Dynamic pricing and lot sizing for a newsvendor problem with supplier selection, quantity discounts, and limited supply capacity
Introduction
High technology products lose value and appeal with time. For instance, the price of a personal computer (PC), on average, decreases by 50–58% in the first year of its lifecycle (Lee et al., 2000, Chen and Xiao, 2011). Firms that deal with products of this sort have to carefully determine their pricing and inventory management policies as both affect profitability (Chen, Pang, & Pan, 2014). Oligopoly firms maximize their revenues by controlling retail prices and, subsequently, to some extent, the demand for their products (Liu, 2005, Liu et al., 2006). Like those that manufacture PCs, firms that produce fashion apparel, farm commodities, and Christmas items, for example, tend to reduce the prices of their products approaching the end of their lifecycle. One could observe here that as product demand slows over time for whatever reason (e.g., obsolescence, perishability, etc.), a firm could reverse this trend through a price discount. Big firms (e.g., Dell and Amazon) today practice dynamic pricing to mitigate the risks of under- or over-stocking items (Chan et al., 2005, Elmaghraby and Keskinocak, 2003, Feng and Shi, 2012, Chao et al., 2008).
Sourcing decisions, single vs. multiple, are also critical to firms. Single sourcing comes with risks such as limited supply capacity, supply disruption, quality issues, or delivery performance, to name a few. Such risks make multiple sourcing preferable to firms, as it also creates competition resulting in lower wholesale prices on average (Aissaoui et al., 2007, Feng and Shi, 2012). Reducing purchasing cost is vital for many manufacturers as raw materials and parts/components make up to 70% of a product’s acquisition cost (Ghodsypour and O’Brien, 1998). Some researchers have also argued that multiple sourcing reduces inventory-related costs (Hong and Hayya, 1992, Aissaoui et al., 2007). The benefits of multi-sourcing are constrained by suppliers having limited supply/production capacity or storage space, which usually leads a buyer to split its order over multiple periods.
Suppliers also consider transportation costs when making production and inventory decisions (Altintas et al., 2008, Parker and Kapuściński, 2011, Parker and Kapuściński, 2004, Federgruen and Zipkin, 1986). Large shipments decrease a supplier’s transportation costs due to economies-of-scale on the one hand and increase the buyer’s inventory-related costs on the other. A supplier can offset a buyer’s inventory-related costs by offering incentives (e.g., all-unit discounts), which decreases the buyer’s purchasing and holding costs (Altintas et al., 2008). For example, H. J. Heinz Company does that by providing its grocery retailers with all-unit discounts (Altintas et al., 2008). There are two popular forms of quantity discounts in the literature, which are all-unit and incremental. The first applies a discount on the entire order once it exceeds a certain quantity, while the second links price to a quantity range, similar in form to one descending stair (e.g., Waters, 2008).
Companies selling high technology products, fashion apparel, farm commodities, and Christmas items, or anything of the sort, may face three main issues. First, the uncertainty of demand, which could be either price- or time-dependent or both. Second, deciding which suppliers to buy from, taking into consideration the supply capacity of those suppliers and whether they offer quantity discounts or not. Third, considering if splitting orders over multiple periods would be cost-efficient. For a sound inventory, sourcing, and pricing policy, a decision-maker should not consider these issues separately. This paper contributes to the literature on the newsvendor problem by developing a model, believed to be the first that does so, that addresses these issues simultaneously. The paper starts first by formulating the problem using mixed-integer nonlinear programming technique and then developing an algorithm to solve it to determine the optimum retail price and quantities to be ordered from the suppliers in each demand period. This study also analyzes the impact of different input parameters, such as suppliers’ price intervals and capacity, demand variation, unit holding costs, etc., on the buyer’s expected profit and decisions to draw some managerial insights.
The supplier selection problem (SSP) is one of the most celebrated topics in the inventory and operations management literature, which has been investigated extensively for different assumptions (e.g., De Boer et al., 2001, Aissaoui et al., 2007). Of interest to this paper are those that studied the SSP in a newsvendor context with quantity discounts and random demand. For example, Dada, Petruzzi, and Schwarz (2007) assumed unreliable suppliers where each supplier has a production capability, where the quantity supplied by a supplier is the minimum between the production capacity of that supplier and the order quantity of the buyer. They did not consider quantity discounts. Yang, Yang, and Abdel-Malek (2007) considered ordering quantities from a set of suppliers with different yields and prices, while Merzifonluoglu and Feng (2014) took into consideration the unreliability of suppliers and their limited supply capacity. A similar problem to the one by Merzifonluoglu and Feng (2014) was investigated by Shu, Wu, Ni, and Chu (2015) who considered a risk-aversive procurement strategy. Also of interest to this paper are those studies that fall within the presented scope, where besides demand being random, it is time and price dependent.
As price becomes dynamic and a decision variable, the problem becomes multi-period in nature, where the buyer has to determine its retail price and order quantities from the suppliers it selects. For example, Hu and Su (2018) assumed a time point before starting the selling season to make a one-purchase from different suppliers and determines the price to satisfy the price-sensitive stochastic demand, making their model a single-period one. A keyword search of the literature in Scopus did not yield a model that considers a multi-period newsvendor problem, with quantity discounts and demand being price- and time-dependent, which the model of this paper does. The model in this paper also considers that the buyer may not place an order up to the maximum capacity of a supplier (call it Supplier #1) with the lowest price. This may happen when that supplier cannot fulfill the order size due to its limited supply/production capacity (call it V1). Assume that there are two suppliers where Supplier #1 offers the lowest price. It is known that (1) the optimum order size of Supplier #1 (call it Q1) is larger than that of Supplier #2 (call it Q2) and (2) if Q1 is larger than V1, the buyer then assigns V1 units to Supplier # 1 and Q2 – V1 units to Supplier #2 (Awasthi, Chauhan, Goyal, & Proth, 2009). We, however, show that when Supplier #2 imposes a minimum purchase quantity (call it V2) that is larger than Q2 – V1, the buyer may then need to assign V2 units to Supplier #2 and reassign the order size of Supplier #1; i.e., min[V1, (Q1 - V2)]. When (Q1 - V2) < V1, the buyer buys less than the maximum capacity of the supplier. This paper tackles such a problem and considers that the buyer reacts to this by suggesting to Supplier #1 to increase its supply capacity to some new level (call it V1,new), such that Q1 ≤ V1,new. To facilitate expanding the supply capacity, the buyer will offer to purchase from Supplier #1 at a higher price than before the expansion only on Q1 - V1 units; termed here as the Reverse Incremental Quantity Discounts (RIQD). This study argues that RIQD can increase the profitability of the buyer and its suppliers. A review of the literature (e.g., Petruzzi and Dada, 1999, Qin et al., 2011, De Yong, 2020) shows that no model in the literature has the same features presented above as those of the one in this paper. This constitutes novelty and a contribution to the literature. Table A1 in the appendix lists and differentiates our work from the relevant ones in the literature. The paper also contributes to the literature by developing an algorithm to simultaneously determine the buyer’s optimum retail price and the order quantities from suppliers. This study produces numerous numerical results to illustrate the behavior of the model and to come up with some managerial insights.
The paper has four remaining sections. Section 2 describes the problem. Section 3 presents a mathematical model for a multi-period SSP and proposes a solution algorithm. Section 4 is for numerical analysis and managerial insights. Section 5 draws some conclusions and suggests future research directions.
Section snippets
Problem description
This paper considers a multi-period two-level supply chain model. In period , a buyer (level #1) purchases a single item from a set of suppliers (level #2), some of which have limited supply capacity and offer all-unit discounts. The delivery lead-time for all suppliers is zero. This paper assumes that the suppliers meet all required criteria, such as product quality, delivery performance, sustainability, etc., except for the quantity discounts and the unit wholesale price. The buyer
Problem formulation and solution algorithm
This section starts by formulating the mathematical model and presents some of its properties that define its behavior. This part is followed by an algorithm to solve the problem.
Numerical analysis
The numerical analysis section has two subsections. Section 5.1 evaluates the performance of the algorithm, starting by considering a single period SSP. In this subsection, demand is not price- and time-dependent; it is referred to as the classical stochastic demand model. It also discusses how to determine the wholesale price of the RIQD. Section 5.2 considers an SSP with a price- and time-dependent random demand where the demand randomness is modeled in the additive form. Appendix B is used
Summary and conclusions
Some products, such as high technology items, fashion apparel, and farm commodities, lose their value, and appeal to customers, sometimes rapidly, with time. To abate the negative impact of time on demand, firms tend to progressively reduce the retail prices of such product to avoid sending them for disposal. Some suppliers, like H. J. Heinz Company, provide buyers with all-unit discounts to minimize their transportation costs. Due to the suppliers’ limited supply capacity, buyers may need to
Acknowledgements
The first author thanks Fondazione Italcementi for the financial support of this research. The second and third authors would like to thank the Natural Sciences and Engineering Research Council of Canada (NSERC) for the financial support of this research.
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