,. . .,Rj t
hjr tl Inventory holding costs for the complete demand of periodt if it is delivered
by supplier jin periodlwith discount levelr, i.e.,hjr tl =hãpjr tãdtã(t−l) gjr t Upper limit of the discount levelr for supplier j in periodt,gj o t= 0
aj t 1, if supplier j is available in periodt
uj t Unit transportation cost for supplier jin periodt The decision variables in this model are as follows:
Yjr tl Proportion of demand in periodt that is delivered by supplier j in periodl
with discount levelr
Xjr t Binary variable for selecting discount levelr in periodtfor supplier j Qjr t Quantity purchased from supplier j in periodt with discount levelr
For the case of all-units quantity discounts, the following mixed-integer linear programming model is formulated:
min T
t=1
T l=t
N j=1
Rj t
r=1
hjr tlYjr tl + T
t=1
N j=1
Rj t
r=1
kj tXjr t+
+ T
t=1
N j=1
Rj t
r=1
(pjr t+uj t)Qjr t (MS-SB-MS-2)
subject to N
j=1
T l=t
Rj t
r=1
Yjr tl =1, ∀t =1,. . . ,T (8.11)
T l=t
Yjr tlDt = Qjr t, ∀j =1,. . .,N;∀t =1,. . . ,T;∀r =1,. . . ,Rj t
(8.12) Qjr t ≤gjr tXjr t, ∀j =1,. . .,N;∀t =1,. . .,T;∀r =1,. . . ,R j t
(8.13)
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142 Burcu B. Keskin
Qjr t ≥(gj,r−1,t+1)Xjr t, ∀j =1,. . .,N; ∀t =1,. . .,T; ∀r =1,. . . ,R j t (8.14)
Rj t
r=1
Xjr t ≤aj t, ∀j =1,. . .,N; ∀t =1,. . . ,T (8.15)
Yjr tl ≤ Xjr t, ∀j =1,. . .,N; ∀t =1,. . . ,T; ∀l =t,. . .,T;
∀r =1,. . ., R j t (8.16)
Qjr t ≥0, ∀j =1,. . .,N; ∀t =1,. . . ,T; ∀r =1,. . ., R j t (8.17) Xjr t ≥0, ∀j =1,. . .,N; ∀t =1,. . . ,T; ∀r =1,. . ., R j t
(8.18)
Yjr tl ∈ {0, 1}, ∀j =1,. . .,N; ∀t =1,. . . ,T; ∀l =t,. . .,T;
∀r =1,. . ., R j t (8.19)
The objective function minimizes the cost of inventory holding, inventory ordering, procurement, and transportation. Constraints (8.11) satisfy demand fulfillment. Constraints (8.12) define the sizes of orders. The upper and lower limits of a discount level is determined with Constraints (8.13) and (8.14), re- spectively. Constraints (8.15) ensure that there will be at most one active dis- count level per delivery period. Constraints (8.16) establish the relation between the binary ordering variables and proportion of demand satisfied. Finally, Con- straints (8.17), (8.18), and (8.19) establish binary and non-negative continuous variables.
This formulation is based on the well-known analogy between plant location and the dynamic lot sizing problem. It can also be easily revised to represent incremental quantity discounts (Tempelmeier 2002). This formulation can be solved with any mixed-integer programming solver such as CPLEX.∗Tempelmeier (2002) also sug- gested a fast and effective heuristic to solve both all-units and incremental quantity discount models.
∗CPLEX is a trademark of ILOG.
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