Analysis of supply chains with quantity based fixed incentives

This paper examines the use of quantity based fixed incentives to coordinate inventory decisions in a decentralized supply chain. We consider a two stage supply chain of autonomous supplier and distributor and prove that the optimal ordering policy for the newsvendor distributor under fixed incentives is an (s,S)$(s\text{,}S)$ type policy. We further show that external and internal quantity based incentives can restore channel coordination in single period and channel members can benefit through arbitrary splitting of the resulting additional chain profit. The single period results are extended to multiple periods and the impact of fixed incentives on the distributor’s optimal stocking policy and channel efficiency are examined under three different multi-period supplier strategies. Numerical examples are used to compare the multi-period strategies and to provide additional managerial insights. The results show that contrary to common belief, incentive plans developed and maintained based only on current inventory data perform poorly in long term and that such incentive plans must be periodically updated to enhance their efficiency. Furthermore, we show that high level of incentives designed to push too much inventory downstream of the supply chain can actually reduce the chain’s efficiency.     

A quantity discount approach to supply chain coordination

Quantity discounts provide a practical foundation for inventory coordination in supply chains. However, typical supply chain participants may encounter difficulties in implementing the coordination policy simply because (1) specified lot size adjustments may deviate from the economic lot sizes and (2) the buying firm may face amplified overstocking risks related to increased order quantities. The main objective of this study is to develop a quantity discount model that resolves the practical challenges associated with implementing quantity discount policies for supply chain coordination between a supplier and a buyer. The proposed Buyer’s Risk Adjustment (B-RA) model allows the supplier to offer discounts that capitalize on the original economic lot sizes and share the buyer’s risk of temporary overstocking under uncertain demand. The analytical results suggest that the proposed B-RA discount approach is a feasible alternative for supply chain coordination under uncertain demand conditions.     

Relating the multiple supply problem to quantity flexibility contracts

We consider two classes of problems in the supply chain management literature: the multiple supply problem and quantity flexibility contracts. We identify the similarities between these two problems and show that for both the total operating cost of the manufacturer is convex with respect to its arguments under mild conditions.     

Optimal inventory policy with supply uncertainty and demand cancellation

We consider a periodic review model where the firm manages its inventory under supply uncertainty and demand cancellation. We show that because of supply uncertainty, the optimal inventory policy has the structure of re-order point type. That is, we order if the initial inventory falls below this re-order point, otherwise we do not order. This is in contrast to the work of Yuan and Cheung (2003) who prove the optimality of an order up to policy in the absence of supply uncertainty. We also investigate the impact of supply uncertainty and demand cancellation on the performance of the supply chain. Using our model, we are able to quantify the importance of reducing the variance of either the distribution of yield or the distribution of demand cancellation. The single, multiple periods and the infinite horizon models are studied.     

A two-stage supply chain coordination mechanism considering price sensitive demand and quantity discounts

This paper explores the coordination between a supplier and a buyer within a decentralized supply chain, through the use of quantity discounts in a game theoretic model. Within this model, the players face inventory and pricing decisions. We propose both cooperative and non-cooperative approaches considering that the product traded experiences a price sensitive demand. In the first case, we study the dynamics of the game from the supplier's side as the leader in the negotiation obtaining a Stackelberg equilibrium, and then show how the payoff of this player could still improve from this point. In the second case, a cooperative model is formulated, where decisions are taken simultaneously, emulating a centralized firm, showing the benefits of the cooperation between the players. We further formulate a pricing game, where the buyer is allowed to set different prices to the final customer as a reaction to the supplier's discount decisions. For the latter we investigate the difference between feasibility of implementing a retail discount given a current coordination mechanism and without it. Finally the implications of transportation costs are analyzed in the quantity discount schedule. Our findings are illustrated with a numerical example showing the difference in the players’ payoff in each case and the optimal strategies, comparing in each case our results with existing work.     

Coordination of supply chains with bidirectional option contracts

In this paper we develop a supply contract for a two-echelon manufacturer–retailer supply chain with a bidirectional option, which may be exercised as either a call option or a put option. Under the bidirectional option contract, we derive closed-form expressions for the retailer’s optimal order strategies, including the initial order strategy and the option purchasing strategy, with a general demand distribution. We also analytically examine the feedback effects of the bidirectional option on the retailer’s initial order strategy. In addition, taking a chain-wide perspective, we explore how the bidirectional option contract should be set to attain supply chain coordination.     

An optimization approach for supply chain management models with quantity discount policy

Owing to the difficulty of treating nonlinear functions, many supply chain management (SCM) models assume that the average prices of materials, production, transportation, and inventory are constant. This assumption, however, is not practical. Vendors usually offer quantity discounts to encourage the buyers to order more, and the producer intends to discount the unit production cost if the amount of production is large. This study solves a nonlinear SCM model capable of treating various quantity discount functions simultaneously, including linear, single breakpoint, step, and multiple breakpoint functions. By utilizing the presented linearization techniques, such a nonlinear model is approximated to a linear mixed 0–1 program solvable to obtain a global optimum.     

Incorporating quantity discounts and their inventory impacts into the centralized purchasing decision

Multi-site organizations must balance conflicting forces to determine the appropriate degree of purchasing centralization for their respective supplies. The ability to garner quantity discounts represents one of the primary reasons that organizations centralize procurement. This paper provides methodologies to calculate optimal order quantities and compute total purchasing and inventory costs when products have quantity discount pricing. Procedures for both all-units and incremental quantity discount schedules are provided for four different strategic purchasing configurations (scenarios): complete decentralization, centralized pricing with decentralized purchasing, centralized purchasing with local distribution, and centralized purchasing and warehousing. For ordering decisions under local distribution, procedures to determine optimal order quantities and costs are presented in a precise form that could be easily implemented into spreadsheets by practicing managers. For the more complicated multi-echelon scenarios, we introduce a single-cycle policy with a tailored aggregation refinement step that performs very well under experimentation when compared to a conservative bound.     

An instrument for measuring supply chain flexibility for the textile and clothing companies

Supply chain flexibility (SCF) represents the capability of firms to respond to unanticipated changes in customer needs and competitor actions. Given the growing research interest in flexibility strategies, the development of a valid and reliable instrument to measure organizational responses toward environmental uncertainties or risks is imperative. However, no systematic and scientific research has been conducted to develop such an instrument. The present study adopts a comprehensive and rigorous procedure to develop a multifaceted scale for SCF through an empirical investigation. The results of a confirmatory factor analysis suggest that SCF can be operationalized as a second-order factor model comprising four dimensions, namely: sourcing flexibility, operating system flexibility, distribution flexibility, and information system flexibility. A series of goodness-of-fit indices further demonstrates that this scale is internally consistent, reliable, and valid. The various findings suggested in the present study provide a more succinct picture of SCF, and the well-validated scale could be used as a basis for further research and theoretical groundwork in the field of supply chain management.     

An economic order quantity with imperfect quality and quantity discounts

Previous studies in the issue of inventory models with imperfect quality assumed the defectives could be sold in a batch by the end of the inspection process and the manufacturing systems were push systems. However, the above assumptions may not be true in the pull system in which buyer is powerful. Therefore, in this paper, we develop a new inventory model for items with imperfect quality and quantity discounts where buyer has exerted power over its supplier. Based on the concept of powerful buyer, there are three considerations included in this new model: (1) the order quantity is manufactured at one setup and is shipped over multiple deliveries, (2) the defectives are screened out by a 100% inspection for each shipment but sold in a batch by the end of inspection at the last shipment of each cycle, and (3) the supplier offers quantity discounts to response the request of the powerful buyer. Further, an algorithm is developed to help the powerful buyer to determine the optimal order policy accurately and quickly. Two numerical examples are available in this paper to illustrate the proposed model and algorithm. Besides, based on the numerical examples, a sensitivity analysis is made to investigate the effects of four important parameters (the inspection rate, the defective rate, the receiving cost, and the ordering cost) on the optimal solution.     

Two-product inventory management with fixed costs and supply uncertainty

This paper determines the optimal ordering policy for a two-product, periodic-review inventory problem in which the probability of supply availability is unknown. Moreover, there are two different fixed costs assigned to each product. Demand rates are random variables with known probability density functions, and the supply availability for each product is updated at the beginning of each time period. We prove the optimality of (s,S) policy with a monotone switching curve that indicates the priority of production, where the order-up-to levels and the reorder points are functions of supply availability information. A simple computation is proposed to calculate the two threshold levels. Bayesian updating helps to manage the optimal ordering policy by updating supply disruption information. Numerical results show that improving the accuracy of the forecast leads to making a better ordering decision and eliminating the negative effect of supply disruption on the total cost.     

Managing capacity flexibility in make-to-order production environments

This paper addresses the problem of managing flexible production capacity in a make-to-order (MTO) manufacturing environment. We present a multi-period capacity management model where we distinguish between process flexibility (the ability to produce multiple products on multiple production lines) and operational flexibility (the ability to dynamically change capacity allocations among different product families over time). For operational flexibility, we consider two polices: a fixed allocation policy where the capacity allocations are fixed throughout the planning horizon and a dynamic allocation policy where the capacity allocations change from period to period. The former approach is modeled as a single-stage stochastic program and solved using a cutting-plane method. The latter approach is modeled as a multi-stage stochastic program and a sampling-based decomposition method is presented to identify a feasible policy and assess the quality of that policy. A computational experiment quantifies the benefits of operational flexibility and demonstrates that it is most beneficial when the demand and capacity are well-balanced and the demand variability is high. Additionally, our results reveal that myopic operating policies may lead a firm to adopt more process flexibility and form denser flexibility configuration chains. That is, process flexibility may be over-valued in the literature since it is assumed that a firm will operate optimally after the process flexibility decision. We also show that the value of process flexibility increases with the number of periods in the planning horizon if an optimal operating policy is employed. This result is reversed if a myopic allocation policy is adopted instead.     

Multi-commodity supply network planning in the forest supply chain

We consider in this paper a two echelon timber procurement system in which the first echelon consists of multiple harvesting blocks and the second echelon consists of multiple mills (e.g., sawmills), both distributed geographically. Demand is put forward by mills in the form of volumes of logs of specific length and species. Due to the impact of log handling and sorting on cut-to-length harvester and forwarder productivity [Gingras, J.-F., Favreau, J., 2002. Incidence du triage sur la productivité des systèmes par bois tronçonnés. Avantage 3], the harvesting cost per unit volume increases as the number of product variety harvested per block increases. The overall product allocation problem is a large scale mixed integer programming problem with the objective of minimizing combined harvesting and aggregated transportation costs, under demand satisfaction constraints. A heuristic is first introduced then, an algorithm based on the branch-and-price approach is proposed for larger scale problems. Experimentations compare solutions found with the heuristic with the corresponding optimal solutions obtained with both Cplex (using the branch-and-bound approach) and the branch-and-price approach. Results demonstrate the good performance level of the heuristic approach for small scale problems, and of the branch-and-price approach for large scale problems.     

The multi-product newsboy problem with supplier quantity discounts and a budget constraint

This paper considers the multi-product newsboy problem with both supplier quantity discounts and a budget constraint, while each feature has been addressed separately in the literature. Different from most previous nonlinear optimization models on the topic, the problem is formulated as a mixed integer nonlinear programming model due to price discounts. A Lagrangian relaxation approach is presented to solve the problem. Computational results on both small and large-scale test instances indicate that the proposed algorithm is extremely effective for the problem. An extension to multiple constraints and preliminary computational results are also reported.     

An integrated revenue sharing and quantity discounts contract for coordinating a supply chain dealing with short life-cycle products

This paper develops a combined contract model for coordinating a two stage supply chain where the demand at the retailer’s end is price sensitive and stock dependent. It has been shown that proposed coordination mechanism achieves perfect coordination and win–win situation for both the members of the supply chain. Further, an extensive sensitivity analysis is performed to examine the impact of various parameters on supply chain performance. It has been found that stock dependency factor has positive impact on order quantity and subsequently on supply chain performance. The paper has also made a comparative statics analysis to see the impact of certain parameters on the pricing and replenishment policies of the retailer.     

Analysis of a two-party supply chain with random disruptions

We study a continuous-review inventory problem of a two-echelon supply chain with random disruptions, identify properties of the optimal cost function, compare the optimal order quantity with the classical economic order quantity, analyze the sensitivity of the optimal solution, and explore the conditions under which zero-inventory ordering policy is preferred.     

A comprehensive note on: An economic order quantity with imperfect quality and quantity discounts

Lin [T.Y. Lin, An economic order quantity with imperfect quality and quantity discounts, Appl. Math. Model. 34 (10) (2010) 3158–3165] recently proposed an EOQ model with imperfect quality and quantity discounts, where the lot-splitting shipments policy is adopted. In this note we first rectify the holding cost terms showed in Lin to obtain a new objective function, then resolve the problem and develop an easy to implement algorithm to find the overall optimal solutions for the model. Besides, we present a new model for items with imperfect quality, where lot-splitting shipments and different holding costs for good and defective items are considered. The closed-form formulas for determining the optimal ordering and shipping policies are derived. Also, the results are examined analytically and numerically to gain more insights of the solutions.     

An optimal solution for the stochastic version of the Wagner–Whitin dynamic lot-size model

We present an algorithm for determining the optimal solution over the entire planning horizon for the dynamic lot-size model where demand is stochastic and non-stationary. The optimal solution to the deterministic problem is the well-known Wagner–Whitin algorithm. The present work contributes principally to knowledge building and provides a tool for researchers. One potentially useful contribution to practice is the solution to an important special case, where demand follows normal distributions. Other contributions to practice will likely flow from the development of improved heuristics and the improved basis to evaluate heuristic performance.     

Profit maximization with quantity discount: An application of geometric programming

Profit maximization is an important issue to the firms that pursue the largest economic profit possible. Traditionally, profit maximization problem is solved by differentiating with respect to input prices. The total differentiation of the first-order conditions might give complicated equations difficult to handle. Different from traditional studies, this paper considers input quantity discount and employs geometric programming technique to derive the objective value for the profit-maximization problem. The geometric programming approach not only gives the global optimum solution but also provides the information that is able to discover the relationship between profit maximization and returns to scale in the solution process. No differentiation is required. Moreover, geometric programming can provide a computationally attractive view of sensitivity analysis for the changes in parameters. Examples are given to illustrate the idea proposed in this paper.