The newsvendor problem with capacitated suppliers and quantity discounts

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Production, Manufacturing and Logistics

The newsvendor problem with capacitated suppliers and quantity discounts Roshanak Mohammadivojdan a, Joseph Geunes b,∗ a b

University of Florida, Gainesville, FL 32611,United States Department of Industrial & Systems Engineering, Texas A&M University, College Station, TX 77843, United States

a r t i c l e

i n f o

Article history: Received 7 September 2017 Accepted 8 May 2018 Available online xxx Keywords: Inventory Newsvendor problem Knapsack problem Asymptotically optimal heuristic

a b s t r a c t We consider a seller who stocks an item in anticipation of a single selling season in which demand for the item is uncertain. The seller may order stock of the item from multiple suppliers, each of which offers a quantity discount pricing structure and has production volume limits. The seller seeks to minimize its total procurement plus expected overstock and understock costs, resulting in an objective function that is neither convex nor concave in the decision variables in general. We provide an algorithmic approach that permits solving this non-convex problem in pseudopolynomial time by solving a set of 0–1 multiple choice knapsack subproblems. We also provide an efficient heuristic solution algorithm and demonstrate the algorithm’s asymptotic optimality in the number of suppliers under mild assumptions on the problem data and under certain quantity discount structures. The results of a set of computational tests demonstrate the superior performance of the knapsack-based algorithms when compared with a commercial solver. © 2018 Elsevier B.V. All rights reserved.

1. Introduction The single-period newsvendor problem is perhaps the most well-studied problem in the area of inventory planning under uncertain demand. This problem considers a decision maker who stocks an item in order to meet uncertain demand during a single selling period or season. The decision maker incurs a purchasing cost associated with the amount of stock acquired, in addition to costs and/or revenues that arise when the stock level does not equal the demand observed in the selling season. If the stock level exceeds demand, any remaining stock either produces revenue via an associated salvage value, or a disposal (or holding) cost if the remaining stock cannot be salvaged. If demand exceeds the stock level, then in addition to the opportunity cost associated with lost sales, a cost associated with a loss of customer goodwill may be incurred, reflecting an attempt to quantify an impact on future lost sales. The decision maker seeks to maximize the expected profit (or, equivalently, minimize the expected cost when opportunity cost is included) associated with stocking and selling an item, which depends on the stock level. The applicability of this model to numerous contexts involving products with short life cycles, such as fashion and technology goods, has led to an ∗

Corresponding author. E-mail addresses: rmohammadivojdan@ufl.edu (R. Mohammadivojdan), [email protected] (J. Geunes).

extensive volume of literature on the newsvendor problem and its many generalizations and extensions. Under the classical approach to this problem, the newsvendor seeks an optimal order quantity, Q, for the selling season. The period’s demand is characterized by a random variable X with a continuously defined probability density function (pdf) f and cumulative distribution function (CDF) F. For each unit stocked, the newsvendor pays a unit cost of c to an external supplier with unlimited capacity. If demand exceeds the stock level, a penalty cost of p is incurred for each unit of demand in excess of the stock level; if the stock level exceeds demand, a unit cost of h is incurred for each unsold unit at the end of the selling period. Then, under the standard expected-cost-minimization approach, the optimal order quantity satisfies F (Q ) = ( p − c )/( p + h ). This paper considers a practical generalization of the newsvendor problem to account for multiple interchangeable suppliers, each of which has finite supply capacity, violating the assumption of a single supplier with unlimited capacity who can supply any quantity at a cost of c per unit. Clearly, when the available supply of the item is unlimited and the inventory procurement cost is linear in the quantity stocked, the resulting problem is the classical newsvendor problem described above. In a variety of practical settings, however, supply availability and costs may violate the assumption of an unlimited supply that is linear in the quantity ordered. In particular, it is often the case that the inventory planner has multiple suppliers from which it can choose.

https://doi.org/10.1016/j.ejor.2018.05.015 0377-2217/© 2018 Elsevier B.V. All rights reserved.

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Consider, for example, a product for which most of the demand occurs relatively early after its initial appearance in the market, such as a cellphone or other electronic device, which may become obsolete in a short time (as newer versions arrive to the market). Such products may have various contract producers available to manufacture the item, perhaps in different geographical areas. These suppliers are not unlikely to have different production costs, and consequently might offer different pricing terms for production. Moreover, each such supplier typically faces some limit on the amount it can supply prior to the selling period, and may offer incentives to its customers, including quantity discounts to encourage high capacity utilization. This work revisits the newsvendor problem in order to account for these practical features, including capacitated suppliers and quantity discounts. We note that Burke, Carillo, and Vakharia (2007) considered a related version of this problem without quantity discounts, focusing on the benefits of diversification through multiple suppliers, while Zhang and Zhang (2011) considered a variant with a fixed plus linear cost associated with each supplier. Zhang and Chen (2013) generalized this latter model to account for all-unit discounts and applied a Bender’s decomposition algorithm to solve the resulting large-scale mixed integer nonlinear optimization problem. In this paper we consider a general class of supplier cost structures that encompasses three well-known quantity discount structures in practice (marginal, all-units, and carload discounts, which we later define more precisely) and show that this class of problems can be solved via a sequence of 0–1 multiple choice knapsack subproblems which, although not trivial, constitute a manageable problem class with respect to solution time requirements. This paper’s contributions are as follows. We provide a model for the broadly applicable newsvendor problem with multiple capacity-constrained suppliers who offer a variety of different kinds of quantity discounts. The resulting model has an objective that is non-convex, implying that the problem is not generally well behaved. Despite this, we show that the problem can be solved via the solution of a polynomial number of 0–1 multiple choice knapsack problems under a broad set of supplier pricing structures. Thus, we show that the problem can be solved in pseudopolynomial time in the worst case for several common quantity discount pricing structures. Based on this exact solution approach, we derive an extremely fast and simple heuristic solution method and demonstrate the asymptotic optimality of this heuristic as the problem size grows under concave quantity discounts. The result is a very fast solution method that provides near-optimal solutions for what appears to be a daunting problem class. The rest of this paper is organized as follows. Section 2 next reviews related literature on the newsvendor and knapsack problems. Section 3 introduces the general case of our model, defines three different common quantity discount structures used in practice, and provides optimization problem formulations for a class of multiple choice knapsack subproblems that can be used to solve the problem under these discount structures. This section also presents a dynamic programming solution approach for these subproblems, and characterizes the worst-case complexity required to solve the class of problems we consider. In Section 4, we consider the marginal and carload quantity discount structures in greater detail, and demonstrate how the special structural properties of these problems lead to additional efficiencies in their solution. Section 5 next discusses a fast heuristic solution method based on a linear underestimation of each supplier’s quantity discount structure, and shows that under certain assumptions, this heuristic solution is asymptotically optimal as the number of suppliers increases. Section 6 provides the results of a set of computational tests intended to characterize the performance of our algorithms, while Section 7 concludes.

2. Related work This section provides a review of relevant literature on the newsvendor problem and related extensions. We also provide a brief review of the well-known class of knapsack and multiple choice knapsack problems, which, as we later discuss, will become relevant to our solution approach for the class of newsvendor problems we consider. 2.1. The newsvendor problem The newsvendor problem has been under study for decades. The problem was first discussed mathematically in 1888 by Edgeworth (1888), who used the concepts and tradeoffs inherent in the newsvendor model in a banking-related context, using a normal distribution to set the level of cash reserves to meet customer cash demands. Morse and Kimball (1951) were the first to use the term “newsboy” in referring to an example of the problem in their book. Qin, Wang, Vakharia, Chen, and Seref (2011) and Khouja (1999) provided literature surveys for this general problem class. Work on variants of the newsvendor problem has considered various problem aspects. Some of this effort has focused on demand specifications and assumptions, including the degree of information available about demand prior to the selling season. Many other extensions to the newsvendor problem have been suggested, focusing on various problem dimensions such as objective function forms, utility functions and risk, and multi-product cases. 2.2. Order cost assumptions One important factor in broadening the applicability of the newsvendor model lies in considering more practical and general classes of order costs. The suppliers from which a newsvendor might order often observe economies-of-scale in production and distribution operations. Consequently, quantity discounts are sometimes employed in an effort to increase total sales and profit, while increasing capacity utilization. Thus, a segment of the literature has focused on various order cost and discount structures. Jucker and Rosenblatt (1985) considered quantity discounts from the supplier’s perspective, including “all-unit quantity discounts” (when the order quantity exceeds a threshold, all units ordered are discounted), “incremental quantity discounts” (when the order quantity exceeds a threshold, only units ordered in excess of the threshold are discounted), and “carload-lot discounts” (wherein some number of units may effectively be ordered for free if ordering a full carload of capacity). They argued that the behavior of the newsvendor in the case of “all-unit quantity discounts” is closely affected by disposal costs of the excess inventory, and showed that this type of discount affects the newsvendor’s behavior in more complicated ways than previous literature suggests. Pantumsinchai and Knowles (1991) considered a different structure in which the order consists of a number of standard sized containers. They proposed algorithms to find the optimal order quantity as well as the optimal number of containers. Khouja (1996) analyzed a newsvendor problem with an emergency supply option and showed that the existence of an emergency supply option can improve the objective. Lin and Kroll (1997) considered a dual performance measure system that seeks to “maximize the expected profit subject to a constraint that the probability of achieving a target profit level is no less than a predetermined risk level,” while assuming quantity discounts from the supplier. Arcelus, Kumar, and Srinivasan (2005) analyzed optimal ordering and pricing policies under the assumption that the supplier offers a discount or a rebate directed to end customers. Qin et al. (2011) analyzed three types of quantity discount structures

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offered by a supplier: “Linear quantity discounts,” “All-units quantity discounts,” and “Incremental-units quantity discounts,” and provided algorithms to determine the optimal order quantity in each case. Burnetas, Gilbert, and Smith (2007) studied a system in which the supplier has less information on the demand distribution than the newsvendor, and designed discount structures to offer to the buyer. Altintas, Erhun, and Tayur (2008) and Lau, Lau, and Wang (2007) also worked on variations of a discount model from the supplier’s perspective. The problem we explore in this paper considers the existence of multiple suppliers, each with limited capacity, under three different commonly employed order cost structures, (1) Marginal quantity discounts, (2) All-units discounts, and (3) Carload discounts. To the best of our knowledge, the problem with multiple, capacitated suppliers with various quantity discount structures has not been addressed in the literature. Our work is related to a stream of work on supplier selection strategies considered by Burke et al. (2007), who considered the problem with uncertain demand when suppliers do not offer quantity discounts. These authors subsequently considered heuristic approaches for supplier selection problems with quantity discounts, but under deterministic demand (Burke, Carillo, & Vakharia, 2008a), as well as the generalization of their earlier work to permit general supplier reliability functions (Burke, Carillo, & Vakharia, 2009). Burke, Geunes, Romeijn, and Vakharia (2008b) provided an exact approach for the problem with deterministic demand and general concave quantity discounts. Each of these cases resulted in a problem with either a concave or convex objective function value. Zhang and Zhang (2011) considered the problem with uncertain demand, when the cost of buying from any supplier takes a fixed plus linear cost form, with lower and upper bounds on the quantity allocated to each supplier. Thus, in this case, the objective function is neither concave nor convex in general. They provide a customized branch-and-bound algorithm for this problem class. Zhang and Chen (2013) generalized this problem to account for suppliers who offer all-units quantity discount structures, and proposed a Benders’ decomposition algorithm for solving the resulting large-scale mixed-integer nonlinear optimization problem. When considering capacity-constrained suppliers who offer different types of quantity discount structures and uncertain demand, as we later show, the resulting objective function is neither convex nor concave, leading to a global optimization problem. 2.3. Knapsack problems As we will later show, we can solve the newsvendor problem with multiple capacitated suppliers by solving a set of binary multiple choice knapsack problems. The binary multiple choice knapsack problem is a well-known optimization problem because of its extensive applications in operations research; it often appears as a subproblem in more complicated problems. This problem considers a set I of n items, where the kth variant of item i has size aki and reward cik , and seeks to insert a subset of the items and a variant of each selected item into a knapsack with capacity U, while maximizing the total reward from the inserted items. The binary multiple choice knapsack problem (MCKP) in its simplest form may be formulated as follows.

maximize subject to:



m k k i∈I k=1 ci xi  m k k i∈I k=1 ai xi ≤ U, m k k=1 xi ≤ 1 xi ∈ {0, 1},

i ∈ I, i ∈ N = {1, 2, . . . , n}.

The MCKP was introduced by Sinha and Zoltners (1979) and has been studied extensively and in various settings, including nonlinear objectives and/or a nonlinear form of the constraint,

3

or under an assumption that the sizes and/or rewards may be non-deterministic. The case in which a single variant exists for each item is known simply as the 0–1 knapsack problem (KP) and this special case will arise later in our analysis of marginal discount functions. We also note that our primary problem of interest falls into the category of nonlinear knapsack problems which, as we will see, can be solved via a series of 0–1 knapsack problems. The text by Martello and Toth (1990) focuses on linear versions of the KP and appropriate solution methods and algorithms for solving problems in this class. A dynamic programming approach is often an efficient method for solving different variants of this problem. Bretthauer and Shetty (2002a) provided a thorough review of nonlinear knapsack problems. Some of these nonlinear knapsack problems are defined with special structural characteristics, such as convex or concave objective functions, or under separability of the decision variables. Readers interested in nonlinear variants of the knapsack problem are referred to the works of Moré and Vavasis (1991), Hochbaum and Hong (1995), Billionnet and Calmels (1996), Kiwiel (2007), Bretthauer and Shetty (2002b), Klastorin (1990), and Sharkey, Romeijn, and Geunes (2011). As noted, the class of newsvendor problems we consider results in a class of nonlinear KPs in which the objective function is neither convex nor concave, and is non-separable. In contrast, past work on nonlinear knapsack problems by Zipkin (1980) and Bitran and Hax (1981), for example, assumes separable and convex functions associated with each decision. The knapsack subproblems ultimately solved within our algorithm are closely related to the problem considered by Marchand and Wolsey (1999), which introduces a single continuous variable to the classical binary knapsack problem (this generalization utilizes an upper bound on the continuous variable’s value). Marchand and Wolsey (1999) focus on characterizing the convex hull of integer solutions for this problem by deriving strong valid inequalities that serve as facets. Similar approaches have been developed for the closely-related class of fixed-charge single-node flow problems (see, for example, Atamtürk (2001)). As we will show later, the solution approach we provide requires solving a set of MCKP subproblems with both upper and lower bounds on the total capacity consumed by items in the knapsack. To the best of our knowledge, a solution method for the MCKP with both lower and upper bound capacity limits on the knapsack capacity has not previously appeared in the literature. We develop dynamic programming algorithms for solving these subproblems, as well as an overall algorithmic approach for solving the newsvendor problem of interest. 3. Model description and analysis As in the standard approach to the newsvendor problem, we let Q denote the quantity stocked at the inventory stage immediately prior to the selling period. Let X denote a random variable for demand in the selling period, with probability density function (pdf) f and cumulative distribution function (CDF) F; we assume that X is continuous on X˜ ⊆ R1 and f > 0 for x ∈ X˜ . Define p as the unit penalty cost when demand exceeds the stock level, and let h denote a unit cost associated with an unsold item at the end of the selling period. If an unlimited quantity of supply is available at a unit cost of c, independent of the quantity ordered, then, as noted in the introduction, the minimization of expected singleperiod cost results in the optimality of an order quantity determined by the familiar newsvendor formula F (Q ) = ( p − c )/( p + h ). Suppose, on the other hand, that multiple qualified suppliers exist for the item, and let I denote the set of n such suppliers, indexed by i. Let ui denote an integer-valued maximum amount that can be ordered from supplier i ∈ I, and let qi equal the amount ordered from supplier i ∈ I. The cost associated with ordering qi units

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Fig. 1. Piecewise linear function with one breakpoint. Fig. 3. Carload discount function.

Fig. 2. All-Units discount function with one breakpoint.

from supplier i is determined by the function gi (qi ). The goal of the inventory planner is to minimize the expected total cost, including  the cost of ordering a total of Q units, where i∈I qi = Q, as well as the expected penalty and excess inventory costs after demand realization. Thus, we wish to solve the following problem:

(P1) minimize s.t.



i∈I

gi ( qi ) + G ( Q )

Proposition 3.1. If each gi (qi ) function is a piecewise-linear function m defined for all 0 ≤ qi ≤ ui with breakpoints 0 = b0i < b1i < · · · < bi i < m +1

bi i = ui , then an optimal solution exists for P1 with at most one qi value strictly between two breakpoints.

0 ≤ qi ≤ ui , i ∈ I,  qi = Q, i∈I

where G(Q ) = hE [(Q − X )+ ] + pE [(X − Q )+ ], and (x )+ = max{x, 0}. Note that G(Q) is a continuous and strictly convex function.1 We next discuss various functional forms of the ordering cost function gi that often arise in practice and are associated with quantity discounts. 3.1. Quantity discount structures For each of the problems we consider, the function gi (qi ) will take the form of a piecewise-linear function with mi breakm

an all-units cost value that is higher than the cost at the next higher breakpoint is set equal to the cost at the next higher breakpoint. This structure discourages unnecessarily inflating the shipment quantity to obtain the cost at the breakpoint. The solid lines in Fig. 3 illustrate a carload discount structure imposed on an original all-units structure (shown using the dashed lines in Fig. 3). Observe that even numbered intervals have unit cost values equivalent to the corresponding all-units structure, while odd numbered intervals have zero slopes. We next state two key results that will facilitate developing our solution approaches for problem P1 under each of the quantity discount structures we have described. The first of these generalizes a property Zhang and Chen (2013) showed for the all-units case to apply to any piecewise-linear function, while the second property helps to reduce the search for an optimal value of Q.

m +1

points, where 0 = b0i < b1i < · · · < bi i < bi i = ui , and ui denotes the maximum capacity of supplier i for each i ∈ I. We first consider the case in which each supplier offers a marginal units discount structure, where gi (qi ) is a continuous piecewise-linear funci tion with slope cri for bri ≤ qi ≤ br+1 and cri > cr+1 for r = 0, . . . , mi . i Fig. 1 shows a marginal discount structure with a single breakpoint. The second functional form we will consider is known as an all-units discount structure, such that gi (qi ) = cri qi if bri ≤ qi ≤ br+1 , i i with cri > cr+1 for r = 0, . . . , mi . Fig. 2 illustrates an all-units discount structure with a single breakpoint. The third and final discount structure we will consider is known as a carload discount. This structure is nearly the same as an all-units structure with the exception that any quantity with

Proof. This can be shown using an interchange argument as in Zhang and Chen (2013). Suppose an optimal solution exists with bli < qi < bl+1 and bkj < q j < bkj +1 , and we increase qi by some ari bitrarily small  > 0 while decreasing qj by  . Such a change is feasible and the corresponding change in objective function value j equals (cli − ck ) . If this quantity is negative, then this contradicts the optimality of the original solution. If this quantity is positive, then a solution in which qi is decreased by  while qj is increased by  results in lower cost, again contradicting the optimality of j the solution. If this quantity equals zero, then cli = ck , and an alternative optimal solution therefore exists with either qi or qj (or both) at its nearest breakpoint. This argument may be repeatedly applied until arriving at an optimal solution with at most one qi value strictly between breakpoints.  , then Proposition 3.2. If an optimal solution satisfies bri < qi < br+1 i the optimal value of Q satisfies F (Q ) = ( p − cir )/( p + h ). Proof. Suppose we have an optimal solution in which bri < qi < br+1 , and note that the objective function is differentiable with rei spect to qi at this point. The partial derivative with respect to qi at this point equals cri + hF (Q ) + p(F (Q ) − 1 ). A necessary condition for the optimality of the given solution requires this partial derivative to equal zero, which implies the condition stated in the proposition is satisfied.  3.2. Multiple choice knapsack subproblems

To see this, note that it can be shown that G (Q ) = hF (Q ) + p(F (Q ) − 1 ) (see, e.g., Nahmias & Olsen, 2015), while G (Q ) = (h + p) f (Q ), which is strictly positive for any Q ∈ X˜ . 1

We next formulate two subproblems that can be used to solve problem P1 under any piecewise-linear functional form of the

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gi (qi ) functions. The first of these subproblems assumes that exactly one qi variable takes a value strictly between two breakpoints, which implies that every other variable takes a value at a breakpoint (note that the set of breakpoints for a supplier includes 0 and the supplier’s capacity limit). We will refer to such a variable as a fractional variable. Given a supplier i, let Cri equal the total cost at breakpoint bri . Next, let f denote the index of the variable selected to take a value strictly between breakpoints, and suppose these are breakpoints brf and br+1 . Proposition 3.2 indicates that f

the optimal value of Q should satisfy Q = F −1 (( p − crf )/( p + h )), which we denote by Q¯ f . In order to formulate this subproblem, let xir denote a binary variable equal to one if qi = bri and zero otherwise. We can then formulate our first subproblem as follows.

Minimize

Subject to:



m i +1

i∈I\{ f } r=1



Cri xir + crf (q f − brf )

m i +1

i∈I\{ f } r=1

(1)

bri xir + q f = Q¯ f ,

(2)

brf ≤ q f ≤ br+1 , f m i +1 r=1

(3)

xir ≤ 1, i ∈ I\{ f },

(4)

Subject to: Q¯ f − br+1 ≤ f m i +1 r=1



m i +1

i∈I\{ f } r=1



m i +1

i∈I\{ f } r=1

C˜ri xir

(6)

bri xir ≤ Q¯ f − brf ,

(7)

xir ≤ 1, i ∈ I\{ f },

(8)

m i +1 r=1

bri xir = Q,

(11)

xir ≤ 1, i ∈ I,

(12)

xir ∈ {0, 1}, i ∈ I, r = 1, . . . , mi + 1.

(13)

In principle, we would require solving problem MCKPE(Q) for a potentially large number of candidate values of Q. However, the dynamic programming approach we discuss in the next section permits solving this problem once using an upper bounding value of Q = Qmax , which provides an optimal solution for all smaller values of Q as a byproduct (the worst-case complexity of this approach is O (nMQmax )). Thus, solving one instance of MCKPE(Qmax ) and one instance of MCKP2(f, r) for each choice of fractional variable (of which we have n candidate values) and every interval of the corresponding piecewise-linear function (of which we have mi choices) allows us to solve problem P1 under any arbitrary piecewise-linear functional form for the gi functions in O (n2 M2 Qmax ). As we show in Section 4, under the marginal and carload discount cost structures, we can reduce the dimension of the required subproblems by exploiting special properties of these discount cost structures.

Problem MCKP2(f, r) is a multiple choice knapsack problem with mi + 1 choices for each i ∈ I\{f} and lower and upper bounds on the knapsack capacity, while formulation MCKPE(Q) is a multiple choice equality-constrained knapsack problem with mi + 1 choices for each i ∈ I. For simplicity, we define L and U as the lower and upper bounds on the knapsack capacity in MCKP2(f, r) and we note that MCKPE(Q) is a special case of MCKP2(f, r) in which L = U. Define W = U − L and we define fi (l, l + W ) as the minimum cost solution containing items 1 through i with lower and upper bounds of l and l + W, respectively. We also let ki (l, u) denote the capacity consumption associated with the solution that gives fi (l, l + W ). We first initialize f1 (0, u ) = k1 (0, u ) = 0 for u = 0, . . . , W . For br1 ≤ u < br+1 , if C˜r1 < 0 then f1 (0, u ) = C˜r1 and k1 (0, u ) = br1 . To 1 compute f1 (l, l + W ) for l = 1, . . . , L, if b11 > l + W or u1 < l, then f1 (l, l + W ) = ∞. Otherwise, from the set of r values for which br1 ∈ [l, l + W ], we choose the one with smallest C˜r1 value (called  r ) and we set k1 (l, l + W ) = br and f1 (l, l + W ) = C˜1 . 1

xir ∈ {0, 1}, i ∈ I\{ f }, r = 1, . . . , mi + 1.

(9)

f In the formulation MCKP(f, r), we have defined C˜ri = Cri − cr bri and f f have suppressed the constant term Cr + cr (Q¯ f − br ) in the objecf

tive function. Section 3.3 provides a dynamic programming algorithm for solving MCKP2(f, r). Letting Qmax denote an upper bound on the order quantity2 and M = maxi∈I {mi } with n = |I|, this dynamic program has worst-case complexity of O (nMQmax ). The second type of subproblem we will consider assumes that no variable takes a value strictly between breakpoints. For a candidate value of Q, this problem is an equality version of the MCKP and is formulated as follows.

MCKPE(Q)

i∈I r=1

(5)

Note that we have omitted a constant from the objective function f equal to Cr , the total cost associated with supplier f at breakpoint r. Substituting qf out of the formulation using (2), we obtain the two-sided multiple choice knapsack problem (MCKP2) below (we say two-sided because both lower and upper limits exist on the knapsack capacity). We refer to this subproblem as MCKP2(f, r) to denote the dependence of the problem on the choice of the fractional variable index f and the choice of breakpoint interval [brf , br+1 ]. f

Minimize

i +1  m

3.3. Solution methods for MCKP2(f, r) and MCKPE(Q)

xir ∈ {0, 1}, i ∈ I\{ f }, r = 1, . . . , mi + 1.

MCKP2( f, r)

Subject to:

5

Minimize

i +1  m

i∈I r=1

Cri xir

(10)

2 We note that a valid and useful value of Qmax is given by Qmax = F −1 ( p/( p + h )), the optimal order quantity when an unlimited quantity can be obtained at zero unit cost.

r

Next, for i > 1 we compute fi (0, u) for u = 0, . . . , W as follows, defining fi (l, u ) = ∞ for any u < 0:

fi (0, u ) = min{ fi−1 (0, u );

min

{ fi−1 (0, u − bri ) + C˜ri }}.

r=1,...,mi +1

If the first term gives the minimum, then ki (0, u ) = ki−1 (0, u ). ∗ If the second term gives the minimum, then ki (0, u ) = bri + ∗ r ∗ ki−1 (0, u − bi ), where r is the minimizing term. For l > 0, we consider fi (l, l + W ), computing:





fi−1 (l, l + W ); fi (l, l + W ) = min minr=1,...,mi +1 { fi−1 (max{0; l − bri }, . l + W − bri ) + C˜ri } As before, if the first term gives the minimum, then ki (l, l + W ) = ki−1 (l, l + W ). If the second term gives the minimum, then ki (l, l + ∗ ∗ ∗ W ) = bri + ki−1 (max{0; l − bri }, l + W − bri ) where r∗ is the minimizing term. Note that the number of values of l that must be considered in computing fi (l, l + W ) is no more than U, which is bounded by the maximum possible value of Q = Qmax . Letting M = maxi=1,...,n {mi }, and n = |I|, the worst-case complexity of this

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dynamic program equals O (nMQmax ). We can apply this same approach to solve MCKPE(Qmax ) with W = 0, with a worst-case complexity that is also equal to O (nMQmax ).

 associated with the constraint i∈I qi = Q, we can write the generalized KKT conditions associated with problem P1 as follows.

∂ gi (qi ) + vi − vi + w 0

i∈I

(14)

4. Analysis of special cases As noted in the previous section, we focus primarily on the three kinds of quantity discount structures that are often found in practice: marginal, all-units, and carload discounts. This section considers the marginal and carload cases in greater detail, as we are able to reduce the number of subproblems that must be considered for these cases by exploiting their special structure.

hF (Q ) − p(1 − F (Q )) − w = 0

(15)

vi ( qi − ui ) = 0

(16)

vi (−qi ) = 0

4.1. Marginal discounts Under the marginal discount structure, each gi function is concave in qi . When this is the case, the following proposition applies that shows that an interior breakpoint is never optimal for any supplier under a marginal quantity discount structure. Proposition 4.1. If gi (qi ) is a concave function for each i ∈ I, a solution in which qi = bri for some i ∈ I cannot be optimal. Proof. Suppose we have an optimal solution with q∗i = bri for some i ∈ I. Let Q∗ denote the optimal value of Q in this optimal solution, and recall that G(Q) is a strictly convex function. The left-hand directional derivative of the objective function with respect to qi at   q∗i = bri , equal to cri + G (Q ∗ ), must satisfy cri + G (Q ∗ ) ≤ 0 in order for an optimal solution to occur at q∗i = bri and Q∗ (otherwise 

cri + G (Q ∗ ) > 0, and an improving solution exists at qi = q∗i −  and Q = Q ∗ −  for some  > 0). The right-hand directional deriva i tive with respect to qi at q∗i equals cr+1 + G (Q ∗ ) < cri + G (Q ∗ ) ≤ 0. This implies that a strictly improving solution exists at qi = q∗i +  and Q = Q ∗ +  for some  > 0, contradicting the optimality of the given solution. 

The above property along with the implications of Proposition 3.1 indicates that multiple choice constraint set (8) is not necessary, and the subproblem becomes equivalent to a 0–1 knapsack problem with lower and upper knapsack capacity limits. Thus, the required instances of MCKP2(f, r) that must be solved are of far smaller size. Similarly, the multiple choice constraints (12) are unnecessary in the MCKPE(Q) subproblem, and this subproblem corresponds to a 0–1 knapsack problem. We next show how additional optimality conditions can serve to further reduce the dimension of the subproblems we need to solve under a marginal discount structure for every supplier. Because G(Q) is everywhere differentiable, each of the functions gi is a locally Lipschitz continuous function on R1+ , and the feasible region is defined by a linear set of constraints, the generalized Karush–Kuhn-Tucker (KKT) conditions are necessary for optimality for problem P1 (see Hiriart-Urruty, 1978) under marginal discounts. Define ∂ gi (qi ) as the set of subgradients3 of the function gi at qi . If gi is not differentiable at qi , ∂ gi (qi ) corresponds to an interval [∂ l gi (qi ), ∂ u gi (qi )]; otherwise ∂ l gi (qi ) = ∂ u gi (qi ), and both of these values correspond to the derivative of gi at qi . For the marginal discount functions we have defined, note that the smallest subgradient value of the function gi (qi ) equals the slope of the i , while the largest subgradient last piecewise-linear segment, cm i c0i .

value equals the slope of the first segment, Letting vi (vi ) denote the KKT multiplier associated with the constraint qi ≤ ui (−qi ≤ 0), and letting w denote the multiplier 3 Because we assume each gi is concave, a subgradient ξ at qi = q0i satisfies g(qi ) ≤ g(q0i ) + ξ (qi − q0i ) for all qi ≥ 0.

qi − ui ≤ 0

i∈I

i∈I

(17)

i∈I

−qi ≤ 0

i∈I

vi , vi ≥ 0

i∈I

(18)

(19)

For a candidate value of the total order quantity Q = Q˜ , the necessary KKT conditions force some variables to their lower or upper bound values, reducing the search for such solutions. In particular, the above KKT conditions imply that the following must be satisfied at a KKT point for a candidate value of Q = Q˜ . Proposition 4.2. Given a candidate value of Q = Q˜ , a corresponding KKT solution requires qi = 0 for all i such that F (Q˜ ) > quires qi = ui for all i ∈ I such that F (Q˜ ) <

i p−cm

p+h

i

p−c0i , p+h

and re-

.

Proof. KKT conditions (15) and (14) imply −hF (Q˜ ) + p(1 − F (Q˜ )) ∈ ∂ gi (qi ) + vi − vi . The set of subgradients of gi (qi ), ∂ gi (qi ) lies in i ]. If −hF (Q ˜ ) + p(1 − F (Q˜ )) < ci , then a correthe interval [c0i , cm 0 i sponding KKT point must have vi > 0, which implies qi = 0 by complementary slackness condition (17). If, on the other hand, i , then a corresponding KKT point must −hF (Q˜ ) + p(1 − F (Q˜ )) > cm i have vi > 0, which implies qi = ui via (16).  As a result of these conditions, when we solve each instance of MCKP2(f, r), for each variable in I\{f}, we can use the above conditions to fix some of the variable values and reduce the size of the associated optimization problem. 4.2. Carload discounts Under a carload discount structure, illustrated in Fig. 3, we can apply the same logic used in the proof of Proposition 4.1 to show the following. Proposition 4.3. Under a carload discount structure, qi = bri for odd values of r in any optimal solution, i.e., a solution in which qi lies at the left-most end of a flat segment of the piecewise-linear curve cannot be optimal. This proposition implies that we only need breakpoint choice variables xir for the even valued breakpoints for the carload discount structure. It is possible, however, for the single fractional variable to fall on the interior of one of these flat segments. Thus, while the subproblems MCKP2(f, r) and MCKPE(Q) have the same number of binary choice variables as in the all-units case, the carload discount case requires solving twice as many instances of subproblem MCKP2(f, r) as in the all-units case.

Please cite this article as: R. Mohammadivojdan, J. Geunes, The newsvendor problem with capacitated suppliers and quantity discounts, European Journal of Operational Research (2018), https://doi.org/10.1016/j.ejor.2018.05.015

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5. Heuristic relaxation solution This section proposes a fast heuristic solution approach based on a linear underestimation of each of the piecewise-linear quantity discount structures for the gi functions in Section 3. We first note that the special case of problem P1 in which each procurement cost function gi (qi ) is linear in qi with slope ci was considered by Burke et al. (2007). In particular, suppose gi (qi ) = ci qi for 0 ≤ qi ≤ ui for each i ∈ I, and suppose suppliers are sorted in nondecreasing order of these ci values. Burke et al. (2007) showed that an optimal solution exists such that Q takes a form of either  Q = F −1 (( p − ci )/( p + h )) for some i ∈ I, or Q = ki=1 ui for some k between 1 and n. In order to find a solution which satisfies this property, we sort the suppliers in nondecreasing order of ci values, then starting from the first supplier and Q = 0, we go through the suppliers one by one and add to the value of Q until we either  j−1 find a supplier j for which Q = i=1 ui + q j can be equal to the above fraction or we get to the end and the least cost solution  with Q = ki=1 ui for some k between 1 and n will be the optimal solution. This solution approach requires sorting suppliers in nondecreasing order of ci values, and then evaluating each supplier which takes O (n ); thus, the corresponding algorithm requires O (n log n ) time. The heuristic solution approach we propose replaces each gi function with a linear underestimating function in problem P1 and then solves this relaxation. For the marginal discount case, supplier i’s marginal discount cost function is replaced by a linear function with slope cui = gi (ui )/ui (note that this linear underestimation equals the actual marginal cost function at its endpoints, 0 and ui , and is strictly less than the marginal cost function at all points in between). For both the all-units and carload discount cases, we simply replace each gi function with a linear function with slope i , and solve the resulting relaxation (note that this linear undercm i estimation equals the actual function value at 0 and for all qi such m that bi i ≤ qi ≤ ui ; at all other qi between 0 and ui , this linear function is strictly less than the actual function value). After solving this relaxation, note that the resulting solution is of course feasible for problem P1, and this therefore provides a fast and efficient heuristic solution method for solving problem P1. Our computational test results, presented in Section 6, show that this heuristic approach is extremely fast and effective. For marginal quantity discount structures, the following proposition provides theoretical support for the effectiveness of this approach. Proposition 5.1. The linear underestimation heuristic solution method is asymptotically optimal under bounded distributions of marginal cost function parameters and a demand distribution belonging to a scale family. ∗ ZP1

∗ ZLUR

Proof. Let and denote the optimal solution values for problem P1 and the linear underestimating relaxation (LUR), ∗ ∗ . Note that Proposition 3.1 holds respectively; note that ZLUR ≤ ZP1 under linear gi functions, and an optimal solution exists in the linear cost case with at most one fractional variable value. If there is no fractional variable value in the optimal solution of LUR, then ∗ ∗ , i.e., this this solution is feasible for P1 and, therefore, ZLUR = ZP1 solution is optimal for P1. Suppose, however, that a fractional variable j exists in the optimal solution to LUR. Let Z¯ H denote the objective function value of the solution to LUR in problem P1, i.e., when applying the correct marginal cost function of the original ∗ problem P1. The value of the difference between Z¯ H and ZLUR equals the difference in cost for supplier j in problem LUR and the cost for supplier j in problem P1. This difference is bounded by the difference in cost for supplier j at some breakpoint brj in the marginal cost function, g j (brj ), and in the linear underestimation of j

this function at this point, cu brj = (g j (u j )/u j )brj . Note that we can

write g j (brj ) =



m j +1

g j (b j

r−1

k=1 m +1

)/b j

j

7

j (ckj −1 − ckj )bkj + cr−1 brj , while (g j (u j )/u j )brj =

brj =

m j

k=1

m j +1

(ckj −1 − ckj )bkj (brj /b j

) + cmj j brj ,

and

j j j the difference between g j (brj ) and cu brj equals (cr−1 − cm j )brj +   m +1 m +1 m (1 − brj /b j j ) r−1 (c j − ckj )bkj − (brj /b j j ) k=j r (ckj −1 − ckj )bkj . k=1 k−1 We assume that the parameters cri and bri are drawn from bounded distributions with finite supports, i.e., cri ∈ [c, c] and bri ∈

[b, b] for all i ∈ I and r = 1, . . . , mi + 1, and that mi ≤ M for all i ∈ I for some finite M. As a result, we can bound the differj ence between g j (brj ) and cu brj by the finite value (c − c )b(M + 1 ). ∗ Thus, Z¯ H − ZLUR ≤ (c − c )b(M + 1 ), which also implies that Z¯ H −

∗ ≤ (c − c )b(M + 1 ), independent of the number of suppliers, n. ZP1 We next consider the problem as the number of suppliers, n, increases, assuming the scale of the demand distribution increases linearly in n (in other words, as the scale the demand distribution increases, additional capacity is required, which increases the number of suppliers needed). That is, if X denotes the random variable for demand for the problem with a single supplier, Y = nX gives the corresponding demand random variable for the problem with n suppliers. We assume that the distribution of X, F, is a scale family, implying that if Fn is the distribution of Y, then Fn = F (y/n ) for y ∈ R and Fn−1 (α ) = nF −1 (α ) for α ∈ (0, 1) (well-known scale families of distributions include the normal, exponential, gamma, uniform, and Weibull distributions). We would like to demonstrate the asymptotic optimality of the heuristic solution obtained by solving the linear relaxation LUR. Let ∗ (n ) denote the heuristic solution obtained by solvZ¯ H (n ) and ZP1 ing the relaxation and the optimal solution for problem P1, respectively, with n suppliers. We would like to show that

lim

n→∞

∗ (n ) Z¯ H (n ) − ZP1 = 0. ∗ ZP1 (n )

∗ (n ) ≤ (c − c )b(M + 1 ) < ∞ inBecause the numerator Z¯ H (n ) − ZP1 ∗ (n ) = ∞. Let dependent of n, we need to show that limn→∞ ZP1 Z ∗LB (n ) denote the optimal solution obtained by minimizing Gn (Q) over Q ≥ 0, where Gn (Q ) = hE[(Q − Y )+ ] + pE[(Y − Q )+ ], and note ∗ (n ). Let Q ∗ denote the optimal solution for this that Z ∗LB (n ) ≤ ZP1 n lower bounding problem with n suppliers, and observe that Qn∗ = Fn−1 (α ∗ ) = nF −1 (α ∗ ), where α ∗ = p/( p + h ). Thus,

Gn (Qn∗ ) = hE[(Qn∗ − Y )+ ] + pE[(Y − Qn∗ )+ ] = hE[(nF −1 (α ∗ ) − nX )+ ] + pE[(nX − nF −1 (α ∗ ))+ ] = nE[(F −1 (α ∗ ) − X )+ ] + npE[(X − F −1 (α ∗ ))+ ] = nG1 (Q1∗ ). This shows that limn→∞ Gn (Qn∗ ) = limn→∞ nG1 (Q1∗ ) = ∞. Because ∗ (n ), this implies lim ∗ Z ∗LB (n ) ≤ ZP1 n→∞ ZP1 (n ) = ∞, i.e., the solution of the linear relaxation LUR is asymptotically optimal.  6. Computational test results This section presents the results of a set of computational experiments intended to compare the efficiency of the proposed dynamic programming based algorithm and also the heuristic approach with a state-of-the-art commercial solver. These experiments are designed for marginal quantity discount and carload discount cases. Section 6.1 discusses the performance of the proposed algorithm compared to a global optimizer for nonconvex MINLPs, BARON, for problems with normally distributed demand.4 In Section 6.2, we present computational tests for our heuristic

4 Although the normal distribution admits negative values, we choose parameter values such that the probability of negative demand is negligible.

Please cite this article as: R. Mohammadivojdan, J. Geunes, The newsvendor problem with capacitated suppliers and quantity discounts, European Journal of Operational Research (2018), https://doi.org/10.1016/j.ejor.2018.05.015

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R. Mohammadivojdan, J. Geunes / European Journal of Operational Research 000 (2018) 1–11 Table 1 Parameters used in random problem generation: Marginal discount case.

Table 2 Parameters used in random problem generation: Carload discount case

Fixed parameters Unit shortage cost Unit cost of an unsold item

Fixed parameters p h

300 50

Unit shortage cost Unit cost of an unsold item Ratio for c0i (c0i = r0 × c4i ) Ratio for c2i (c2i = r2 × c4i ) Ratio for bi1 (bi1 = r1 × ui ) Ratio for bi2 (bi2 = r2 × ui ) Ratio for bi3 (bi3 = r3 × ui ) Ratio for bi4 (bi4 = r4 × ui ) Parameter c4i ui

Randomly generated parameters (U[a, b]) Parameter c1i r (c0i = r × c1i ) ui r ( b i = r  × ui )

a 100 1.5 20 0.3

b 150 2 30 0.8

algorithm, and compare the solutions found by the heuristic approach with the optimal solutions. These tests were implemented on a Intel® CoreTM i7-2760QM CPU, 2.4 gigahertz processor with 8 gigabytes RAM.

6.1. Performance of dynamic programming algorithm In this section, we evaluate the efficiency of the proposed dynamic programming (DP) algorithm (coded in C++) against the mixed-integer nonlinear optimizer, BARON. We assume that the demand follows a normal distribution with mean μd and standard deviation σ d , denoted as N(μd , σ d ). The reason for choosing the normal distribution is its ability to describe many practical situations in which the demand that the retailer is trying to satisfy is the aggregation of the independent random demands of many individual customers. In particular, if each of n buyers makes a purchase with probability p, then total demand follows a binomial distribution with parameters n and p. The normal distribution provides a very close approximation of the binomial distribution when np ≥ 5 and n(1 − p) ≥ 5 (see, e.g., Hayter, 2012). Moreover, by the central limit theorem, when demand arises from some number n of independent and identically distributed random demand sources, then as n becomes large, the distribution of total demand tends to a normal distribution. In order to analyze the performance of the algorithm, we consider a set of scenarios for the demand distribution parameters μd and σ d , together with various values of the number of suppliers (n). We also analyze the effect of the discount structure offered by the suppliers by investigating the marginal discount and carload discount cases. As mentioned earlier in Section 3, the worst-case complexity of the DP algorithm is O (n2 M2 Q max ), and our goal here is to compare the efficiency of our algorithm with the solver for various values of n and Qmax (letting c1min = mini∈I {c1i }, we let  Q max = F −1 ( p − c1min )/( p + h ) ; we wish to capture the influence of Qmax on performance by considering various potential values of this parameter). First, we assume that each supplier offers a marginal quantity discount structure. The unit cost of an unsold item, h, and the unit cost of unsatisfied demand, p, are assumed constant, as shown in Table 1. Supplier i ∈ I offers a marginal quantity discount with capacity ui and breakpoint bi . The unit price for qi < bi is c0i and for qi ≥ bi equals c1i . All parameters were generated randomly using uniform distributions, with the bounds of these distributions shown in Table 1. We first generated c1i values and then calculated c0i as the product of c1i and a random coefficient ri between 1.5 and 2. Similarly, bi is a random proportion of ui between 0.3 and 0.8. Tables 3–7 compare the performance of the DP algorithm with BARON for various demand distribution parameters and values of n when marginal discount structure is used. The worst-case com-

p h r0 r2 r1 r2 r3 r4 a 10 0 0 50

30 0 0 500 1.8 1.2 2/9 1/3 5/9 2/3 b 1500 100

Table 3 Demand ∼ Normal(μd = 50, σd = 10 ) Avg Run time (seconds)

Std Dev (seconds)

n

DP

BARON

DP

BARON

15 20 30 40 50

0.454 0.592 1.391 2.266 3.986

2.323 2.656 3.403 4.509 4.639

0.147 0.305 0.491 0.501 1.609

0.354 0.924 0.648 2.264 0.834

Table 4 Demand ∼ Normal(μd = 100, σd = 30 ). Avg Run time (seconds)

Std Dev (seconds)

n

DP

BARON

DP

BARON

15 20 30 40 50

1.372 1.487 2.941 8.066 15.97

18.614 46.604 3.545 4.259 3.367

0.629 0.556 1.025 1.907 1.899

19.512 39.646 0.672 1.169 0.461

Table 5 Demand ∼ Normal(μd = 150, σd = 50 ). Avg Run time (seconds)

Std Dev (seconds)

n

DP

BARON

DP

BARON

15 20 30 40 50

2.390 2.767 14.304 15.039 25.862

78.893 40.862 6.870 32.248 29.298

1.489 0.982 4.403 5.095 6.772

67.727 69.240 16.775 114.607 56.942

Table 6 Demand ∼ Normal(μd = 200, σd = 50 ). Avg Run time (seconds)

Std Dev (seconds)

n

DP

BARON

DP

BARON

15 20 30 40 50

1.680 4.306 13.404 21.455 36.878

87.395 225.385 442.511 129.127 110.546

0.622 1.833 8.066 6.187 8.056

56.903 170.697 414.454 295.360 218.828

Table 7 Demand ∼ Normal(μd = 250, σd = 50 ). Avg Run time(seconds)

Std Dev (seconds)

n

DP

BARON

DP

BARON

15 20 30 40 50

4.558 7.050 10.035 30.430 46.885

149.813 532.715 669.462 317.045 273.947

2.672 2.530 3.052 5.301 8.935

99.605 363.894 522.567 443.887 442.829

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9

Table 9 Demand ∼ Normal(μd = 100, σd = 30 ) Avg Run time (seconds)

Std Dev (seconds)

n

DP

BARON

DP

BARON

15 20 30 40 50

6.435 10.525 41.261 110.808 217.372

11.470 30.180 413.421 1129.797 2633.118

1.230 2.784 6.099 4.636 11.009

22.311 84.986 659.206 787.439 2027.278

Table 10 Demand ∼ Normal(μd = 150, σd = 50 ).

Fig. 4. Marginal discount case: Average run time for DP and BARON as a function of demand expectation. Table 8 Demand ∼ Normal(μd = 50, σd = 10 ).

n 15 20 30 40 50

Avg Run time (seconds)

Std Dev (seconds)

DP

DP

1.596 3.464 8.542 45.097 50.926

BARON 12.080 27.236 119.351 249.726 488.660

0.313 0.418 0.960 7.024 12.390

Avg Run time (seconds)

Std Dev (seconds)

n

DP

BARON

DP

BARON

15 20 30 40 50

14.743 33.806 122.813 200.273 407.633

4.651 20.754 168.595 855.140 1657.834

2.478 4.632 7.900 7.853 44.377

4.531 46.968 383.157 917.589 1790.416

Table 11 Demand ∼ Normal(μd = 200, σd = 50 ). Avg Run time (seconds)

Std Dev (seconds)

n

DP

BARON

DP

BARON

15 20 30 40 50

22.836 42.810 194.621 230.974 469.524

9.250 30.611 169.578 1083.121 2226.068

4.642 0.553 14.937 13.360 17.959

17.545 108.754 454.883 1360.722 2082.083

BARON 13.600 21.101 136.318 260.449 335.933

plexity of the DP algorithm for this cost structure is O (n2 Q max )5 . The termination tolerance used in solving the model using BARON was set to 0.005, which means that the solver stops when the proportional difference between the best solution found and the best bound on the objective function is less than or equal to 0.005. The average and standard deviation of run times in the tables are in seconds, and each value in the table represents an average across 20 sample problems with the specified properties. For example, Table 4 shows the average run time for various values of n (number of suppliers) when the demand follows a normal distribution with μd = 100 and σd = 30. The expected value of demand increases in Table 3–Table 7. These tables show how run time increases as Qmax increases, and can be used to compare the performance of the DP algorithm with BARON. With smaller values of Qmax , the performance of the DP algorithm is similar to that of BARON, on average. As Qmax increases, however, the average solution time for BARON increases drastically. When the mean demand, μd , equals 250 (Table 7), the DP algorithm performs 19 times faster than BARON on average. This is also apparent in Fig. 4, which depicts the impact of increasing the mean demand (μ) on average solution time for the DP algorithm and BARON. Next, we assume that each supplier offers a carload discount structure with four breakpoints, similar to the structure represented in Fig. 3. The associated parameters and constant values used to generate this type of problem are shown in Table 2 and the steps of producing the sample tests for this problem type are similar to those used in the marginal discount case. Tables 8–12 compare the performance of the DP algorithm with BARON for various demand distribution parameters and values of 5 In the marginal discount case, recall that M = 1, i.e., we require only a single binary variable associated with each i ∈ I\{f}, and the resulting problem is a 0–1 KP and not a 0–1 MCKP.

Table 12 Demand ∼ Normal(μd = 250, σd = 50 ). Avg Run time(seconds)

Std Dev (seconds)

n

DP

BARON

DP

BARON

15 20 30 40 50

39.674 62.558 314.092 332.046 660.460

4.261 12.593 57.560 1196.347 2398.138

4.718 11.767 11.012 18.312 16.656

2.613 36.859 190.270 1862.987 2392.059

n when the suppliers offer carload discount structures. Other experiment settings such as termination tolerance and the number of samples are similar to the marginal discount case. The expected value of the demand distribution and therefore Qmax increases in Table 8–12. Under the carload discount structure, due to the complexity of the mathematical model and the large number of variables required to model the carload discount function, the average performance of the DP algorithm is always better than BARON. Fig. 5 also compares the average solution time for the DP algorithm and BARON for various values of the mean demand (μ). 6.2. Performance of the heuristic algorithm This section evaluates the performance of the proposed heuristic approach, which solves the problem using a linear relaxation of the objective, when the demand is assumed to be normally distributed. Since the worst-case complexity of the algorithm is O (n log n ), which is independent of Qmax , we can handle problems with various values of n and Qmax very quickly. The parameters used are the same as those in Section 6.1, using the data in Table 1. Tables 13–17 show the average difference in the solutions found by the heuristic algorithm and BARON (the heading H corresponds

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R. Mohammadivojdan, J. Geunes / European Journal of Operational Research 000 (2018) 1–11 Table 16 Run time results, heuristic approach vs. BARON, Demand ∼ Normal(200, 50). Avg Run time (seconds)

Std Dev (seconds)

n

H

BARON

H

BARON

% Opt Gap Avg

Std Dev

15 20 30 40 50

< 0.001 < 0.001 < 0.001 < 0.001 0.001

123.004 220.734 229.567 148.594 6.058

< 0.001 < 0.001 < 0.001 < 0.001 0.003

117.009 198.247 302.037 343.282 2.780

0.434 0.470 0.391 0.390 0.338

0.210 0.338 0.305 0.442 0.352

Table 17 Run time results, heuristic approach vs. BARON, Demand ∼ Normal(250, 50).

Fig. 5. Carload discount case: Average run time for DP and BARON as a function of demand expectation.

Avg Run time (seconds)

Std Dev (seconds)

% Opt Gap

n

H

BARON

H

BARON

Avg

Std Dev

15 20 30 40 50

< 0.001 0.001 < 0.001 < 0.001 0.001

139.696 577.894 511.623 62.260 448.825

< 0.001 0.003 < 0.001 < 0.001 0.003

112.269 336.483 519.591 219.237 536.086

0.508 0.121 0.241 0.267 0.241

0.434 0.340 0.419 0.287 0.337

Table 13 Run time results, heuristic approach vs. BARON, Demand ∼ Normal(50, 10). Avg Run time (seconds)

Std Dev (seconds)

% Opt Gap

n

H

BARON

H

BARON

Avg

Std Dev

15 20 30 40 50

< 0.001 < 0.001 0.001 0.001 < 0.001

2.531 3.07 3.557 3.99 4.769

< 0.001 < 0.001 0.003 0.003 < 0.001

0.518 2.611 1.001 0.765 0.931

0.600 0.676 0.565 0.817 0.750

0.748 1.100 0.540 0.791 0.778

Table 14 Run time results, heuristic approach vs. BARON, Demand ∼ Normal(100, 30). Avg Run time (seconds)

Std Dev (seconds)

n

H

BARON

H

BARON

% Opt Gap Avg

Std Dev

15 20 30 40 50

< 0.001 0.001 < 0.001 < 0.001 0.001

9.660 45.314 3.713 4.017 3.266

< 0.001 0.003 < 0.001 < 0.001 0.003

8.747 50.619 1.335 0.876 0.193

0.581 0.583 0.415 0.460 0.787

0.656 0.554 0.572 0.631 0.543

Table 15 Run time results, heuristic approach vs. BARON, Demand ∼ Normal(150, 50).

than 0.5 percent. Fig. 6 demonstrates the asymptotic optimality of the heuristic solution method as the problem size grows. 7. Conclusion

Avg Run time (seconds)

Std Dev (seconds)

% Opt Gap

n

H

BARON

H

BARON

Avg

Std Dev

15 20 30 40 50

< 0.001 < 0.001 < 0.001 < 0.001 0.001

60.292 61.793 1.854 2.652 4.978

< 0.001 < 0.001 < 0.001 < 0.001 0.003

49.673 94.560 0.601 0.420 0.578

0.614 0.566 0.625 0.520 0.451

0.357 0.560 0.306 0.439 0.392

to the heuristic method). Each figure corresponds to the average ∗ across 20 sample problems. If ZH and ZP1M denote the heuristic solution and the optimal solution for problem P1 under marginal  discounts, respectively, the gap is calculated as 100 ×

Fig. 6. Average % optimality gap of heuristic solution as a function of demand expectation.

∗ ZH −ZP1M ∗ ZP1M

(shown under the heading % Opt Gap in the tables). The results show that the heuristic approach clearly outperforms the solver in terms of solution time for various values of the number of suppliers and Qmax , with small optimality gaps. Again, we note that as the demand mean increases, the average BARON solution time increases drastically. The average BARON run time across all of these sample problems is more than 30 0,0 0 0 times the average run time of the heuristic, while the average heuristic optimality gap is less

This paper considers a newsvendor problem in which the seller has the option of procuring stock from a set of interchangeable suppliers. Each supplier has a capacity limit on the amount it can provide, and may offer quantity-based discounts. This setting reflects numerous practical contexts involving short life products. We considered three different supply cost structures including marginal quantity discounts, all-units discounts, and carload discounts. We proposed an algorithmic approach that exploits special structural properties of the problem and solves it through the consideration of a set of candidate KKT points. We compared this algorithm to a global optimization solver and found it to be significantly more efficient. We also proposed a heuristic algorithm based on a linear relaxation of the problem that solves the problem very quickly with small gaps from optimality. This study provides a new perspective on the well-known newsvendor problem involving multiple suppliers, and discusses the option of choosing among suppliers who offer different cost structures and incentives. Future research may account for other supplier performance dimensions in addition to the prices they offer. Our model focuses on the price factor, while other differences

Please cite this article as: R. Mohammadivojdan, J. Geunes, The newsvendor problem with capacitated suppliers and quantity discounts, European Journal of Operational Research (2018), https://doi.org/10.1016/j.ejor.2018.05.015

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Please cite this article as: R. Mohammadivojdan, J. Geunes, The newsvendor problem with capacitated suppliers and quantity discounts, European Journal of Operational Research (2018), https://doi.org/10.1016/j.ejor.2018.05.015