Analysis of supply contracts with commitments and flexibility

In this article we address an important class of supply contracts called the Rolling Horizon Flexibility (RHF) contracts. Under such a contract, at the beginning of the horizon a buyer has to commit requirements for components for each period into the future. Usually, a supplier provides limited flexibility to the buyer to adjust the current order and future commitments in a rolling horizon manner. We present a general model for a buyer's procurement decision under RHF contracts. We propose two heuristics and derive a lower bound. Numerically, we demonstrate the effectiveness of the heuristics for both stationary and non‐stationary demands. We show that the heuristics are easy to compute, and hence, amenable to practical implementation. We also propose two measures for the order process that allow us to (a) evaluate the effectiveness of RHF contracts in restricting the variability in the orders, and (b) measure the accuracy of advance information vis‐a‐vis the actual orders. Numerically we demonstrate that the order process variability decreases significantly as flexibility decreases without a dramatic increase in expected costs. Our numerical studies provide several other managerial insights for the buyer; for example, we provide insights into how much flexibility is sufficient, the value of additional flexibility, the effect of flexibility on customer satisfaction (as measured by fill rate), etc. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Auctions in the post‐change‐order period

A change order is frequently initiated by either the supplier or the buyer, especially when the contract is long‐term or when the contractual design is complex. In response to a change order, the buyer can enter a bargaining process to negotiate a new price. If the bargaining fails, she pays a cancellation fee (or penalty) and opens an auction. We call this process the sequential bargaining‐auction (BA). At the time of bargaining, the buyer is uncertain as to whether the bargained price is set to her advantage; indeed, she might, or might not, obtain a better price in the new auction. To overcome these difficulties, we propose a new change‐order‐handling mechanism by which the buyer has an option to change the contractual supplier after bargaining ends with a bargained price. We call this the option mechanism. By this mechanism, the privilege of selling products or services is transferred to a new supplier if the buyer exercises the option. To exercise the option, the buyer pays a prespecified cash payment, which we call the switch price, to the original supplier. If the option is not exercised, the bargained price remains in effect. When a switch price is proposed by the buyer, the supplier decides whether or not to accept it. If the supplier accepts it, the buyer opens an auction. The option is exercised when there is a winner in the auction. This article shows how, under the option mechanism, the optimal switch price and the optimal reserve price are determined. Compared to the sequential BA, both the buyer and the supplier benefit. Additionally, the option mechanism coordinates the supply chain consisting of the two parties. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 248–265, 2015     

Supplier‐buyer contracting: Asymmetric cost information and cutoff level policy for buyer participation

We consider the problem of designing a contract to maximize the supplier's profit in a one‐supplier–one‐buyer relationship for a short‐life‐cycle product. Demand for the finished product is stochastic and price‐sensitive, and only its probability distribution is known when the supply contract is written. When the supplier has complete information on the marginal cost of the buyer, we show that several simple contracts can induce the buyer to choose order quantity that attains the single firm profit maximizing solution, resulting in the maximum possible profit for the supplier. When the marginal cost of the buyer is private information, we show that it is no longer possible to achieve the single firm solution. In this case, the optimal order quantity is always smaller while the optimal sale price of the finished product is higher than the single firm solution. The supplier's profit is lowered while that of the buyer is improved. Moreover, a buyer who has a lower marginal cost will extract more profit from the supplier. Under the optimal contract, the supplier employs a cutoff level policy on the buyer's marginal cost to determine whether the buyer should be induced to sign the contract. We characterize the optimal cutoff level and show how it depends on the parameters of the problem. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48: 41–64, 2001     

Strategic dynamic sourcing from competing suppliers with transferable capacity investment

We study the supplier relationship choice for a buyer that invests in transferable capacity operated by a supplier. With a long‐term relationship, the buyer commits to source from a supplier over a long period of time. With a short‐term relationship, the buyer leaves open the option of switching to a new supplier in the future. The buyer has incomplete information about a supplies efficiency, and thus uses auctions to select suppliers and determine the contracts. In addition, the buyer faces uncertain demand for the product. A long‐term relationship may be beneficial for the buyer because it motivates more aggressive bidding at the beginning, resulting a lower initial price. A short‐term relationship may be advantageous because it allows switching, with capacity transfer at some cost, to a more efficient supplier in the future. We find that there exists a critical level of the switching cost above which a long‐term relationship is better for the buyer than a short‐term relationship. In addition, this critical switching cost decreases with demand uncertainty, implying a long‐term relationship is more favorable for a buyer facing volatile demand. Finally, we find that in a long‐term relationship, capacity can be either higher or lower than in a short‐term relationship. © 2009 Wiley Periodicals, Inc. Naval Research Logistics 2009     

Contractual agreements for coordination and vendor‐managed delivery under explicit transportation considerations

We consider the coordination problem between a vendor and a buyer operating under generalized replenishment costs that include fixed costs as well as stepwise freight costs. We study the stochastic demand, single‐period setting where the buyer must decide on the order quantity to satisfy random demand for a single item with a short product life cycle. The full order for the cycle is placed before the cycle begins and no additional orders are accepted by the vendor. Due to the nonrecurring nature of the problem, the vendor's replenishment quantity is determined by the buyer's order quantity. Consequently, by using an appropriate pricing schedule to influence the buyer's ordering behavior, there is an opportunity for the vendor to achieve substantial savings from transportation expenses, which are represented in the generalized replenishment cost function. For the problem of interest, we prove that the vendor's expected profit is not increasing in buyer's order quantity. Therefore, unlike the earlier work in the area, it is not necessarily profitable for the vendor to encourage larger order quantities. Using this nontraditional result, we demonstrate that the concept of economies of scale may or may not work by identifying the cases where the vendor can increase his/her profits either by increasing or decreasing the buyer's order quantity. We prove useful properties of the expected profit functions in the centralized and decentralized models of the problem, and we utilize these properties to develop alternative incentive schemes for win–win solutions. Our analysis allows us to quantify the value of coordination and, hence, to identify additional opportunities for the vendor to improve his/her profits by potentially turning a nonprofitable transaction into a profitable one through the use of an appropriate tariff schedule or a vendor‐managed delivery contract. We demonstrate that financial gain associated with these opportunities is truly tangible under a vendor‐managed delivery arrangement that potentially improves the centralized solution. Although we take the viewpoint of supply chain coordination and our goal is to provide insights about the effect of transportation considerations on the channel coordination objective and contractual agreements, the paper also contributes to the literature by analyzing and developing efficient approaches for solving the centralized problem with stepwise freight costs in the single‐period setting. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006     

Rolling‐horizon replenishment: Policies and performance analysis

We consider a rolling‐horizon (RH) replenishment modeling framework under which a buyer can update demand information and inventory status, modify order quantities committed previously, place an advanced order for a new period at the end of the RH, and move along in time seamlessly. We show that the optimal order policy for the two‐period RH problem is a dual‐threshold type for updating period(s) plus a base‐stock type for the advanced order. We provide analytical formulas and algorithms to compute the optimal thresholds and the optimal base‐stock level exactly. With our analytical results and numerical procedures, we demonstrate the significant value of RH replenishment in matching supplies to demands more closely. We also show that with RH updating (flexibility), the value of additional demand information beyond the RH diminishes quickly. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

Multiperiod resource allocation with a smoothing objective

We consider a multiperiod resource allocation problem, where a single resource is allocated over a finite planning horizon of T periods. Resource allocated to one period can be used to satisfy demand of that period or of future periods, but backordering of demand is not allowed. The objective is to allocate the resource as smoothly as possible throughout the planning horizon. We present two models: the first assumes that the allocation decision variables are continuous, whereas the second considers only integer allocations. Applications for such models are found, for example, in subassembly production planning for complex products in a multistage production environment. Efficient algorithms are presented to find optimal allocations for these models at an effort of O(T²). Among all optimal policies for each model, these algorithms find the one that carries the least excess resources throughout the planning horizon. © 1995 John Wiley & Sons, Inc.     

An inventory model with search for best ordering price

This paper presents a single-item inventory model with deterministic demand where the buyer is allowed to search for the most favorable price before deciding on the order quantity. In the beginning of each period, a sequential random sample can be taken from a known distribution and there is a fixed cost per search. The decision maker is faced with the task of deciding when to initiate and when to stop the search process, as well as determining the optimal order quantity once the search process is terminated. The objective is to minimize total expected costs while satisfying all demands on time. We demonstrate that a set of critical numbers determine the optimal stopping and ordering strategies. We present recursive expressions yielding the critical numbers, as well as the minimal expected cost from the beginning of every period to the end of the horizon.     

Supply contracts in manufacturer‐retailer interactions with manufacturer‐quality and retailer effort‐induced demand

We consider a decentralized distribution channel where demand depends on the manufacturer‐chosen quality of the product and the selling effort chosen by the retailer. The cost of selling effort is private information for the retailer. We consider three different types of supply contracts in this article: price‐only contract where the manufacturer sets a wholesale price; fixed‐fee contract where manufacturer sells at marginal cost but charges a fixed (transfer) fee; and, general franchise contract where manufacturer sets a wholesale price and charges a fixed fee as well. The fixed‐fee and general franchise contracts are referred to as two‐part tariff contracts. For each contract type, we study different contract forms including individual, menu, and pooling contracts. In the analysis of the different types and forms of contracts, we show that the price only contract is dominated by the general franchise menu contract. However, the manufacturer may prefer to offer the fixed‐fee individual contract as compared to the general franchise contract when the retailer's reservation utility and degree of information asymmetry in costs are high. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Joint inventory and pricing decisions for perishable products with two‐period lifetime

We consider a periodic review model over a finite horizon for a perishable product with fixed lifetime equal to two review periods. The excess demand in a period is backlogged. The optimal replenishment and demand management (using price) decisions for such a product depend on the relative order of consumption of fresh and old units. We obtain insights on the structure of these decisions when the order of consumption is first‐in, first‐out and last‐in, first‐out. For the FIFO system, we also obtain bounds on both the optimal replenishment quantity as well as expected demand. We compare the FIFO system to two widely analyzed inventory systems that correspond to nonperishable and one‐period lifetime products to understand if demand management would modify our understanding of the relationship among the three systems. In a counterintuitive result, we find that it is more likely that bigger orders are placed in the FIFO system than for a nonperishable product when demand is managed. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013     

Pricing and inventory management during new product introduction when shortage creates hype

In this study, we analyze the joint pricing and inventory management during new product introduction when product shortage creates additional demand due to hype. We develop a two‐period model in which a firm launches its product at the beginning of the first period, before it observes sales in the two periods. The product is successful with an exogenous probability, or unsuccessful with the complementary probability. The hype in the second period is observed only when the product is successful. The firm learns the actual status of the product only after observing the first‐period demand. The firm must decide the stocking level and price of the product jointly at the beginning of each of the two periods. In this article, we derive some structural properties of the optimal prices and inventory levels, and show that (i) firms do not always exploit hype, (ii) firms do not always increase the price of a successful product in the second period, (iii) firms may price out an unsuccessful product in the first period if the success probability is above a threshold, and (iv) such a threshold probability is decreasing in the first‐period market potential of the successful product. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 304–320, 2015     

Optimal policies for a multi-product inventory system with negotiable lead times

The objective of this paper is to determine the optimum inventory policy for a multi-product periodic review dynamic inventory system. At the beginning of each period two decisions are made for each product. How much to “normal order” with a lead time of λ_n periods and how much to “emergency order” with a lead time of λ_e periods, where λe = λ_n - 1. It is assumed that the emergency ordering costs are higher than the normal ordering costs. The demands for each product in successive periods are assumed to form a sequence of independent identically distributed random variables with known densities. Demands for individual products within a period are assumed to be non-negative, but they need not be independent. Whenever demand exceeds inventory their difference is backlogged rather than lost. The ordering decisions are based on certain costs and two revenue functions. Namely, the procurement costs which are assumed to be linear for both methods of ordering, convex holding and penalty costs, concave salvage gain functions, and linear credit functions. There is a restriction on the total amount that can be emergency ordered for all products. The optimal ordering policy is determined for the one and N-period models.     

Switching costs,dynamic uncertainty,and buyer–seller relationships

We analyze strategic relationships between buyers and sellers in markets with switching costs and dynamic uncertainty by investigating the scenario wherein a representative buyer trades with two foreign sellers located in the same foreign country. We show that, under exchange rate uncertainty, switching costs may lead to switching equilibria where both sellers co‐exist in the market with the buyer, or no‐switching equilibria where either seller captures the market. The presence of exchange rate uncertainty facilitates competition by allowing the sellers to co‐exist in the market with the buyer. However, if the level of uncertainty is beyond a threshold, the only viable equilibria are those where one of the sellers captures the market. Further, depending on the level of exchange rate uncertainty and the sellers' variable costs, switching costs may either raise or lower the level of prices in long‐term contracts between the buyer and the sellers. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Base‐stock policies in capacitated assembly systems: Convexity properties

We study an assembly system with a single finished product managed using an echelon base‐stock or order‐up‐to policy. Some or all operations have capacity constraints. Excess demand is either backordered in every period or lost in every period. We show that the shortage penalty cost over any horizon is jointly convex with respect to the base‐stock levels and capacity levels. When the holding costs are also included in the objective function, we show that the cost function can be written as a sum of a convex function and a concave function. Throughout the article, we discuss algorithmic implications of our results for making optimal inventory and capacity decisions in such systems.© 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

Commitment decisions with partial information updating

In this paper, we extend the results of Ferguson M. Naval Research Logistics 8 . on an end‐product manufacturer's choice of when to commit to an order quantity from its parts supplier. During the supplier's lead‐time, information arrives about end‐product demand. This information reduces some of the forecast uncertainty. While the supplier must choose its production quantity of parts based on the original forecast, the manufacturer can wait to place its order from the supplier after observing the information update. We find that a manufacturer is sometimes better off with a contract requiring an early commitment to its order quantity, before the supplier commits resources. On the other hand, the supplier sometimes prefers a delayed commitment. The preferences depend upon the amount of demand uncertainty resolved by the information as well as which member of the supply chain sets the exchange price. We also show conditions where demand information updating is detrimental to both the manufacturer and the supplier. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005     

Selecting T for a periodic review inventory model with staggered deliveries

Consider a single‐item, periodic review, infinite‐horizon, undiscounted, inventory model with stochastic demands, proportional holding and shortage costs, and full backlogging. Orders can arrive in every period, and the cost of receiving them is negligible (as in a JIT setting). Every T periods, one audits the current stock level and decides on deliveries for the next T periods, thus incurring a fixed audit cost and—when one schedules deliveries—a fixed order cost. The problem is to find a review period T and an ordering policy that satisfy the average cost criterion. The current article extends an earlier treatment of this problem, which assumed that the fixed order cost is automatically incurred once every T periods. We characterize an optimal ordering policy when T is fixed, prove that an optimal review period T** exists, and develop a global search algorithm for its computation. We also study the behavior of four approximations to T** based on the assumption that the fixed order cost is incurred during every cycle. Analytic results from a companion article (where μ/σ is large) and extensive computational experiments with normal and gamma demand test problems suggest these approximations and associated heuristic policies perform well when μ/σ ≥ 2. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 329–352, 2000     

Collaborative procurement among competing buyers

Collaborative procurement emerged as one of the many initiatives for achieving improved inter‐firm coordination and collaboration. In this article, we adopt a game‐theoretical approach to study the interaction between two firms who procure jointly, but produce independently and remain competitors in a product market characterized by price‐sensitive demand. We study the underlying economics behind collaborative procurement, examine the effects of collaboration on buyer and supplier profitability, and derive conditions under which collaboration is beneficial to each participant. We find that a necessary and sufficient condition for a buyer to collaborate is to increase its sales. We identify the conditions that lead equal size buyers (i.e., consortia consisting of only large buyers or only small buyers) versus different size buyers to collaborate. We also determine the conditions that make collaboration profitable for the supplier, and show that rather than selling a large quantity to a single buyer, the supplier prefers to sell to multiple buyers in smaller quantities. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Vertically restrictive pricing in supply chains with price‐dependent demand

We examine the behavior of a manufacturer and a retailer in a decentralized supply chain under price‐dependent, stochastic demand. We model a retail fixed markup (RFM) policy, which can arise as a form of vertically restrictive pricing in a supply chain, and we examine its effect on supply chain performance. We prove the existence of the optimal pricing and replenishment policies when demand has a linear additive form and the distribution of the uncertainty component has a nondecreasing failure rate. We numerically compare the relative performance of RFM to a price‐only contract and we find that RFM results in greater profit for the supply chain than the price‐only contract in a variety of scenarios. We find that RFM can lead to Pareto‐improving solutions where both the supplier and the retailer earn more profit than under a price‐only contract. Finally, we compare RFM to a buyback contract and explore the implications of allowing the fixed markup parameter to be endogenous to the model. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006.     

Formulations for dynamic lot sizing with service levels

In this article, we study deterministic dynamic lot‐sizing problems with a service‐level constraint on the total number of periods in which backlogs can occur over a finite planning horizon. We give a natural mixed integer programming formulation for the single item problem (LS‐SL‐I) and study the structure of its solution. We show that an optimal solution to this problem can be found in \begin{align*}\mathcal O(n^2\kappa)\end{align*} time, where n is the planning horizon and \begin{align*}\kappa=\mathcal O(n)\end{align*} is the maximum number of periods in which demand can be backlogged. Using the proposed shortest path algorithms, we develop alternative tight extended formulations for LS‐SL‐I and one of its relaxations, which we refer to as uncapacitated lot sizing with setups for stocks and backlogs. {We show that this relaxation also appears as a substructure in a lot‐sizing problem which limits the total amount of a period's demand met from a later period, across all periods.} We report computational results that compare the natural and extended formulations on multi‐item service‐level constrained instances. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013     

Revenue sharing contracts in a supply chain with uncontractible actions

We consider a supplier–customer relationship where the customer faces a typical Newsvendor problem of determining perishable capacity to meet uncertain demand. The customer outsources a critical, demand‐enhancing service to an outside supplier, who receives a fixed share of the revenue from the customer. Given such a linear sharing contract, the customer chooses capacity and the service supplier chooses service effort level before demand is realized. We consider the two cases when these decisions are made simultaneously (simultaneous game) or sequentially (sequential game). For each game, we analyze how the equilibrium solutions vary with the parameters of the problem. We show that in the equilibrium, it is possible that either the customer's capacity increases or the service supplier's effort level decreases when the supplier receives a larger share of the revenue. We also show that given the same sharing contract, the sequential game always induces a higher capacity and more effort. For the case of additive effort effect and uniform demand distribution, we consider the customer's problem of designing the optimal contract with or without a fixed payment in the contract, and obtain sensitivity results on how the optimal contract depends on the problem parameters. For the case of fixed payment, it is optimal to allocate more revenue to the supplier to induce more service effort when the profit margin is higher, the cost of effort is lower, effort is more effective in stimulating demand, the variability of demand is smaller or the supplier makes the first move in the sequential game. For the case of no fixed payment, however, it is optimal to allocate more revenue to the supplier when the variability of demand is larger or its mean is smaller. Numerical examples are analyzed to validate the sensitivity results for the case of normal demand distribution and to provide more managerial insights. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008