An optimal replenishment policy for deteriorating items with stock-dependent demand and relaxed terminal conditions under limited storage space

This paper deals with the problem of determining the optimal replenishment policy for deteriorating items with stock-dependent demand in which the terminal condition of zero-ending inventory is relaxed. In the model, shortages are allowed and partial backlogging/lost sales. That is, the zero/non-zero ending inventory models are considered simultaneously. The items in stock are displayed to the customers in shelves with limited storage capacity. In theoretical analysis, the necessary and sufficient conditions of the existence and uniqueness of the optimal solutions under various cases are shown. We then also provide a simple algorithm to find the optimal solutions for various situations. Further, a couple of numerical examples are presented to demonstrate the developed model and solution procedure, and several management insights are obtained from the numerical examples. Finally, sensitivity analysis of the optimal solution with respect to major parameters is also carried out.     

Continuous-review,lost-sales inventory models with Poisson demand,a fixed lead time and no fixed order cost

This paper considers continuous-review lost-sales inventory models with no fixed order cost and a Poisson demand process. There is a holding cost per unit per unit time and a lost sales cost per unit. The objective is to minimise the long run total cost. Base stock policies are, in general, sub-optimal under lost sales. The optimal policy would have to take full account of the remaining lead times on all the orders currently outstanding and such a policy would be too complex to analyse, let alone implement. This paper considers policies which make use of the observation that, for lost sales models, base stock policies can be improved by imposing a delay between the placement of successive orders. The performance of these policies is compared with that of the corresponding base stock policy and also with the policy of ordering at fixed and regular intervals of time.     

Make-or-buy service capacity decision in a supply chain providing after-sales service

We consider a supply chain comprising a manufacturer and a retailer. The manufacturer supplies a product to the retailer, while the retailer sells the product bundled with after-sales service to consumers in a fully competitive market. The sales volume is affected by the retailer’s service-level commitment. The retailer can build service capacity in-house at a deterministic price before service demand is realized, or buy the service from an outsourcing market at an uncertain price after service demand realization. We find that the outsourcing market encourages the retailer to make a higher level of service commitment, while prompting the manufacturer to reduce the wholesale price, resulting in more demand realization. We analyze how the expected cost of the service in the outsourcing market and the retailer’s risk attitude affect the decisions of both parties. We derive the conditions under which the retailer is willing to build service capacity in-house and under which it will buy the service from the outsourcing market. Moreover, we find that the manufacturer’s sharing with the retailer the cost to build service capacity improves the profits of both parties.     

Flexible capacity strategy with multiple market periods under demand uncertainty and investment constraint

We establish a flexible capacity strategy model with multiple market periods under demand uncertainty and investment constraints. In the model, a firm makes its capacity decision under a financial budget constraint at the beginning of the planning horizon which embraces n market periods. In each market period, the firm goes through three decision-making stages: the safety production stage, the additional production stage and the optimal sales stage. We formulate the problem and obtain the optimal capacity, the optimal safety production, the optimal additional production and the optimal sales of each market period under different situations. We find that there are two thresholds for the unit capacity cost. When the capacity cost is very low, the optimal capacity is determined by its financial budget; when the capacity cost is very high, the firm keeps its optimal capacity at its safety production level; and when the cost is in between of the two thresholds, the optimal capacity is determined by the capacity cost, the number of market periods and the unit cost of additional production. Further, we explore the endogenous safety production level. We verify the conditions under which the firm has different optimal safety production levels. Finally, we prove that the firm can benefit from the investment only when the designed planning horizon is longer than a threshold. Moreover, we also derive the formulae for the above three thresholds.     

Optimal inventory policy under power demand pattern and partial backlogging

In this paper, we study an inventory model with a power demand pattern that allows shortages. It is assumed that only a fraction of demand is backlogged during the shortage period and the remainder is considered lost sales. The aim of the paper is to determine the lot size and the length of the inventory cycle that maximize the total inventory profit per unit time. A general approach to obtain the optimal solution of the inventory problem and the maximum associated profit is developed. Some inventory models proposed in the literature are particular cases of the model analyzed here. Numerical examples are included to complement the theoretical results.     

Optimal policies for inventory systems with discretionary sales, random yield and lost sales

We determine replenishment and sales decisions jointly for an inventory system with random demand, lost sales and random yield. Demands in consecutive periods are independent random variables and their distributions are known. We incorporate discretionary sales, when inventory may be set aside to satisfy future demand even if some present demand may be lost. Our objective is to minimize the total discounted cost over the problem horizon by choosing an optimal replenishment and discretionary sales policy. We obtain the structure of the optimal replenishment and discretionary sales policy and show that the optimal policy for finite horizon problem converges to that of the infinite horizon problem. Moreover, we compare the optimal policy under random yield with that under certain yield, and show that the optimal order quantity （sales quantity） under random yield is more （less） than that under certain yield.     

An inventory model with partial backordering and unit backorder cost linearly increasing with the waiting time

As the implementation of JIT practice becomes increasingly popular, each echelon in a supply chain tends to carry fewer inventories, and thus the whole supply chain is made more vulnerable to lost sales and/or backorders. The purpose of this paper is to recast the inventory model to be more relevant to current situations, where the penalty cost for a shortage occurrence at a downstream stage in a supply chain is continually transmitted to the upstream stages. The supplier, in this case, at the upstream of the supply chain is responsible for all the downstream shortages due to the chain reaction of its backlog. The current paper proposes a model in which the backorder cost per unit time is a linearly increasing function of shortage time, and it claims that the optimal policy for the supplier is setting the optimal shortage time per inventory cycle to minimize its total relevant cost in a JIT environment.     

An inventory model where backordered demand ratio is exponentially decreasing with the waiting time

We analyze an inventory system with a mixture of backorders and lost sales, where the backordered demand rate is an exponential function of time the customers wait before receiving the item. Stockout costs (backorder cost and lost sales cost) include a fixed cost and a cost proportional to the length of the shortage period. A procedure for determining the optimal policy and the maximum inventory profit is presented. This work extends several inventory models of the existing literature.     

Research note on the criteria for the optimal solution of the inventory model with a mixture of partial backordering and lost sales

Perishable products are commonly seen in inventory management. By allowing shortages and backlogging, the impact on the cost from the decay of the products can be balanced out. In a recent paper published in Computers and Industrial Engineering [P.L. Abad, Optimal lot size for a perishable good under conditions of finite production and partial backordering and lost sale, Comput. Ind. Eng. 38 (2000) 457–465] considered a problem in such context. However, his algorithm was incomplete due to flaws in his solution procedure. The purpose of this note is to explore the same production inventory models with a mixture of partial backordering and lost sales for deteriorated items. We find the criteria for the optimal solution for different cases and derive a formulated minimum value. By theoretical analysis, we develop a few lemmas to reveal parameter effects and optimal solution procedure. The solutions are illustrated by solving the same examples from Abad’s paper to illustrate the accuracy and completeness of our procedure.     

Multiproduct single-machine production system with stochastic scrapped production rate, partial backordering and service level constraint

In this paper, a multiproduct single-machine production system under economic production quantity (EPQ) model is studied in which the existence of only one machine causes a limited production capacity for the common cycle length of all products, the production defective rates are random variables, shortages are allowed and take a combination of backorder and lost sale, and there is a service rate constraint for the company. The aim of this research is to determine the optimal production quantity, the allowable shortage level, and the period length of each product such that the expected total cost, including holding, shortage, production, setup and defective items costs, is minimized. The mathematical model of the problem is derived for which the objective function is proved to be convex. Then, a derivative approach is utilized to obtain the optimal solution. Finally, two numerical examples in each of which a sensitivity analysis is performed on the model parameters, are provided to illustrate the practical usage of the proposed methodology.     

Advanced supply chain planning with mixtures of backorders,lost sales,and lost contract

This paper investigates a supply chain system in which a supplier prepares for the selling season by building stock levels prior to the beginning of the season and shortages realized at the beginning of the season are represented as mixtures of backorders and lost sales. Backlogged items are replenished as soon as possible through an emergency procurement as opposed to waiting until the next scheduled delivery as in many continuous review scenarios, and the backorder rate is modeled as a piecewise linear function of the magnitude of the shortage. The often intangible cost associated with lost sales and customer goodwill is also quantified. In particular, the buyer and supplier are engaged in a contractual agreement and the loss of customer goodwill from the supplier’s perspective is represented as the expected cost associated with violating the conditions of the contract. The likelihood of contract cancellation is also represented as a function of the magnitude of shortage. The optimal solution is derived in closed form for the case of exponential demand distribution, and an example problem is illustrated with numerical data in order to demonstrate calculation of the optimal solution and corresponding sensitivity analysis for demand distributions in which the solution cannot be expressed in closed form.     

The general optimal market area model

Market area models determine the optimal size of market for a facility. These models are grounded in classical location theory, and express the fundamental tradeoff between economies-of-scale from larger facilities and the higher costs of transport to more distant markets. The simpler market area models have been discovered and rediscovered, and applied and reapplied, in a number of different settings. We review the development and use of market area models, and formulate a General Optimal Market Area model that accommodates both economies-of-scale in facilities costs and economies-of-distance in transport costs as well as different market shapes and distance norms. Simple expressions are derived for both optimal market size and optimal average cost, and also the sensitivity of average cost to a non-optimal choice of size. The market area model is used to explore the implications of some recently proposed distance measures and to approximate a large discrete location model, and an extension to price-sensitive demands is provided.The author thanks S. Jacobsen, M. O'Kelly, C. Tang, J.-F. Thisse, and S.C. Wirasinghe for their helpful comments.     

Optimal Contract Design of Supplier-Led Outsourcing Based on Pontryagin Maximum Principle

This paper provides a supplier-led outsourcing model to maximize the supplier’s profits based on a principal-agent framework with both asymmetric cost information and uncertain market demand information described by continuous random variables. The salvage value of the unsold product is processing-cost dependent. By converting the proposed model, which is a dynamic optimization problem, to an optimal control problem, we obtain the analytical form of the optimal supplier outsourcing contract composed of the wholesale price and the transfer payment by applying Pontryagin’s maximum principle. It is shown that the optimal contract is directly related to the supplier’s beliefs about the manufacturer’s unit cost and the salvage value function. The Pontryagin’s maximum principle-based solution method serves as a powerful tool to support the decision making for the best sourcing strategy, and it provides analytical insights for outsourcing management. Finally, numerical examples are presented to illustrate the validness of the theoretical results.     

An economic order quantity model with partial backlogging under general backorder cost function

In this work, we study a single-item inventory model where shortages are allowed. A known constant fraction of the demand during the stockout period is backlogged, and the rest are lost sales. Usually, in the literature on inventory control, the unit backorder cost is considered to be a linear function of the waiting time until the customer gets the item. However, in some real-world situations, the unit cost of a backorder may not be linear. To model this situation, we develop a new approach by considering that the backlogging unit cost is a nondecreasing, continuous, and positive function of the amount of time the customers wait before receiving the item. Our objective is to maximize the average profit per unit time. An effective solution procedure to determine the optimal policy and the maximum average profit is developed. Numerical examples, which help us to understand the theoretical results, are also presented.     

Optimal inventory replenishment policy for a queueing system with finite waiting room capacity

We treat an inventory control problem in a facility that provides a single type of service for customers. Items used in service are supplied by an outside supplier. To incorporate lost sales due to service delay into the inventory control, we model a queueing system with finite waiting room and non-instantaneous replenishment process and examine the impact of finite buffer on replenishment policies. Employing a Markov decision process theory, we characterize the optimal replenishment policy as a monotonic threshold function of reorder point under the discounted cost criterion. We present a simple procedure that jointly finds optimal buffer size and order quantity.     

Note on inventory model with a mixture of back orders and lost sales

Competitiveness is an important means of determining whether a company will prosper. Business organizations compete with one another in a variety of ways. Among these competitive methods are time and cost factors. The purpose of this paper is to examine the inventory models presented by Padmanabhan and Vrat [International Journal of Systems Sciences 21 (1990) 1721] with a mixture of back orders and lost sales. We develop the criterion for the optimal solution for the total cost function. If the criterion is not satisfied, this model will degenerate into one cycle inventory model with a finite inventory period. This implies an extension of shortage period as long as possible to produce lower cost. However, we know that time is another important factor in company competitiveness. Customers will not indefinitely wait for back orders. A tradeoff will be made between the two most important factors; time and cost. The minimum total cost is evaluated under the diversity cycle time and illustrations are applied to explain the calculation process. This work provides a reference for decision-makers.     

On a production-inventory system of deteriorating items subject to random machine breakdowns with a fixed repair time

This paper considers the impact of random machine breakdowns on the classical Economic Production Quantity (EPQ) model for a product subject to exponential decay and under a no-resumption (NR) inventory control policy. A product is manufactured in batches on a machine that is subject to random breakdowns in order to meet a constant demand over an infinite planning horizon. The product is assumed to have a significant rate of deterioration and time to deterioration is described by an exponential distribution. Also, the time-to-breakdown is a random variable following an exponential distribution. Under the NR policy, when a breakdown occurs during a production run, the run is immediately aborted. A new run will not be started until all available inventories are depleted. Corrective maintenance of the production system is carried out immediately after a breakdown and it takes a fixed period of time to complete such an activity. The objective is to determine the optimal production uptime that minimizes the expected total cost per unit time consisting of setup, corrective maintenance, inventory carrying, deterioration, and lost sales costs. A near optimal production uptime is derived under conditions of continuous review, deterministic demand, and no shortages.     

The optimality of myopic stocking policies for systems with decreasing purchasing prices

This paper studies the stocking/replenishment decisions for inventory systems where the purchasing price of an item decreases overtime. In a periodic review setting with stochastic demands, we model the purchasing prices of successive periods as a stochastic and decreasing sequence. To minimize the expected total discounted costs (purchasing, inventory holding and shortage penalty) for systems with backlogging and lost sales, we derive conditions, regarding the cost parameters, under which myopic stocking policies are optimal.     

Flexible long-term capacity planning in closed-loop supply chains with remanufacturing

We deal with long-term demand-driven capacity planning policies in the reverse channel of closed-loop supply chains (CLSCs) with remanufacturing, under high capacity acquisition cost coupled with uncertainty in actual demand, sales patterns, quality and timing of end-of-use product returns. The objective is to facilitate the decision-making when the management faces the dilemma of implementing either a strategy of early large-scale investments to benefit from economies of scale and capacity readiness, or a flexible strategy of low volume but more frequent capacity expansions. We consider a CLSC with two sequential product-types. We study the system’s response in terms of transient flows, actual/desired capacity level, capacity expansions/contractions and total supply chain profit, employing a simulation-based system dynamics optimization approach. Extensive numerical investigation covers a broad range of real-world remanufacturable products under alternative scenarios in relation to the market preference over product-types. The key findings propose flexible policies as improved alternatives to large-scale capacity expansions/contractions in terms of adaptability to the actual pattern of end-of-use product returns and involved risk in the investments’ turnover. Flexible policies are also proposed as practices to avoid overcapacity phenomena in collection and remanufacturing capacity and as robust policies to product demand. Their implementation is revealed to be even more important for the case of remanufacturing, when a high capacity acquisition unit-cost ratio (remanufacturing/collection) is coupled with strong economies of scale. Finally, results under different information sharing structures show changes in remanufacturing policies, thus justifying the importance of coordination between the decision-maker and the distributor.     

Theory of exchange

In the context of production activity, several parameters play an important role in the total cost estimation and its optimization. These parameters include facility setup cost, inventory carrying cost, and manufacturing cost for the basic model. Shortages can be incorporated in certain environment and costs associated with shortages need to be included in the analysis. It is expected that the industries will run their manufacturing facility at an optimum level. In the multi-product manufacture, optimum common cycle time approach is usually adopted and all the items are produced in each cycle. A situation may occur in the real world, in which a particular parameter concerning an item is exchanged with that of another item. It is of interest to examine the aftereffects. Otherwise also, for the purpose of internal benchmarking, a deliberate exchange of parameters can take place. This can be implemented in case of cost improvement. A generalized approach is presented and discussion is made with reference to various parameters.