No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
An incentive contract for leadtime reduction in an (S-1, S) inventory system
Abstract
We consider a single-item, infinite-horizon, continuous-review (S-1,S) inventory system with Poisson demand and stochastic leadtimes. In the business scenario examined, the supplier exerts a one-time effort that reduces the mean and/or variance of replenishment leadtimes, and the inventory manager of the system makes periodic payments to the supplier based on realized leadtimes. Assuming an exponential utility function for the supplier and a normal leadtime distribution, we provide the parameters of a contract, under which the supplier periodically receives a fixed amount plus the sum of linear incentive payments for the replenishment orders delivered during that period based on their leadtimes. Extensive numerical experiments suggest that the cost impact of the leadtime reduction can be large, and that the proposed fixed-amount-plus-linear-incentive contract is very effective.
... Our research focuses on another dimension of lead time incentive in addition to a fixed-price contract to enhance the system uptime and reduce the cost of on-site spare parts. Lee [14] analyzes an incentive contract for lead time reduction over an (S − 1, S) inventory system and proposes a fixed-amount-pluslinear-incentive contract for the supplier. Accordingly, the cost effect of lead time reduction can be demonstrated by extensive quantitative experiments. ...
... In the operations, when compared with CIPN and CIPRL1, CIPRL2 has the additional decision variable of the inventory reserve level r i p,t over the time period t of service contract i for part p. This optimal reserve level can be identified by minimizing over-reserved and under-reserved inventory by considering the holding cost of part p and the backlog costs of same-day and next-day orders (14) where G p,t+1 refers to the demand of the same-day service contract of part p minus the replenishment quantity on the next day, t + 1; r sd * p,t refers to the reserve level of part p for the sameday demand in the period t; b nd p and b sd p refer to the backlog costs of missing the same-day and next-day requests of part p; h p refers to the holding cost of part p. ...
... During the cutoff period, the remaining part will be used to fulfill the nextday order under CIPRL1. However, under CIPRL2, whether it will be used to fulfill the next-day order or not depends on the result of (14). ...
Spare parts support services have received increasing management attention due to the growing number of critical systems in many business sectors. In this paper, we examine an integrated system design approach to customize spare parts support services based on response time with inventory pooling strategies. To provide customized services that meet user requirements for spare part response time, we depart from the traditional spare parts management and develop a systematic approach to design service parts support services based on axiomatic design theory. In particular, we focus on pricing discrimination decisions in service parts contracts for two-tier users under a mechanism design framework. Distinguishing between users of next-day and same-day contracts for service parts operations, we further evaluate the effect of various inventory pool structures with reserve strategies through a simulation model for the objective of cost minimization. These analytical results of this new approach provide guidance for managers in customizing spare parts support services with the holistic consideration of pricing scheme, response time, and inventory policy.
... To determine the optimal warranty period as a pricing decision, Kim and Park (2008) set up a two-stage optimal control theory model to deal with the conflicting decision among sale price, inventory holding cost, and spare parts manufacturing cost. Perhaps, the most related work is by Lee (2009), who studied an incentive contract for lead time reduction over an (S-1, S) inventory system and proposed a fixed amount plus linear incentive contract for a user. In addition to the cost-plus pricing scheme, Tong et al. (2014) studied the design and pricing decision of a two-dimensional extended warranty service for users to purchase at the point of sale and the expiration of the base warranty. ...
This study examines user outsourcing of spare parts management to vendors through a service contract. The user’s selection of a fixed-price service parts contract is formulated as a stochastic integer programming model that decides multiple response times and on-site spare parts, while considering component breakdown with uncertain failure rates. We analytically derive the optimality conditions for the continuous case and subsequently design an efficient algorithm. Numerical illustrations and analyses are conducted to evaluate decisions under various scenarios. Our analysis shows that when both failure rate
and expedited contract cost are high, coupled with low part cost, users would prefer the purchase of spare parts for all components to expedited contracts. A fixed-price expedited contract has a lower marginal cost with respect to failure rate than a fixed-price next day contract and a usage-based contract. We also examine inventory behaviour for a single part, multiple types of parts, and multiple groups of parts. It is shown that there is a cost-saving pooling effect in spare parts for identical items, which significantly raises the likelihood of having on-site stored parts. The problem becomes more complex for multiple items, reflecting bundling effects between items for a given contract.
... The demand is often modelled as a Poisson process for several reasons (Zipkin, 2000), and numerous researchers adopt this assumption (e.g., Schmidt and Nahmias, 1985;Moinzadeh and Schmidt, 1991;Perry and Posner, 1998;Dekker et al., 2002;Lee, 2009). Under this condition, the lead-time demand is Poissonian as well, and, in special cases, it can be approximated quite well by a well-suited Gaussian random variable (Zipkin, 2000). ...
In this paper, we consider the continuous review (S − 1, S) policy with complete backordering, deterministic and constant lead-time, and with a Poissonian lead-time demand. Moreover, we take into account two different expressions of the total expected cost depending on the customer-service criterion adopted. First, we approximate the Poissonian lead-time demand with a well-suited Gaussian random variable. Although this system can be easily solved with a simple algorithmic approach, it is interesting to be aware of the direct analytical functional relationship between the optimal value of the decision variable and the parameters of the model. Hence, the second step consists in providing some approximated closed-form minimum-cost solutions for both cost models considered under a Gaussian lead-time demand. An extensive numerical study is finally shown to characterise the precision achieved by the approximations developed.
This research brings together two important research streams: lead-time management and the inventory billboard effect. While traditional inventory theory, which assumes that demand is independent of the average on-hand inventory, recommends that lead-time be reduced to the lowest level possible, it is clearly not the case when inventory exhibits the billboard effect, which refers to the demand stimulating effect of inventory. We use analytical models to examine the firm's optimal lead-time decision in the presence of the inventory billboard effect in two scenarios: with and without inventory based competition. We begin with the single firm scenario, where a firm employs the base stock policy for the infinite horizon continuous-review inventory problem with a deterministic lead-time and inventory dependent demand. The firm's decisions are to choose the optimal lead-time and the corresponding base stock level to maximize its expected profit rate. We find that the inventory billboard effect favors a long lead-time, because a long lead-time results in a higher inventory level, which in turn induces more demand. We then consider the case where two firms compete on inventory for customer demand. We completely characterize the unique Nash equilibrium and provide closed-form solutions, which allow us to conclude that inventory based competition pressures both firms to increase the lead-time. Our numerical studies show that the extent to which demand depends on inventory amplify the impact of competition on the optimal decisions and the associated profit. Our findings suggest that lead-time reduction, although widely advocated by popular production philosophies such as Just-in-Time, has to be carefully evaluated when inventory exhibits billboard effect.
It is possible to evaluate production system performance measurement and organizational alternative structures with dynamic simulation methods for efficiency of cost and productivity. The parameters which are vary for different organizational structures and scope of production systems are examined in a mutual simulation-based performance measurement system. In general, the simulation-based performance measurement criteria are lead time, lead time deviation, utilization rate, and work-in-process as goal achievements of logistics and delivery rate in performance evaluation module.In this study, an evaluation module is developed and integrated to system simulation results as a part of the research project of “Development of Simulation Software for Facility Organizing, Production System Structuring, and Performance Measuring (Faborg-Sim)” at Suleyman Demirel University Industrial Engineering Department which is supported by The Scientific and Technological Research Council of Turkey (TUBITAK).
In a recent paper, Ben-Daya and Raouf presented a continuous review inventory model in which they considered both lead time and the order quantity as decision variables where the shortages are neglected. We assume here that the shortages are allowed and we extend the Ben-Daya and Raouf model by adding the stockout cost. Furthermore, the effects of parameters are also included.
In a recent paper, Ben-Daya and Raouf presented a continuous review inventory model in which they considered both lead time and the order quantity as decision variables where the shortages are neglected. We assume here that the shortages are allowed and we extend the Ben-Daya and Raouf model by adding the stockout cost. Furthermore, the effects of parameters are also included.
This paper focuses on a firm selling a make-to-stock product with a constant customer demand rate. The firm follows an exact (Q, r ) policy for raw material inventory control and faces a random eplenishment lead time. Through this research, we wish to gain a better understanding of the impact of investing in reducing supply lead time when the investment costs have to be borne, partly or fully, by the firm. This work is motivated by the recognition that lead time reduction is now of strategic importance in the successful operation of a firm. We examine different types of investment schemes in replenishment lead time reduction and the different cost models they generate.We present analytical and numerical results and insights for each type of model, compute the optimal (Q, r ) policy and the associated investment levels. The work presents new results, and sheds light on some apparently counter-intuitive observations.
We consider (S - 1, S) policies for a single item whose lifetime is fixed and known with certainty. Demands are generated by a stationary Poisson process and there is a positive leadtime for replenishment. We believe this study gives the only analysis for perishables with a positive order leadtime. The analysis involves the derivation of the stationary distribution of the S-dimensional stochastic process corresponding to the time elapsed since the last S orders were placed. This distribution is then used to obtain an expression for the expected cost rate of operating the system in steady state as a function of S. A computer program has been developed to compute optimal S values and expected annual costs. We report a computation for a variety of system parameters which show some of the unusual features of the problem. Finally, we show how this model can be used in the context of a problem of optimizing availability of operating equipment subject to scheduled maintenance as well as random failure.
The Japanese experience of Just-in-Time (JIT) production has shown that there are advantages and benefits associated with the efforts to reduce inventory lead time and the associated inventory cost. The length of lead time directly affects the customer service level, inventory investment in safety stock, and the competitive abilities of a business. In most of the literature dealing with inventory problems, either a deterministic model or probabilistic model, lead time is viewed as a prescribed constant or a stochastic variable, and is not subject to control. However, in many practical situations, lead time can be reduced by an additional cost. Moreover, the successful implementation of JIT production in today's supply chain enviromnent requires a new spirit of cooperation between the buyer and the vendor (Goyal and Srinivasan 1992). A desirable condition in long time purchase agreements in such a manufacturing environment is the frequent delivry of small quantities of items so as to minimize inventory holding cost for the buyer. The vendor also needs to minimize his or her total inventory costs. An integrated inventory model that allows the two trading parties to form a strategic alliance for profit sharing may prove helpful in breaking down the traditional barriers. This paper presents an integrated inventory model with controllable lead time. The model is shown to provide a lower total cost and shorter lead time compared with those of Banerjee (1986) and Goyal (1988), and is useful for practical inventory problems.
In this article we analyze a lost sales (S−1,S) perishable system, under Poisson demands and exponential lifetimes, in which the reorders are placed at every demand epoch so as to take the inventory position back to its maximum level S. The items are replenished one at a time and the resupply time has arbitrary distribution. The various operating characteristics are obtained using Markov renewal techniques. A matrix recursive scheme is developed to determine the stationary distribution of the underlying Markov chain. The efficiency of this procedure in the determination of optimal S that minimizes the long run expected cost rate is discussed. Sensitivity analysis of various system parameters is also carried out.
This paper derives the important measures of supply performance for an inventory model in which the replenishment policy is to reorder a unit whenever demand occurs. The performance measures considered are: the expected number of back-orders; the expected resupply and stockout times; and the probability distribution for resupply times. All demands, described by an arbitrary distribution, are captured, but need not necessarily be satisfied immediately. The delivery times are assumed constant. Also considered is the problem of trade-offs between spares and the delivery time to achieve a given level of supply performance at least cost.
Almost all inventory models assume that lead time is prescribed and
thus is not subject to control. In many practical situations, however,
lead time is controllable; that is, lead time can be shortened, at the
expense of extra costs, so as to improve customer service, reduce
inventory investment in safety stocks, and improve system
responsiveness. Although some authors recognise the advantage of short
lead time and suggest that it should be considered a variable for
management to control instead of a given, there is a lack of a suitable
inventory model for determining the optimal lead time. A probabilistic
inventory model in which the lead time is a decision variable is
presented. It is assumed that the demand follows normal distribution and
the lead time consists of n components each having a different
cost for reduced lead time. The objective is to determine the lead time
that minimises the sum of the expected holding cost and the additional
cost.
Most inventory and production planning models in the academic literature treat lead times either as constants or random variables with known distributions outside of management control. However, a number of recent articles in the popular press have argued that reducing lead times is a dominant issue in manufacturing strategy. The benefits of reducing customer lead times that are frequently cited include increased customer demand, improved quality, reduced unit cost, lower carrying cost, shorter forecast horizon, less safety stock inventory, and better market position.
Although the costs of reducing lead times in the long term may be relatively insignificant compared with the benefits, in the short term these costs can have a significant impact on the profitability of a firm. This article develops a conceptual framework within which the costs and benefits of lead time reduction can be compared. Mathematical models for optimal lead time reduction are developed within this framework. The solutions to these models provide methods for calculating optimal lead times, which can be applied in practice. Sensitivity analysis of the optimal solutions provides insight into the structure of these solutions.
The principal-agent paradigm, in which a principal has a primary stake in the performance of some system but delegates operational control of that system to an agent, has many natural applications in operations management (OM). However, existing principal-agent models are of limited use to OM researchers because they cannot represent the rich dynamic structure required of OM models. This paper formulates a novel dynamic model that overcomes these limitations by combining the principal-agent framework with the physical structure of a Markov decision process. In this model one has a system moving from state to state as time passes, with transition probabilities depending on actions chosen by an agent, and a principal who pays the agent based on state transitions observed. The principal seeks an optimal payment scheme, striving to induce the actions that will maximize her expected discounted profits over a finite planning horizon. Although dynamic principal-agent models similar to the one proposed here are considered intractable, a set of assumptions are introduced that enable a systematic analysis. These assumptions involve the "economic structure" of the model but not its "physical structure." Under these assumptions, the paper establishes that one can use a dynamic-programming recursion to derive an optimal payment scheme. This scheme is memoryless and satisfies a generalization of Bellman's principle of optimality. Important managerial insights are highlighted in the context of a two-state example called "the maintenance problem".
The agency theory approach to understanding salesforce compensation plans is modified to incorporate the intratemporal nature of the salesperson's effort-rate decision, i.e., the decision about the effort-rate at any given point in time potentially depends upon the sales performance up to that point in time in the accounting period. Under the assumptions considered in this paper, Holmstrom and Milgrom (1987) have shown that the optimal compensation plan is linear in total sales over the accounting period. The comparative statics results obtained here corroborate most of the corresponding results in the salesforce compensation literature; moreover, we derive many additional results not available in the literature. It is demonstrated that the commission income as a fraction of total compensation goes up with an increase in the effectiveness of the sales-effort or an increase in base sales. On the other hand, the salary component of the total compensation goes up with increases in uncertainty, absolute risk aversion, marginal cost of production, perceived cost of effort, and/or alternative job opportunities for the salesperson. We provide a discussion of different selling situations where our results may be more or less applicable. An examination of empirical studies already available in the literature reveals support for our findings regarding the relative emphasis of salary and incentive pay in the compensation plan. We also extend the agency theory approach to compare commission rates across products for a multiproduct salesperson. Here it is shown that commission rates are higher for products with higher sales-effort effectiveness, lower levels of uncertainty, and/or lower marginal costs.
Stereotypically, marketing is mainly concerned about satisfying customers and manufacturing is mainly interested in factory efficiency. Using the principal-agent (agency) paradigm, which assumes that the marketing and manufacturing managers of the firm will act in their self-interest, we seek incentive plans that will induce those managers to act so that the owner of the firm can attain as much as possible of the residual returns. One optimal incentive plan can be interpreted as follows: The owner subcontracts to pay the manufacturing manager a fixed rate for all capacity he delivers. Each marketing manager receives all of the returns from his product. In turn, all managers pay a fixed fee to the owner. Under this plan, the marketing managers will often complain about the stock level decisions, even though these levels are announced in advance. Under a revised plan, the owner can eliminate such complaints by delegating the stocking decisions to the respective marketing managers, without any loss. This plan is interpreted as requiring the owner to make a futures market for manufacturing capacity, paying the manufacturing manager the expected marginal value for each unit of capacity delivered, receiving the realized marginal value from the marketing managers, and losing money on average in the process.
In this paper we consider an (S - 1, S) inventory system with Poisson demand and constant resupply times. An arriving customer facing stockout will either backorder or withdraw its order. Expressions describing steady state operating characteristics of the system and the expected total cost rate are developed. Furthermore, the behavior of the expected total cost rate as the function of the relevant parameters is explored.
This paper considers the problem of finding optimal stock levels for an inventory system with Poisson demands, arbitrary replacement time distribution, and emergency handling of shortages at a premium cost. The ordering policy is assumed to be of the (S - 1, S) type, i.e., a replacement item is ordered as soon as a unit of stock is used. A criterion for selecting the optimal stock level is formulated by minimizing the expected cost per unit time in steady state. Based on the fact that the probability of shortage for this model is given by the Erlang loss formula, it is possible to derive an approximate stocking level formula, which appears to be accurate over a wide range of parameter values. Comparisons with other related models and determination of achieved service levels are also discussed.
This paper derives the simple analytic solution to the special but important inventory problem in which the optimal policy is to reorder whenever units are demanded. The demand distribution can be any compound Poisson; the resupply distribution is arbitrary. Both the backorder case and the lost sales case are solved by generalizing a queueing theorem due to Palm.
The steady state probabilities for the number of units in resupply (or repair) completely describe the item's long term behavior, and are simply the normalized values of the compound Poisson demand distribution based on the mean of the resupply distribution but not on the distribution itself. Knowledge of these state probabilities enables us to compute several measures of item supply performance as a function of the spare stock, s. Traditional inventory analysis can then be applied to minimize total cost based on estimates of holding cost and supply performance cost.
The appendices contain a description of the algorithm and the computer program for calculating stuttering Poisson state probabilities and the measures of effectiveness for the backorder case. Numerical illustrations are also provided.
In this paper we determine optimal reduction in the procurement lead time duration for some stochastic inventory models, jointly with the optimal ordering decisions. The models are developed with complete and partial information about the lead time demand distribution. The stochastic models analyzed in this paper are the classical continuous and periodic review models with a mixture of backorders and lost sales and the base stock model. For each of these models, we provide sufficient conditions for the uniqueness of the optimal operating policy. We also develop algorithms for solving these models and provide illustrative numerical examples.
This paper considers inventory models of the order-quantity/order-point type, or (Q,r) models. In general, the control parameters (Q and r) depend on both the demand process and the replenishment lead time. Although many studies have treated lead time as constant, focusing solely on demand, a (Q,r) model with stochastic lead time could be a building block in Supply Chain Management. Variability in lead time between successive stages is often what disturbs supply chain coordination.In a two-stage system with a constant demand rate, we will concentrate on lead time as a random variable, and develop two probabilistic models. In the first, lead time T is exogenous. Lead time is made endogenous in the second stochastic model through an “expediting factor” τ, the constant of proportionality between random variables T̃ (the expedited lead time) and T (ordinary lead time): . For expedited orders (τ<1), shorter-than-average lead time can be obtained at a cost. Similarly, longer mean lead times result in a rebate for the customer when τ>1. The second model thus has three decision variables (Q,r,τ).For each model, we show that the expected cost per unit time is jointly convex in the decision variables and obtain the global minimizer. Numerical examples are given. Sensitivity analyses are conducted with respect to the cost parameters, and suggestions are made for future research
The figures for inventory make up a huge proportion of a company's working capital. Because of this, we formulated the optimal replenishment policy considering the time value of money to represent opportunity cost. In this article, we provide a mixed inventory model, in which the distribution of lead time demand is normal, to consider the time value. First, the study tries to find the optimal reorder point and order quantity at all lengths of lead time with components crashed to their minimum duration. Secondly, we develop a method to insure the uniqueness of the reorder point to locate the optimal solution. Finally, some numerical examples are given to illustrate our findings.
This research is motivated by the realization that Japanese manufacturers have devoted much time and effort to establish a long term partnership with their suppliers in order to reduce lead-time uncertainty. It is very common for a Japanese manufacturer to advance money to finance its suppliers and help them meet the rigid delivery standards imposed. However, there have been very few mathematical analyses of the advantages of investing in such efforts. The main objective of this paper is to provide an analytical model to quantify the tradeoffs associated with lower lead-time uncertainty. In particular, we consider the option of investing in order to reduce the lead-time variance in an inventory model with stochastic lead time. The paper also considers the option of investing in both setup cost and lead-time variability reduction. Numerical results are presented to demonstrate the use of the models. Sensitivity analysis is performed to indicate under what conditions investment is warranted.
We examine the problem of designing and implementing a continuous-review (Q,r) inventory system from an agency perspective, in which the agent's effort influences the item's replenishment leadtime. Our results are as follows: if the agent is risk-neutral, a linear or quadratic contract achieves first-best. For a risk-averse agent with an exponential utility function, assuming a normal leadtime distribution, we determine the optimal linear contract. Extensive numerical experiments suggest that ignoring the possible influence of the agent on the replenishment leadtime can be costly, but that the cost penalty of ignoring agency can be significantly reduced by a simple contract.
We consider the selection of optimal lead time variability in continuous review inventory models. The variability level is optimized jointly with the lot size and reorder point, where lower variability can be achieved at an extra cost. Low variabilities may correspond to more reliable sources or production processes, and their attainment is one of the implicit goals of JIT logistics. We analyze the resulting tri-variate models, providing sufficient conditions for joint convexity, and hence unique optimum, an algorithm and illustrative examples.
In this article, both lead time and the order quantity are considered as the decision variables of a mixture inventory model. Instead of having a stockout term in the objective function, a service level constraint, which implies that the stockout level per cycle is bounded, is added to the model. In our studies, we first assume that the lead time demand follows a normal distribution, and then we relax the assumption about the form of the distribution function of lead time demand and apply the minimax distribution free procedure to solve the problem. We develop an algorithm procedure, respectively, to find the optimal order quantity and optimal lead time. Further, the effects of parameters are also included.
This article tries to find the optimal order quantity and optimal lead time of a mixture inventory model. First, this article establishes a principle to compare the optimal order quantities at all the lengths of lead times with components crashed to their minimum duration. Second, some lemmas are developed to reveal the effects of parameters and to simplify the solution procedure of Ouyang, Yeh and Wu. Finally, some numerical examples are given to illustrate all lemmas in this article.Scope and purposeThis article explores the problem of a continuous review inventory model in which both lead time and the order quantity are treated as decision variables and the shortages are allowed. In a recent paper, Ouyang, Yeh and Wu considered such problems. The purpose of this article is twofold. First, this article examines the effect of the fraction of the demand backordered during the stockout period on the minimum total expected annual costs at all the lengths of lead times with components crashed to their minimum duration. Second, a simple method to locate the optimal solution is developed to simplify the solution procedure.
Most of the literature dealing with inventory problems assumes lead time as prescribed, whether deterministic or probabilistic. In certain cases, lead time can be reduced but at an added cost. In this article we discuss inventory models where lead time is one of the decision variables.
In this thesis we examine several business scenarios in which the owner (principal) of an inventory system, who is risk-neutral, delegates its design and/or implementation to an agent (internal manager, external supplier, or consultant) whose hidden effort influences the duration of the item's replenishment leadtimes. In the (Q, r) agency model, we examine two scenarios: the one-time-effort scenario and the combined-effort scenario. For both scenarios we demonstrate that if the agent is risk-neutral, then a simple contract achieves first-best. In the one-time-effort scenario, if the agent has an exponential utility function and the leadtime is drawn from a normal distribution, then we provide the parameters of the optimal linear contract. In the (S-1, S) agency model with Poisson demand and stochastic replenishment leadtimes, we provide the first-best optimal payment scheme, in which the agent is completely insured against the random timing and the random size of his payoffs. For the second-best case, if the agent has an exponential utility function and the leadtimes are drawn from a normal distribution, then we provide a salary-and-linear-bonus heuristic payment scheme, in which the agent receives a fixed payment in each of his consumption periods and a history-independent and linear incentive payment for each replenishment order. In the periodic-review agency model, we consider a standard, T-period inventory system, where a replenishment order is delivered immediately or one-period later, depending probabilistically on the effort expended by the agent. We provide the form of optimal payment scheme, and suggest that a state-dependent base-stock policy is optimal. We also find that, unlike in the corresponding traditional multi-period newsvendor model, given the same initial inventory, the optimal order-up-to levels may not be monotonic across time periods. For each of the three agency models, we conduct a computational study, and observe that ignoring the possible influence of the agent on replenishment leadtime can be significant; and that the magnitude of agency loss for the principal due to the agent's hidden effort appears to be small except in the periodic-review agency model with an agent who has a high risk-aversion.
The authors develop two themes in the theory of incentive schemes. First, one need not always use all of the information available in an optimal incentive contract. Accounting information, which aggregates performance over time, is sufficient for optimal compensation schemes in certain classes of environments. Second, optimal rules in a rich environment must work well in a range of circumstances and cannot, therefore, be complicated functions of the observed outcome. The authors illustrate these ideas in a particular model where the agent has a rich space of controls, showing that the unique optimal compensation scheme is a linear function of profits. Copyright 1987 by The Econometric Society.
Analysis of Inventory Systems
- Hadley G Whitin
- Tm
Hadley G, Whitin TM. Analysis of Inventory Systems. Englewood Cliffs, NJ: Prentice-Hall; 1963.
S-1,S) policies for perishable inventory. Management Science. v31 i4
- C P Schmidt
- S Nahmias
Optimal inventories for an (S-1,S) system with no backorders. Management Science. v23 i5
- S A Smith