Production, Manufacturing and Logistics
Procurement strategies for lost-sales inventory systems with all-units discounts

https://doi.org/10.1016/j.ejor.2018.06.053Get rights and content

Highlights

  • The ordering cost under all-units discount is neither concave nor convex.

  • A Q-jump (s, S) policy is optimal for the single model.

  • A generalized Q-jump (s, S) policy is optimal for the multi-period model.

  • We provide a sufficient condition for simplifying the optimal policy.

Abstract

Despite the prevalence of all-units discounts in procurement contracts, these discounts pose a technical challenge to analyze procurement strategies due to neither concave nor convex ordering costs. In this paper, we consider the optimal procurement strategies with all-units discounts under the lost-sales setting. By assuming log-concave demands, we find that the optimal procurement strategies have a generalized Q-jump (s, S) structure by introducing a new notion of Q-jump single-crossing. In particular, a sufficient condition is provided for degenerating the optimal procurement strategies from a generalized Q-jump (s, S) structure into a Q-jump (s, S) structure, which is definitely optimal for the single-period problem. Extensive numerical results suggest that the Q-jump (s, S) policy as a heuristic performs considerably well when its optimality sufficient condition is violated. Our results can be extended to systems with multi-break all-units discounts, and systems with all-units discounts on batch ordering.

Introduction

A supplier would often offer an all-units discount scheme to his sellers by providing a regular price as well as a discount price. The discount price is lower than the regular price but can be obtained for every unit only if the purchase amount equals or exceeds some quantity threshold. A more general all-units discount scheme may involve multiple breaks (i.e., quantity thresholds), and the seller gets a discount for every unit when the purchase amount equals or exceeds each threshold. According to the empirical study of Munson and Rosenblatt (1998), among all the quantity discount schemes that they investigate, 95% of them are in the form of all-units discounts. Also as mentioned in Kolay, Shaffer, and Ordover (2004), all-units discounts are common in intermediate-goods markets (e.g., sugar, steel, rubber, etc). For example, Coca-Cola uses them in their contracts with its retailers; Porsche uses them in their contracts with its car dealers, and Michelin uses them in their contracts with its tire dealers; Nike would also uses them in their contracts with its authorized resellers. In some cases, a seller can only enjoy quantity discounts from a supplier if the seller orders in batches. Such a strategy can be widely found in practice, especially in retail industry, where the supplier would offer a wholesale price for batch ordering as well as a retail price (see Hu, Pavlin, Shi, 2013, Munson, Rosenblatt, 1998). Indeed, a Google search using the keywords “all-units discounts” yields 10 million results that include a wide range of products, as of August 10, 2016.

In spite of prevalence in many industries, all-units discounts raise a variety of important and challenging questions on managing procurement. The main reason is that the ordering cost function under an all-units discount scheme becomes neither concave nor convex in the order quantity, and can be strictly decreasing near the break. As a result, those well-planned and well-designed procurement strategies for procurement problems with concave or convex ordering cost in the existing literature are no longer applicable. Common wisdom tells us that the optimal procurement strategies under all-units discounts should be much more complex and it is also technically challenging to identify them. Perhaps due to these reasons, the available academic literature on the procurement problems with all-units discounts and multiple periods is still in its infancy. In fact, the majority of the studies in the literature study a single-period setting and only few studies consider the heuristic policy for the multi-period setting, e.g.,Altintas, Erhun, and Tayur (2008). Our paper addresses this gap in the existing literature.

Under an all-units discount contract, a seller has to decide the optimal procurement strategy to take advantage of it. From a seller’s perspective, our research questions include: (1) What is the structure of the optimal procurement strategy? (2) Can a simple policy be optimal? What is the optimality condition of the simple policy? (3) What is the performance of the simple policy in general? The answers of these questions provide us with benchmarks on how to manage procurement under all-units discounts for the seller.

We do so by considering a stochastic inventory problem with an all-units discount scheme over a finite planning horizon. Unmet demand is lost. Demand in each period is assumed to follow a log-concave distribution. This class of log-concave densities includes Gamma and Weibull distributions with shape parameter k ≥ 1, Beta distribution with two shape parameters p ≥ 1 and q ≥ 1, normal and truncated normal distributions (see Dharmadhikari & Joag-Dev, 1988). There is a single break Q in the all-units discount scheme. Specifically, the supplier provides a discount price c2 when the order quantity reaches Q, as well as a regular price c1 when the order quantity is strictly less than Q, with 0 < c2 < c1. In Section 7, we shall extend our results to systems with multi-break all-units discounts. The objective of the seller is to minimize the total expected discounted cost (the discount factor 0 < α ≤ 1), which is the sum of ordering costs, inventory holding and penalty costs over the planning horizon by deciding on the optimal procurement strategy in each period.

Our main results are summarized as follows: (1) for a single-period problem, we find that the optimal procurement strategy has a Q-jump (s, S) structure, which we refer to as the simple policy. (2) For a multi-period problem, the optimal procurement strategies have a generalized Q-jump (s, S) structure in general. In particular, we show that this policy can be degenerated into a Q-jump (s, S) policy under a sufficient condition, i.e., h+c2αc1. (3) Our numerical results reveal that the performance of the Q-jump (s, S) as a heuristic, when its optimality sufficient condition is violated, is quite well, i.e., the relative difference is within 1%. (4) Finally, we also extend our results to (i) systems with multi-break all-units discounts, and (ii) systems with all-units discounts on batch ordering. For the former, its optimal policy becomes more complicated but its overall structure mimics that for the single-break case. For the later, its optimal policy has a generalized (r, Q) structure.

The rest of this paper is organized as follows. In Section 2, we provide a brief review of related literature. In Section 3, we introduce our model and formulation. We then present several preliminary results in Section 4. In Section 5, we first characterize the structure of the optimal procurement strategies for the single-period and multi-period problems, respectively, and then provide a sufficient condition for the optimality of a simple policy for the multi-period problem. In Section 6, we presents the numerical studies. In Section 7, we extend our results to systems with multi-break all-units discounts, and systems with all-units discounts on batch ordering. Finally, we provide concluding remarks in Section 8. All proofs of this paper can be found in Appendix A of the Online Supplement.

Section snippets

Related literature

There are three streams of related literature. The first stream is on quantity discounts, including the studies of the economic motivation for the supplier to offer these discounts, and the impacts of quantity discounts on procurement strategies for the seller and channel coordination for the supply chain (i.e., both the supplier and the seller). The second stream is on inventory models utilizing quasi-convexity. Finally, our paper is also related to the stream of literature on inventory models

Model framework

We consider a single-item periodic-review inventory system where a seller replenishes his inventory from a supplier, who has no capacity restrictions, with all-units discount and lost-sales over a finite planning horizon, which consists of T periods. Periods are indexed by t=1,,T. The seller faces a nonnegative random demand Dt in period t. We assume that demands in different periods are independently (but not necessarily identically) distributed and each demand distribution has a log-concave

Preliminary results

Due to the technically challenging nature of the procurement problems with all-units discounts, we need to introduce several versions of generalized convexity that facilitate our subsequent analysis. Throughout this paper, we use “increasing” in place of “nondecreasing” and “decreasing” in place of “nonincreasing”.

To begin with, we first review the definition of ordinary quasi-convexity:

Definition 1

A function f: RR is said to be quasi-convex if for any x,yR and λ ∈ [0, 1], the inequality f(λx+(1λ)y)max

Optimal procurement strategies

In this section, we first characterize the structure of the optimal procurement strategy for the single-period problem, then extend our characterization for the multi-period problem, and finally provide a sufficient condition for the optimality of a simple policy for the multi-period problem.

Numerical studies

In this section, we propose the Q-jump (s, S) policy as a heuristic, whose performance we compare with that of the optimal policy, for the multi-period problem over an infinite horizon when Condition 1 does not hold. The heuristic consists of three parameters (S1, S2, s) of the Q-jump (s, S) policy. For a lost-sales inventory system with (Q, c1, c2) scheme and starting inventory state x, define L(y)=E[h[yD]++p[Dy]+] as the expected single-period inventory cost. Let v*(x) be the optimal cost

Extensions

In this section, we discuss how to extend our results to various systems under the lost-sales setting, which include (i) systems with multi-break all-units discounts and (ii) systems with all-units discounts on batch ordering.

Concluding remarks

In this paper, we investigate the optimal procurement strategy for the lost-sales inventory problem with an all-units discount scheme under log-concave demands over a finite planning horizon. The technical challenge for such a problem is that the ordering cost function is neither concave nor convex in the order quantity and can be strictly decreasing near the break. Hence, we adopt several versions of generalized convexity to facilitate our analysis.

For the single-period problem, we find that a

Acknowledgments

The authors are grateful to Ruud Teunter, the editor, and the two anonymous referees for their valuable and constructive comments on this paper. This research was supported by the NSFC (National Natural Science Foundation of China) projects (71571158, 71320107004), Research Grants Council of the Hong Kong Special Administrative Region, China (GRF 11506015, T32-102/14-N and T32-101/15-R), CityU 7005014, and the development funding of teaching and scientific research for teachers of Liberal Arts

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