A newsvendor model with fuzzy price-dependent demand

Price-dependence is an important characteristic for some inventory problems. This paper proposes a newsvendor model with fuzzy price-dependent demand, and discusses the conditions to determine the optimal pricing and inventory decisions jointly so that the expected profit could be maximized. Then an algorithm combining the method of ranking fuzzy numbers is developed to tackle the problem. Furthermore, comparison is made between the fuzzy model and the deterministic model to study the effect of the uncertain price-dependent demand, and the sensitivity properties of the joint optimal decisions are illustrated through numerical examples.     

Coordination of a socially responsible two-stage supply chain under price-dependent random demand

This paper investigates the coordination problem of a supply chain (SC) composed of a manufacturer exhibiting corporate social responsibility (CSR) and a retailer faced with random demand. The random demand is made up of the multiplication of price-dependent demand and random demand factor (RDF), plus the CSR-dependent demand. The centralized decision problem of the SC is an extension of the existing price setting newsvendor problem (PSNP). It is found that the sufficient condition for the quasi-concavity of expected profit (EP) on PSNP can not ensure the quasi-concavity of EP of the SC. Then, the concavity condition of EP related to the CSR effect factor is presented in the case of uniformly distributed RDF and linear demand in price, and the concavity of EP is proven under centralized decision. For decentralized decision under manufacturer’s Stackelberg game, the manufacturer determines wholesale price and its CSR investment, and then the retailer decides the order quantity and the retail price. The standard revenue-sharing (RS) contract is found not able to coordinate the SC, so a modified RS (MRS) contract is proposed to coordinate the SC. Finally, numerical examples illustrate the validity of the theoretical analysis and the coordination effectiveness of the MRS contract via Matlab.     

Coordination mechanism for the supply chain with leadtime consideration and price-dependent demand

We study a coordination contract for a supplier–retailer channel producing and selling a fashionable product exhibiting a stochastic price-dependent demand. The product’s selling season is short, and the supply chain faces great demand uncertainty. We consider a scenario where the supplier reserves production capacity for the retailer in advance, and permits the retailer to place an order not exceeding the reserved capacity after a demand information update during a leadtime. We formulate a two-stage optimization problem in which the supplier decides the amount of capacity reservation in the first stage, and the retailer determines the order quantity and the retail price after observing the demand information in the second stage. We propose a three-parameter risk and profit sharing contract that coordinates the supply chain. The proposed contract permits any agreed-upon division of the supply-chain profit between the channel members.     

On coordinating an assembly system under random yield and random demand

We consider a decentralized assembly system in which the customer demand and the yield of the suppliers are random. We establish the concavity of expected supply chain profit for arbitrary number of suppliers. We propose two contracts and show that they coordinate the chain under forced compliance. The contracts are mixed type of contracts that include payments from different contract schemes. Particularly, a payment or a penalty to the worst performing supplier seems inevitable. Apart from providing a coordinating contract, we also provide qualitative insights based on a numerical illustration of centralized and decentralized solutions.     

Buyback contracts with price-dependent demands: Effects of demand uncertainty

We explore buyback contracts in a supplier–retailer supply chain where the retailer faces a price-dependent downward-sloping demand curve subject to uncertainty. Differentiated from the existing literature, this work focuses on analytically examining how the uncertainty level embedded in market demand affects the applicability of buyback contracts in supply chain management. To this end, we seek to characterize the buyback model in terms of only the demand uncertainty level (DUL). With this new research perspective, we have obtained some interesting new findings for buyback. For example, we find that (1) even though the supply chain’s efficiency will change over the DUL with a wholesale price-only contract, it will be maintained constantly at that of the corresponding deterministic demand setting with buyback, regardless of the DUL; (2) in the practice of buyback, the buyback issuer should adjust only the buyback price in reaction to different DULs while leave the wholesale price unchanged as that in the corresponding deterministic demand setting; (3) only in the demand setting with an intermediate level of the uncertainty (which is identified quantitatively in Theorem 5), buyback provision is beneficial simultaneously for the supplier, the retailer, and the supply chain system, while this is not the case in the other demand settings. This work reveals that DUL can be a critical factor affecting the applicability of supply chain contracts.     

An EOQ model with two-parameter Weibull distribution deterioration and price-dependent demand

An Inventory replenishment policy is developed for a deteriorating item and price-dependent demand. The rate of deterioration is taken to be time-proportional and the time to deterioration is assumed to follow a two-parameter Weibull distribution. A power law form of the price dependence of demand is considered. The model is solved analytically and is illustrated with a numerical example.     

Serial production systems with random yield and rigid demand: A heuristic

We consider a heuristic for serial production systems with random yields and rigid demand: all usable units exiting a stage move forward. We calculate optimal lots and corresponding expected costs for binomial, interrupted-geometric, and all-or-nothing yields. Our method is that it makes it easy to analyze large systems.     

Multi-period modeling of two-way price commitment under price-dependent demand

This paper examines the use of price-commitment policies in dynamic contracting in multiple-period, finite-time horizons. Two specific forms of price commitment are considered: one on the part of the retailer through a retail-fixed-markup contract and one on the part of the manufacturer through a price-protection contract. Optimal policies for each form of price commitment are analytically derived, as are optimal policies for the traditional price-only and centralized supply chain scenarios that we use as comparisons. We prove that optimal retail price and order size solutions exist in each period under the assumption of non-increasing price-dependent demand. We show that the existence of retailer inventory between periods causes the optimal policies to differ from a static single-period model. Further, we show that a supplier offers a price-protection policy as a signal to the retailer to resolve the gaming that naturally occurs under price-only; this effectively decouples the multi-period dynamic contracting setting into repeated single-period scenarios. However, the resulting behavior can actually inhibit supply chain performance. On the retail commitment side, we find that retail-fixed-markup policies are quite effective in improving supply chain efficiency. We show that such policies can lead to Pareto-improvement over price-only contracts and can even coordinate the supply chain in some situations.     

Optimal ordering and pricing policy with supplier quantity discounts and price-dependent stochastic demand

We study a joint ordering and pricing problem for a retailer whose supplier provides all-unit quantity discount for the product. Both generalized disjunctive programming model and mixed integer nonlinear programming model are presented to formulate the problem. Some properties of the problem are analysed, based on which a solution algorithm is developed. Two numerical examples are presented to illustrate the problem, which are solved by our solution algorithm. Managerial analysis indicates that supplier quantity discount has much influence on the ordering and pricing policy of the retailer and more profit can be obtained when the supplier provides quantity discount.     

Combined pricing and supply chain operations under price-dependent stochastic demand

In this study, we determined product prices and designed an integrated supply chain operations plan that maximized a manufacturer’s expected profit. The computational results of this study revealed that as the variance of the demand distribution increases, a manufacturer will increase its inventory to levels that are greater than the anticipated demand to prevent the potential loss of sales and will simultaneously raise product prices to obtain a greater profit. In the cost minimization approach, the manufacturer may earn the highest possible profits, as determined by the profit optimization approach, only if this firm precisely forecasts the mean market demand for its products. Greater inaccuracies in this forecast will produce lower levels of expected profit.     

Ordering and pricing of service products in an advance sales system with price-dependent demand

This investigation addresses a service inventory control problem in which a firm orders and sells a service which will be used or consumed by customers on a specific future date. The firm sells the product through an advance booking system, aiming to optimize product price to maximize the total expected profit. Considering situations in which product demand is price-dependent and customers with reservations may cancel advance orders, this work develops a continuous-time model to simultaneously determine the order quantity and selling prices. The analytical results reveal that the optimal ordering quantity and prices are derived via closed-form solutions. In addition, sensitivity analysis of the optimal prices with respect to the system parameters is also conducted to illustrate optimal decision characteristics.     

Concepts for safety stock determination under stochastic demand and different types of random production yield

We consider a manufacturer’s stochastic production/inventory problem under periodic review and present methods for safety stock determination to cope with uncertainties that are caused by stochastic demand and different types of yield randomness. Following well-proven inventory control concepts for this problem type, we focus on a critical stock policy with a linear order release rule. A central parameter of this type of policy is given by the safety stock value. When non-zero manufacturing lead times are taken into account in the random yield context, it turns out that safety stocks have to be determined that vary from period to period. We present a simple approach for calculating these dynamic safety stocks for different yield models. Additionally, we suggest approaches for determining appropriate static safety stocks that are easier to apply in practice. In a simulation study we investigate the performance of the proposed safety stock variants.     

Manufacturer's pricing strategies in a single-period framework under price-dependent stochastic demand with asymmetric risk-preference information

This paper considers a single-period problem designed to analyse the pricing strategy of a manufacturer who does not possess full information about the retailer's risk-preferences. The retailer, who faces a price-dependent stochastic demand, is a maximizer of the risk-adjusted expected profit, rather than of the expected profit. The paper first evaluates the implication of the various risk-preferences of the retailer on the manufacturer's policy under a full-information scenario. Then, it considers a partial information scenario and computes the expected value of perfect information. Finally, it assesses the impact on the manufacturer's profit of sharing the retailer's risk through the introduction of a buyback policy. Linear or iso-elastic demand functions and additive or multiplicative demand error structures capture the demand distributions. Analytical results as well as numerical examples illustrate the main features of the model.     

Coordinating a decentralized supply chain with customer returns and price-dependent stochastic demand using a buyback policy

We investigate a decentralized supply chain that consists of a manufacturer and a retailer where the retailer simultaneously determines the retail price and order quantity while experiencing customer returns and price dependent stochastic demand. We propose an agreement between the manufacturer and the retailer that includes two buyback prices, one for unsold inventory and a second for customer returns, and show that this type of easy-to-implement agreement can achieve perfect supply chain coordination and be a win-win for both manufacturer and retailer when a complementary profit-sharing agreement is included.     

Two-period pricing and decision strategies in a two-echelon supply chain under price-dependent demand

This paper presents a two-period supply chain model which is comprised of one manufacturer and one retailer who are involved in trading a single product. The demand rate in each period is dependent on the selling prices of the current period and the previous period. We assume that the manufacturer acts as the Stackelberg leader and declares wholesale price(s) to the retailer who follows the manufacturer’s decision and sets his selling prices for two consecutive periods. The manufacturer adopts one of the two pricing options: (1) setting the same wholesale price to both the selling periods (2) setting different wholesale prices to two different selling periods. Based on these pricing options, we develop four decision strategies of the manufacturer and the retailer and compare them. For a numerical example, we study the effects of these decision strategies on the optimal results of the supply chain. Further, we graphically analyze under what circumstances a particular decision strategy plays a dominant role.     

A periodic review production and maintenance model with random demand,deteriorating equipment,and binomial yield

In many environments, product yield is heavily influenced by equipment condition. Despite this fact, previous research has either focused on the issue of maintenance, ignoring the effect of equipment condition on yield, or has focused on the issue of production, omitting the possibility of actively changing the machine state. We formulate a Markov decision process model of a single-stage production system in which demand is random. The product yield has a binomial distribution that depends on the equipment condition, which deteriorates over time. The objective is to choose simultaneously the equipment maintenance schedule as well as the quantity to produce in a way that minimizes the sum of expected production, backorder, and holding costs. After proving some results about the structural properties of the optimal policy, numerical problems are used to compare this method to the typical approach of solving the maintenance and production problems sequentially. The results show that the simultaneous solution provides substantial gains over the sequential approach. In the cases studied, the proposed method resulted in an average cost savings of approximately 18%.     

Sourcing decision under disruption risk with supply and demand uncertainty: A newsvendor approach

Multi-sourcing is considered as a common practice to hedge against supply disruption risk. In this context, this paper proposes two models for optimal order allocation in newsvendor setting, where both supply and demand are uncertain. The first model considers a risk neutral decision maker who maximizes the total expected profit under disruption risk. The second one is for a risk averse decision maker who does so under service level constraints. Analytical closed form solutions for both the models are derived. To overcome the computational complexity of the exact optimal solution, two algorithms are developed to generate optimal order quantity and the corresponding set of suppliers. The solutions with exact optimization algorithms and the proposed ones are illustrated and compared with numerical examples. The results show that the proposed algorithms give the exact optimal solution while being tractable. Finally, a case study is used to illustrate the applicability of the proposed model.     

Newsvendor models with dependent random supply and demand

The newsvendor model is perhaps the most widely analyzed model in inventory management. In this single-period model, the only source of randomness is the demand during the period and one tries to determine the optimal order quantity in view of various cost factors. We consider an extention where supply is also random so that the quantity ordered is not necessarily received in full at the beginning of the period. Such models have been well-received in the literature with the assumption of independence between demand and supply. In this setting, we suppose that the random demand and supply are not necessarily independent. We focus on the resulting optimization problem and provide interesting characterizations on the optimal order quantity.     

Heuristics for sourcing from multiple suppliers with alternative quantity discounts

In this paper, we analyze the impact of supplier pricing schemes and supplier capacity limitations on the optimal sourcing policy for a single firm. We consider the situation where the total quantity to be procured for a single period is known by the firm and communicated to the supplier set. In response to this communication, each supplier quotes a price and a capacity limit in terms of a maximum quantity that can be supplied to the buyer. Based on this information, the buyer makes a quantity allocation decision among the suppliers and corresponding to this decision is the choice of a subset of suppliers who will receive an order. Based on industry observations, a variety of supplier pricing schemes from the constituent group of suppliers are analyzed, including linear discounts, incremental units discounts, and all units discounts. Given the complexity of the optimization problem for certain types of pricing schemes, heuristic solution methodologies are developed to identify a quantity allocation decision for the firm. Through an extensive computational comparison, we find that these heuristics generate near-optimal solutions very quickly. Data from a major office products retailer is used to illustrate the resulting sourcing strategies given different pricing schemes and capacity limitations of suppliers in this industry. We find for the case of capacity constrained suppliers, the optimal quantity allocations for two complex pricing schemes (linear discount, and incremental units discount) are such that at most one selected supplier will receive an order quantity that is less than its capacity.     

Optimal ordering policies in inventory systems with random demand and random deal offerings

In this paper, we determine the optimal order policies for a firm facing random demand and random deal offerings. In a periodic review setting, a firm may first place an order at the regular price. Later in the period, if a price promotion is offered by the supplier (with a certain probability), the firm may decide to place another order. We consider two models in the paper. In the first model, the firm does not share the cost savings (due to the promotion offered by the supplier) with its own customers, i.e. its demand distribution remains fixed. In the second model, the cost savings are shared with the final customers. As a result, the demand distribution shifts to the right. For both the models, in a dynamic finite-horizon problem, the order policy structure is divided into three regions and is as follows. If the initial inventory level for the firm exceeds a certain threshold level, it is optimal not to order anything. If it is in the medium range, it is optimal to wait for the promotion and order only if it is offered. The order quantity when the promotion is offered has an ‘order up to’ policy structure. Finally, if the inventory level is below another threshold, it is optimal to place an order at the regular price, and to place a second order if the promotion is offered. The low initial inventory level makes it risky to just wait for the promotion to be offered. The sum of the order quantities in this case has an ‘order up to’ structure. Finally, we model the supplier's problem as a Stackelberg game and discuss the motivation for the supplier to offer a promotion for the case of uniform demand distribution for the firm. In the first model (when the firm does not share the cost savings with its customers), we show that it is rarely optimal for the supplier to offer a promotion. In the second model, the supplier may offer a promotion depending on the price elasticity of the product.