Optimal replenishment policy for product with season pattern demand

This paper investigates a deterministic inventory model in which demand follows a seasonal pattern that repeats itself after a short time interval. An algorithm is developed for determining an optimal replenishment cycle, a shortage length and an order quantity such that the total profit per unit time is maximized.     

Optimal dynamic pricing and inventory control with stock deterioration and partial backordering

This paper studies the optimal dynamic pricing and inventory control policies in a periodic-review inventory system with fixed ordering cost and additive demand. The inventory may deteriorate over time and the unmet demand may be partially backlogged. We identify two sufficient conditions under which (s,S,p) policies are optimal.     

Optimal ordering and transfer policy for an inventory with stock dependent demand

This paper deals with an ordering-transfer inventory model to determine the retailer’s optimal order quantity and the number of transfers per order from the warehouse to the display area. It is assumed that the amount of display space is limited and the demand rate depends on the display stock level. The objective is to maximize the average profit per unit time yielded by the retailer. The proposed models and algorithms are developed to find the optimal strategy by retailer. Numerical examples are presented to illustrate the models developed and the sensitivity analysis is also reported.     

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

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

Optimal policy for inventory systems with capacity commitment and fixed ordering costs

This paper investigates a single-product, periodic-review, non-stationary inventory system with total maximum capacity commitment and fixed ordering costs over a finite planning horizon. We characterize the optimal ordering policy as a state-dependent ($s,S$) policy whose parameters only depend on the sum of the net inventory and the remaining capacity. We show that such policy can degenerate into two simple policies in two special cases respectively. We also derive bounds on parameters of the optimal ordering policy.     

Joint pricing and ordering policy for an advance booking system with partial order cancellations

In this paper, an inventory model for deteriorating items with price-dependent demand is developed. The cycle is divided into two periods, where an advance sales period is followed by a spot sales period. In practice, customers with reservations may cancel their orders before receiving them. During the advance sales period, the rate of reservations which will not be cancelled is dependent on the length of the waiting time for the receiving order. During the spot sales period, all customers receive their orders at the time of the purchase. We prove the existence of the realistic relationship that the advance sales price is smaller than the spot sales price. We also develop some useful properties and provide an iterative procedure for solving the maximization problem. Numerical examples are given to demonstrate the effectiveness of the proposed approach and we conclude the paper with suggestions for possible future research.     

Optimal control policy for a standing order inventory system

In this paper, we consider a standing order inventory system in which an order of fixed size arrives in each period. Since demand is stochastic, such a system must allow for procurement of extra units in the case of an emergency and sell-offs of excess inventory. Assuming the average-cost criterion, Rosenshine and Obee (Operations Research 24 (1976) 1143–1155) first studied such a system and devised a 4-parameter inventory control policy that is not generally optimal. The current paper uses dynamic programming to determine the optimal control policy for a standing order system, which consists of only two operational parameters: the dispose-down-to level and order-up-to level. Either the average-cost or discounted-cost criterion can be assumed in the proposed model. Also, both the backlogged and lost-sales problems are investigated in this paper. By using a convergence theorem, we stop the dynamic programming computation and obtain the two optimal parameters.     

Periodic pricing and replenishment policy for continuously decaying inventory with multivariate demand

This paper deals with the joint decisions on pricing and replenishment schedule for a periodic review inventory system in which a replenishment order may be placed at the beginning of some or all of the periods. We consider a single product which is subject to continuous decay and a demand which is a function of price and time, without backlogging over a finite planning horizon. The proposed scheme may adjust periodically the selling price upward or downward that makes the pricing policy more responsive to structure changes in supply or demand. The problem is formulated as a dynamic programming model and solved by numerical search techniques. An extensive numerical study is conducted to attend qualitative insights into the structures of the proposed policy and its sensitivity with respect to major parameters. The numerical result shows that the solution generated by the periodic policy outperforms that by the fixed pricing policy in maximizing discount profit.     

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.     

Production planning and pricing policy in a make-to-stock system with uncertain demand subject to machine breakdowns

We consider a make-to-stock system served by an unreliable machine that produces one type of product, which is sold to customers at one of two possible prices depending on the inventory level at the time when a customer arrives (i.e., the decision point). The system manager must determine the production level and selling price at each decision point. We first show that the optimal production and pricing policy is a threshold control, which is characterized by three threshold parameters under both the long-run discounted profit and long-run average profit criteria. We then establish the structural relationships among the three threshold parameters that production is off when inventory is above the threshold, and that the optimal selling price should be low when inventory is above the threshold under the scenario where the machine is down or up. Finally we provide some numerical examples to illustrate the analytical results and gain additional insights.     

Dynamic inventory and pricing policy in a capacitated stochastic inventory system with fixed ordering cost

We study a dynamic inventory and pricing optimization problem in a periodic review inventory system with setup cost and finite ordering capacity in each period. We show that the optimal inventory control is characterized by an (s,s^′,p) policy in four regions of the starting inventory level.     

Optimal pricing and production in an inventory model

This paper deals with the problem of simultaneously determining the optimal price policy and production rate over a given planning horizon. For nonlinear demand functions and convex inventory and shortage cost functions the optimal solution paths are derived by using optimal control theory. The treatment of linear nonsmooth cost functions requires the use of a generalized maximum principle. The solution method is a phase portrait analysis providing insight into the optimal pricing and production policies as well as the resulting inventory paths. Moreover, it is shown that in the case of nonsmooth piecewise linear cost functions the equilibrium is approached within finite time although the model is nonlinear in the control variables. Finally it is illustrated that exogenous fluctuations in the demand rate (seasonal demand pattern) amount to cyclical optimal solutions.     

A queuing approach for inventory planning with batch ordering in multi-echelon supply chains

This paper presents stylized models for conducting performance analysis of the manufacturing supply chain network (SCN) in a stochastic setting for batch ordering. We use queueing models to capture the behavior of SCN. The analysis is clubbed with an inventory optimization model, which can be used for designing inventory policies . In the first case, we model one manufacturer with one warehouse, which supplies to various retailers. We determine the optimal inventory level at the warehouse that minimizes total expected cost of carrying inventory, back order cost associated with serving orders in the backlog queue, and ordering cost. In the second model we impose service level constraint in terms of fill rate (probability an order is filled from stock at warehouse), assuming that customers do not balk from the system. We present several numerical examples to illustrate the model and to illustrate its various features. In the third case, we extend the model to a three-echelon inventory model which explicitly considers the logistics process.     

Infinite horizon optimal policy for an inventory system with two types of product sharing common hardware platforms

We consider a periodic review inventory system and present its optimal policy in the infinite horizon setting. The optimal inventory policy that maximizes the infinite horizon expected discounted profit for the model is analytically obtained by relating to the finite horizon setting using results from variational analysis. Results are provided that elucidate the operations of the inventory system in the long run.     

A note on: Optimal ordering policy for stock-dependent demand under progressive payment scheme

In a recent paper, Soni and Shah [Soni, H., Shah, N. H. (2008). Optimal ordering policy for stock-dependent demand under progressive payment scheme. European Journal of Operational Research 184(1), 91–100] developed a model to find the optimal ordering policy for a retailer with stock-dependent demand and a supplier offering a progressive payment scheme to the retailer. This note corrects some errors in the formulation of the model of Soni and Shah. It also extends their work by assuming that the credit interest rate of the retailer may exceed the interest rate charged by the supplier. Numerical examples illustrate the benefits of these modifications.     

Managing stochastic inventory systems with free shipping option

In many industries, customers are offered free shipping whenever an order placed exceeds a minimum quantity specified by suppliers. This allows the suppliers to achieve economies of scale in terms of production and distribution by encouraging customers to place large orders. In this paper, we consider the optimal policy of a retailer who operates a single-product inventory system under periodic review. The ordering cost of the retailer is a linear function of the ordering quantity, and the shipping cost is a fixed constant K whenever the order size is less than a given quantity – the free shipping quantity (FSQ), and it is zero whenever the order size is at least as much as the FSQ. Demands in different time periods are i.i.d. random variables. We provide the optimal inventory control policy and characterize its structural properties for the single-period model. For multi-period inventory systems, we propose and analyze a heuristic policy that has a simple structure, the (s, t, S) policy. Optimal parameters of the proposed heuristic policy are then computed. Through an extensive numerical study, we demonstrate that the heuristic policy is sufficiently accurate and close to optimal.     

Optimal policy for an inventory system with backlogging and all-units discounts: Application to the composite lot size model

This paper examines an inventory model with full backlogging and all-units quantity discounts. The practical scenario of a salesperson offering compensation to a client so as not to lose the sale is considered. The cost of a backorder thus includes both a fixed cost and a further cost which is proportional to the length of time the said backorder exists. A first algorithm is developed to determine the optimal policy while some extensions to this algorithm are obtained that include additional conditions on the model. In particular, the well known composite lot size model, developed by Tersine, is solved, incorporating a new stockout cost and a new all-units discount. Numerical examples are provided to illustrate the application of the algorithms.     

Optimal ordering policy for stock-dependent demand under progressive payment scheme

In this article, a mathematical model is developed to formulate optimal ordering policies for retailer when demand is partially constant and partially dependent on the stock, and the supplier offers progressive credit periods to settle the account. The notion of progressive credit period is as follows:     

An inventory system with unit demand and varying ordering levels

An inventory system with unit demand, varying ordering levels and random lead times is considered in this paper. Ordering level is determined by the number of demands during last lead time. The ordering quantity will be such as to bring back the inventory level to S at the ordering epoch. No backlog is permitted. The time dependent probability distribution of the inventory level is obtained. Correlation between the number of demands during a lead time and the length of the next inventory dry period is obtained and it is illustrated by an example.