Price-dependent Demand Function Research Articles

An integrated stochastic model and algorithm for constrained multi-item newsvendor problems by two-stage decision-making approach

In this paper, we first build a model for a complicated newsvendor problem, which is involved with optimization to preseason production quantity with random yields and outsourcing quantity with supplier quantity discounts. Different from the existing models, a two-stage decision-making strategy is proposed for the retailer such that the possible loss, caused by the uncertainty of production, is reduced in virtue of dynamically choosing the outsourcing quantities and sales prices. Additionally, the cross-elasticity of prices is incorporated into formulating the random and price-dependent demand function in our model. Consequently, the mathematical model is expressed by a parameterized optimization problem, and a cyclic coordinate descent algorithm (CCD), combined with a feasible direction method and a parameters-choosing procedure, is developed to obtain optimal solutions of the original problem. Both substitute and complementary scenarios for different products are further investigated by numerical simulation. Main managerial implications are summarized as follows: (1) The two-stage policy of decision-making brings more benefits than a static policy for the complicated newsvendor problems. (2) The budget available is a critical constraint for the small-and-medium sized business, the maximum expected profit increases as the budget becomes greater, but remains at the same level after it reaches a threshold value. (3) The cross elasticity relationship between the products plays a critical role in choosing optimal quantities of preseason production, sales prices and outsourcing quantities.

An Inventory Model for Perishable Items with Price-, Stock-, and Time-Dependent Demand Rate considering Shelf-Life and Nonlinear Holding Costs

Nowadays, consumers are more health conscious than before, and their demand of fresh items has intensely increased. In this context, an effective and efficient inventory management of the perishable items is needed in order to avoid the relevant losses due to their deterioration. Furthermore, the demand of products is influenced by several factors such as price, stock, and freshness state, among others. Hence, this research work develops an inventory model for perishable items, constrained by both physical and freshness condition degradations. The demand for perishable items is a multivariate function of price, current stock quantity, and freshness condition. Specific to price, six different price-dependent demand functions are used: linear, isoelastic, exponential, logit, logarithmic, and polynomial. By working with perishable items that eventually deteriorate, this inventory model also takes into consideration the expiration date, a salvage value, and the cost of deterioration. In addition, the holding cost is modelled as a quadratic function of time. The proposed inventory model jointly determines the optimal price, the replenishment cycle time, and the order quantity, which together result in maximum total profit per unit of time. The inventory model has a wide application since it can be implemented in several fields such as food goods (milk, vegetables, and meat), organisms, and ornamental flowers, among others. Some numerical examples are presented to illustrate the use of the inventory model. The results show that increasing the value of the shelf-life results in an increment in price, inventory cycle time, quantity ordered, and profits that are generated for all price demand functions. Finally, a sensitivity analysis is performed, and several managerial insights are provided.

Decentralized Supply Chains under Random Price-Dependent Demand: Noncooperative Equilibria vs. Coordination with Cost-Sharing Contracts

It is common for a supplier to sell products to multiple retailers. In this paper, we investigate the equilibrium behavior of a decentralized supply chain with multiple retailers facing a random price-dependent demand in the additive form. Here, we consider two kinds of demand functions: the distribution of the demand depends only on the retailer’s own retail price (noncompeting retailers) and not only on his own retail price but also on that of the other retailers (competing retailers). We present appropriate wholesale price, buy-back, and lost-sales cost-sharing contracts to coordinate the total supply chain, so that when all the retailers adopt their equilibrium response, the supply chain system coordination is also achieved. Furthermore, the coalition formation among retailers is also analyzed. We find that with buy-back and lost-sales cost-sharing contracts and linear price-dependent demand function, retailers always prefer being in the grand coalition to forming any other coalition.

An ordering policy for deteriorating items with price-dependent iso-elastic demand under permissible delay in payments and price inflation

ABSTRACTThis paper considers the problem of dynamic decision-making for an inventory model for deteriorating items under price inflation and permissible delay in payment. In this paper, we adopt an iso-elastic and selling price dependent demand function to model the finite time horizon inventory for deteriorating items. The stocks deteriorate physically at a constant fraction of the on-hand inventory. The objective of this paper is to determine the optimal retail price, number of replenishments, and the cycle time under two different credit periods so that the net profit is maximized. We discuss the optimization properties and develop an algorithm for solving the problem based on dynamic programming techniques. Numerical examples are presented to illustrate the validity of the optimal control policy, and sensitivity analysis on major parameters is performed to provide more managerial insights into deteriorating items.

Multi-item deteriorating two-echelon inventory model with price- and stock-dependent demand: A trade-credit policy

This article is concerned with a multi-item inventory model for deteriorating items. The model is formed on the basis of a two-level supply chain policy, i.e., based on manufacturer's and retailer's perspective. The deterioration rate is considered as constant. The demand factor of any items suffer from a large amount of stock level; so, we consider stock-dependent demand function. The demand of any item is also dependent on its selling price; thus, a price-dependent demand function is introduced here. The retailer adopts the trade-credit policy for his customers in order to promote market competitiveness. He can earn revenue and interest after the customer pays the amount of purchasing cost to the retailer until the end of the trade-credit period, offered by the supplier. Shortages are allowed in the retailer's model as it is a very realistic item, too. A price discount on backordered commodities is offered for those customers who are willing to backorder their demand. Thereafter, we present an easy analytical solution procedure to find the total profit for both manufacturer and retailer. We also use the classical game theory and Nash equilibrium approach to find an optimal solution of the joint profit. The results are discussed with several numerical examples to illustrate our model and to provide some managerial insights related to the model. Furthermore, a parametric sensitivity analysis of the optimal solutions is provided and a concavity figure of our profit function is supplied to stabilize our model. The paper ends with a conclusion and an outlook to future research projects.

Coordinating Supply Chains With a General Price-Dependent Demand Function: Impacts of Channel Leadership and Information Asymmetry

This paper studies the effects of channel leadership and information asymmetry on supply chain coordination (SCC) of a single-manufacturer ( M ) single-retailer ( R ) supply chain, which sells a product with a general stochastic price-dependent-demand function. Three all-units quantity discount (QD) contracts are examined: 1) the manufacturer-led QD (MQD); 2) the retailer-led QD (RQD); and 3) the vendor managed inventory QD (VQD), which represent different channel leaderships. The problems are constructed as multistage sequential games. We find that any VQD that coordinates the supply chain has an equivalent RQD counterpart, and hence only MQD and RQD need to be examined for SCC. For the symmetric information scenario, both MQD and RQD can achieve coordination and the respective conditions for RQD are distribution free, but not for the MQD case. Only RQD, but not MQD, can always achieve Pareto improvement. For the asymmetric information scenario, MQD achieves coordination only if the actual supply chain's optimal quantity is known by M , but this condition is not needed for RQD. There is a unique RQD that achieves SCC, and hence the profit division between channel members is fixed. To overcome this inflexibility issue, we demonstrate that R can offer a menu of two RQDs to M . However, Pareto improvement is not always guaranteed by using RQD. Since QD contracts are widely adopted in practice, this paper lays the foundation to advance our knowledge on how channel leadership affects performance under both the information symmetric and asymmetric cases.

Auction and contracting mechanisms for channel coordination with consideration of participants' risk attitudes

This paper considers a two-supplier one-retailer coordinated supply chain system with auction and contracting mechanism incorporating participants' risk attitudes. The risk attitude is quantified using the value-at-risk (VaR) measure and the retailer faces a stochastic linear price-dependent demand function. In the supply chain, the suppliers (providing identical products) compete with each other in order to win the ordering contract of the retailer. Several auction and contracting mechanisms are developed and compared. It can be analytically shown that the retail price of the risk-averse system is higher than that of the risk-neutral system, but the order quantity is lower than that of the risk-neutral system.

Auction and contracting mechanisms for channel coordination with exponential demand functions

This paper considers a supply chain system with two competing suppliers providing identical products or service to a retailer incorporating risk attitudes. The risk attitude is quantified using the value-at-risk measure and the retailer faces a stochastic exponential price-dependent demand function. Several simple situation and auction mechanisms are developed and compared. The results show that, for auction and two-part contracting mechanisms, the retailer and the low-cost supplier are cooperating together to maximize the total profit of the supply chain with complete information and asymmetric information; however, in this simple situation without complicated auction, the channel-coordinated system optimums cannot be achieved.

Dominance Relationship Among the Retailer’s Strategies Under the Semi-Stackelberg Newsvendor Situation with Quantity Discounts

In this paper, we investigate the retailer’s behavior under the supplier-driven semi-Stackelberg newsvendor situation in which (i) the supplier plays as a semi-Stackelberg leader with or without his discounts schedule being offered to the retailer, (ii) the retailer plays as a Stackelberg follower with or without her discounts schedule being offered to the end customer, and (iii) neither party has perfect information on the endogenous price-dependent demand function or exogenous probability distribution of demand. In this situation, the retailer’s concern is identifying the dominant strategy by which she can safely implement her own scheme for customer discounts regardless of the order quantity rather than finding the best strategy yielding the optimal order quantity that maximizes her expected profit. We show that a consistent dominance relationship exists among the retailer’s strategies when the newsvendor chooses to offer progressive multiple discounts to customers regardless of the supplier’s strategy of offering the retailer either all-units discounts or no-discounts.

Optimal rebate strategies in a two-echelon supply chain with nonlinear and linear multiplicative demands

We examine the pure rebate strategies in a two-echelon supply chain under stochastic demand with multiplicative error. Given exogenous wholesale price and retail price, we characterize the unique Nash equilibrium when both manufacturer and retailer provide rebate policy to consumers under nonlinear and linear price-dependent demand functions, including iso-elastic multiplicative demand function (EMDF) and linear multiplicative demand function (LMDF). Based on a game theoretical framework, we prove that there still exists a unique equilibrium when the price elasticity is rather small with constraint conditions in the former case. We also find that in this case the retailer(manufacturer) may increase its rebate value in reaction to the manufacturer's(retailer's) rebate value in order to stimulate sales, which is contrary to the conventional wisdom that the retailer(manufacturer) will shrink its rebate value to gain an ``extra advantage unfairly. As a result, both parties share the same profit at equilibrium. Further, we compare the expected profit outcomes at equilibrium among joint-rebate game, single-party rebate game and no-rebate game by using numerical examples. It is shown that the joint-rebate policy is not always dominates the others unless the price elasticity is sufficiently flexible.

Optimal ordering and pricing policy for demand functions that are separable into price and inventory age

We formulate and analyze two models for determining the optimal pricing, order quantity and replenishment period for items whose demand function is separable into components of price and inventory age. The first model assumes a multiplicative demand function. We provide conditions, which are satisfied by most common price-dependent demand functions, to reduce the three-variable profit maximization problem into a single-variable problem, which can be solved using an efficient line-search method. Next, we show that a genuine additive model cannot exist, and instead suggest and analyze a pseudo-additive model. However, this model is more limited than the multiplicative model in its ability to incorporate various combinations of price and inventory age effects, and reduction of the maximization problem into a single-variable problem is more complicated, except in the case of a linear price effect, which is further analyzed. For both models, we show that the optimal solution satisfies the first-order condition for equilibrium under a monopoly, with a modification that includes inventory holding costs. We solve numerical examples to illustrate the solution procedures.

A rewarding-punishing coordination mechanism based on Trust in a divergent supply chain

Coordination of decentralized supply chains using contract design is a problem that has been widely addressed in the literature. We consider a divergent supply chain including a supplier and several retailers producing fashion products with short sale seasons. The retailers cooperate with the supplier as sales agents; i.e., they work in the framework of revenue sharing contracts. Because of their proximity to the market, retailers can provide more accurate demand forecasts to the supplier that is used to decide on issues such as capacity building and market prices with regard to retailers stiff due dates, different lead times and different price-dependent demand functions. To ensure abundant supply and cope with the demand variability, the retailers have an incentive to exaggerate their private forecast information. In this study, we propose a new rewarding-punishing coordination mechanism based on trust between supply chain tiers, considered as a differentiation factor between honest and deceptive partners. An optimization model is developed as a building block of this mechanism. An approximation method is used to simplify and solve the problem. The model is then implemented using Monte-Carlo simulation in four different situations, according to 10 different strategies for forecast information sharing. The findings from the tests show that the mechanism including trust as a decisional factor performs better than ‘No Trust’ mechanism in all situations. These results suggest that taking into account Trust in designing coordination mechanism may have significant influence on the financial performance of the supply chain.

The economics of hybrid manufacturing systems in a closed-loop supply chain

This paper deals with the economics of a closed-loop supply chain and the decision-making in a hybrid manufacturing system. We develop analytic models and quantify the potential profits gain generated by such a hybrid system that collects, remanufactures, and remarkets the end-of-use products as perfect substitutes of brand-new products while facing a price-dependent demand function. The research objective behind developing analytic models is to investigate the following research questions: Can the hybrid system incorporating reverse logistics operate cost-effectively? Can it outperform the manufacturing-only system without reverse logistics? If not, does it persist in certain decision bias that leads to a lower profit and inefficiency? If yes, to what extent and under which conditions--lower remanufacturing cost or higher price sensitivity coefficients? We model the decision-making of the system as an appropriate optimization problem. Our analysis shows that the hybrid system does not outperform the manufacturing-only counterpart under generic settings; it only performs better under the condition with a higher price sensitivity coefficient and/or a lower remanufacturing cost. In addition, the pricing decisions for the brand-new and remanufactured products differ significantly, which is consistent with the industrial practice. In addition, numerical results suggest that it may be worth adopting such a hybrid strategy when the remanufacturing cost is low, in comparison with that of the new product manufacturing. Another favorable factor to adopt the hybrid strategy is higher completion intensity between the two versions of the same product. In such scenarios, participating in remanufacturing operations is not only an issue of being environmentally conscious but a profit-improving opportunity.

Price, Rebate, and Returns Supply Contracts for Coordinating Supply Chains with Price Dependent Demand

Channel rebates and returns policies are common mechanisms for manufacturers to entice retailers to increase their order quantities and sales ultimately. However, when the underlying demand depends on the retail price, it has been known that channel coordination cannot be achieved if only one of these mechanisms is deployed. In this paper, we show that a policy that combines the use of wholesale price, channel rebate, and returns can coordinate a channel with both additive and multiplicative price-dependent demands. In addition to determining the sufficient conditions for the contract parameters associated with the equilibrium policy, we show that multiple equilibrium policies for channel coordination exist. We further explore how the equilibrium policy can be adjusted to achieve Pareto improvement. Other issues such as the maximum amount of expected profit that the manufacturer can share under the coordinated channel, the structural properties of the contracts under both the additive and multiplicative price-dependent demand functions are also discussed.

An optimal credit policy to increase supplier's profits with price-dependent demand functions

Abstract This paper deals with the problem of determining an optimal length of credit period from the perspective of supplier. We assume that a retailer jointly determines the unit retail price and order size to maximize profit when he/she purchases a product for which the supplier offers a trade credit. Two widely used demand functions are adopted for the study in which demands are decreasing functions of the retail price. A procedure is presented which shows how to achieve an optimal length of credit period for suppliers. The effects of credit period on the behaviour of retailers are also investigated using an example.

Joint Economic Ordering Policies of Multiple Wholesalers and a Single Manufacturer with Price-dependent Demand Functions

The economic ordering policies for multiple regional wholesalers and the production lot-sizing policy for a single manufacturer have been studied in a joint analysis under the assumption that the yearly demands of each region are functions of their respective retail proces. We obtained optimum EOQs for both linear and constant price elasticity demand functions. Although normally the wholesalers would order in quantities equal to their EOQs, they are encouraged to purchase in different quantities by the producer providing compensation to offset the wholesalers' increased costs. The production lot-size is determined to minimize the overall production cost.

Joint Economic Ordering Policies of Multiple Wholesalers and a Single Manufacturer with Price-Dependent Demand Functions

Joint Economic Ordering Policies of Multiple Wholesalers and a Single Manufacturer with Price-Dependent Demand Functions

Joint Economic Ordering Policies of Multiple Wholesalers and a Single Manufacturer with Price-dependent Demand Functions

The economic ordering policies for multiple regional wholesalers and the production lot-sizing policy for a single manufacturer have been studied in a joint analysis under the assumption that the yearly demands of each region are functions of their respective retail proces. We obtained optimum EOQs for both linear and constant price elasticity demand functions. Although normally the wholesalers would order in quantities equal to their EOQs, they are encouraged to purchase in different quantities by the producer providing compensation to offset the wholesalers' increased costs. The production lot-size is determined to minimize the overall production cost.

Farm Level Demand for Pecans Reconsidered

AbstractPrevious studies have consistently indicated the anomalous result of a price inflexible demand for pecans. However, these efforts did not have an adequate measure of pecan stocks available and, as a result, stocks were either excluded from consideration or a proxy variable was introduced. A time series of pecan stocks is now available. Use of this time series in a price dependent demand function results in a flexible farm level demand for pecans. This points out the danger of excluding an appropriate variable or using a so-called “reasonable” proxy variable.

Optimal production, purchasing and pricing: A differential game approach

This paper deals with a problem of determining optimal production and pricing policies of a manufacturing firm which is supplying a retailer, The latter faces a price-dependent demand function towards the final consumers and wishes to determine optimal purchasing and pricing policies. Both firms carry inventories and backlogging is permitted. The problem is modelled as a two-player nonzero-sum differential game with the inventory levels as the state variables. The controls are the rates of production and purchasing as well as the prices. Assuming linear production costs, but convex ordering and holding costs, open-loop Nash controls are characterized by using switching point analysis as well as phase diagrams.