Manufacturer's pricing and lot-sizing decisions for perishable goods under various payment terms by a discounted cash flow analysis
Introduction
It is a well-known fact that demand of a perishable product (e.g., vegetables, fruits, baked goods, fashion merchandise, etc.) is declining over time due to loss of freshness and quality. To quantify this phenomenon, Ghare and Schrader (1963) investigated an EOQ model with a constant rate of deterioration. Covert and Philip (1973) then expanded the constant rate of deterioration to a Weibull deterioration rate. Research publications in this field are very extensive. The reader is referred to the works of Raafat (1991), Goyal and Giri (2001), Bakker et al. (2012), and Pahl and Voβ (2014). For today's health-conscious consumers, product freshness is an important factor in purchasing decisions. Product freshness degenerates over time and reaches zero at the expiration date. As a result, today's health-conscious consumers prefer a product further from its expiration date because it is fresher and can be stored longer. Sarkar (2012) proposed that deterioration rate reaches 100% at the expiration date. There are numerous other models for perishable goods that consider product freshness linked to expiration date, such as Dye et al. (2014), Teng et al. (2016), Wang et al. (2014), and Wu et al. (2014, 2017).
Both sellers and buyers use a variety of payment schemes to settle their business transactions of goods and services. Basically, there are three payment types in terms of payment time: (1) cash-on-delivery, in which the buyer pays for goods or services as soon as receiving them (i.e., cash payment), (2) permissible-delay-in-payment, in which the seller offers the buyer a short-term interest-free loan for purchasing goods or services (i.e., credit payment), and (3) cash-in-advance, in which the buyer prepays the seller for goods and services prior to the time of delivery (i.e., advance payment).
The assumption of cash payment is most commonly used in the field of inventory management, such as economic order quantity (EOQ) and economic production quantity (EPQ) models. A permissible-delay-in-payment is commonly offered to attract new buyers and to avoid lasting price competition. Actually, the use of credit payment is widespread and popular in today's business transactions. More than 80 percent of companies in the United Kingdom and the United States offer their products on various credit payments to stimulate sales and reduce inventory. To save time and money, Zhang (1996) proposed an advance payment in which a consumer may be better off to prepay $80 for 4 months of water bills than to pay $20 each month.
To reduce default risks with credit-risk buyers, Teng (2009) proposed a cash-credit payment in which a buyer must pay some cash when placing an order and then pay the remainder in credit. In reality, a contractor often requests that a customer prepay a good-faith deposit when signing a contract to install a new roof or to pave a driveway. A cash payment to cover the contractor's materials cost is then required upon delivery of the materials to do the job. Finally, the customer will pay the remainder of the total cost after the work is completed and deemed satisfactory. To solve this real-world problem, Wu et al. (2018a) and Li et al. (2017) explored the use of advance-cash-credit (ACC) payment. In fact, an ACC payment is indeed a general framework that includes all of the above-mentioned payment types as special cases.
It is well-known in economics and marketing theory that the lower the price, the higher the demand. Thus, price is an important factor in consumers’ purchasing decisions. Begum et al. (2012) developed an EOQ model for deteriorating items when demand is an exponential function of price and deterioration is a three-parameter Weibull distribution. Ghasemy Yaghin et al. (2014) expanded the pricing and lot-sizing model from an exponential demand to a logit demand function. Feng et al. (2017) further explored pricing and lot-sizing policies for perishable goods when demand is a function of selling price, displayed stocks and expiration date. Feng and Chan (2019) studied the joint pricing and production decisions for new products when the learning curve effect is most pronounced. Otrodi et al. (2019) investigated the pricing and lot-sizing problem for perishable items with multiple demand classes. Hence, in this paper we develop an EPQ model in a three-echelon supplier-manufacturer-customer chain in which (1) the manufacturer simultaneously determines selling price, production run time and replenishment cycle time as decision variables to maximize the present value of total annual profit, (2) the manufacturer sells a perishable product that degrades over time and cannot be sold after its expiration date, (3) the supplier demands the manufacturer use an ACC payment type while the manufacturer grants customers a cash-credit payment scheme, and (4) for generality, a discounted cash-flow analysis is used to reflect time value of money.
The remainder of this paper is as follows: Section 2 discusses the relevant literature review related to cash, credit and advamce payments. Section 3 defines the notation used and assumptions made in the model. Section 4 describes mathematical models for different cases. Section 5 discusses the necessary and sufficient conditions to find the optimal solution for different cases. In Section 6, several numerical examples are presented to illustrate theoretical results. In addition, a sensitivity analysis is conducted to yield some managerial insights. Section 7 presents the conclusions.
Section snippets
Literature review
In this section, the literature; concerning inventory models with cash payments, credit payments, or advance payments, is reviewed.
Harris (1913) applied a cash payment to determine optimal order quantity for the traditional EOQ model with a constant demand rate. Recently, Chen et al. (2016), and Wu et al. (2016) assumed cash-on-delivery business transactions, and derived optimal order quantity and shelf space for perishable goods when demand is dependent on product freshness and displayed
Notation and assumptions
The following notation and assumptions are introduced to model the EPQ inventory model for perishable goods when the supplier asks the manufacturer for an advance-cash-credit payment scheme and the manufacturer in turn offers customers a cash-credit payment scheme.
Mathematical model
Given the above notation and assumptions, the proposed EPQ inventory model for perishable products is described as follows. The manufacturer pays the supplier proportion of the purchasing cost A in l years prior to the time of delivery. When all raw materials have arrived then the production process starts at time 0. A constant production rate P is considered during the production time [0, ]. In this period, the inventory accumulates because the production rate is higher than the sum of
Theoretical results
In order to determine the optimal selling price , production time , and cycle time which maximize the present value of total profit , we first find the optimal solutions for every case, respectively. Due to the complexity of the problem, we are unable to prove the present value of total profit is joint concave in s, and T. However, we are able to prove that (a) for any given s and is strictly pseudo-concave in T, and has a unique maximum
Numerical examples
In this section, we first use two numerical examples to illustrate the results for cases of and , respectively. Then, a sensitivity analysis is conducted to study the effect on the optimal solution of varying values of some major parameters. In reality, perishable goods must be consumed as soon as possible and cannot be used after its expiration date. As a result, the demand for perishable goods is relatively inelastic, which in turn implies that an exponential demand curve of price is a
Conclusion
In this paper, we have established an EPQ model in a three-echelon supplier-manufacturer-customer chain in which (1) the manufacturer simultaneously determines selling price, production run time and replenishment cycle time as decision variables to maximize the present value of total annual profit, (2) the manufacturer sells a perishable product that degrades over time and cannot be sold after its expiration date, (3) the supplier demands the manufacturer an ACC payment type which is a
Acknowledgment
The authors sincerely appreciate Editor Peter Kelle, and two anonymous reviewers for their insightful suggestions and encouragement. The principal author's research was supported by the National Science Council of the Republic of China under Grant MOST 107-2410-H-032-031. The corresponding author's research was supported by Assigned Released Time for Research from William Paterson University of New Jersey and a research grant from Chaoyang University of Technology in ROC.
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