A Note on the Heuristic for Replenishment of Trended Inventories Considering Shortages

We discuss the inventory replenishment policy for an item having a deterministic demand pattern with a linear (positive) trend and shortages. A heuristic is developed to determine the decision rule for selecting the times and sizes of replenishments over a finite time-horizon so as to keep the total costs minimum. The use of the heuristic is illustrated with a numerical example.     

A Lot-sizing Model for Just-in-time Manufacturing

We consider the problem of determining lot sizes of multiple items that are manufactured by a single capacitated facility. The manufacturing facility may represent a bottleneck processing activity on the shop floor or a storeroom that provides components to the shop floor. Items flow from the facility to a downstream facility, where they are assembled according to a specified mix. Just-in-time (JIT) manufacturing requires a balanced flow of items, in the proper mix, between successive facilities. Our model determines lot sizes of the various items based on available capacity and four attributes of each item: demand rate, holding cost, set-up time and processing time. Holding costs for each item accrue until the appropriate mix of items is available for shipment downstream. We develop a lot-sizing heuristic that minimizes total holding cost per time unit over all items, subject to capacity availability and the required mix of items.     

Optimal production stopping and restarting times for an EOQ model with deteriorating items

A perishable single item production-inventory system is studied in this paper. The objective is to describe a general model in which the production rate, the product demand rate, and the item deterioration rate are all considered as functions of time, and to discuss the optimal production stopping and restarting times which minimise the total relevant cost per unit time. In the general model, demand shortage is allowed, where some of the demand is lost and the rest is backlogged. Popular models, such as the pure inventory system and the zero shortage system, are shown to be special cases of our model. The conditions for a feasible stationary point to be optimal are given. The simplest cases with constant rates of production, demand and deterioration are discussed and shown as illustrative examples.     

Heuristics with guaranteed performance bounds for a manufacturing system with product recovery

We consider a manufacturing system with product recovery. The system manufactures a new product as well as remanufactures the product from old, returned items. The items remanufactured with the returned products are as good as new and satisfy the same demand as the new item. The demand rate for the new item and the return rate for the old item are deterministic and constant. The relevant costs are the holding costs for the new item and the returned item, and the fixed setup costs for both manufacturing and remanufacturing. The objective is to determine the lot sizes and production schedule for manufacturing and remanufacturing so as to minimize the long-run average cost per unit time. We first develop a lower bound among all classes of policies for the problem. We then show that the optimal integer ratio policy for the problem obtains a solution whose cost is at most 1.5% more than the lower bound.     

The static stochastic knapsack problem with normally distributed item sizes

This paper develops exact and heuristic algorithms for a stochastic knapsack problem where items with random sizes may be assigned to a knapsack. An item’s value is given by the realization of the product of a random unit revenue and the random item size. When the realization of the sum of selected item sizes exceeds the knapsack capacity, a penalty cost is incurred for each unit of overflow, while our model allows for a salvage value for each unit of capacity that remains unused. We seek to maximize the expected net profit resulting from the assignment of items to the knapsack. Although the capacity is fixed in our core model, we show that problems with random capacity, as well as problems in which capacity is a decision variable subject to unit costs, fall within this class of problems as well. We focus on the case where item sizes are independent and normally distributed random variables, and provide an exact solution method for a continuous relaxation of the problem. We show that an optimal solution to this relaxation exists containing no more than two fractionally selected items, and develop a customized branch-and-bound algorithm for obtaining an optimal binary solution. In addition, we present an efficient heuristic solution method based on our algorithm for solving the relaxation and empirically show that it provides high-quality solutions.     

考虑外包的单产品再制造批量决策

在已有动态批量决策问题的基础上,考虑了再制造与外包这两个因素建立了单产品的再制造批量决策模型。对考虑外包的单产品再制造批量决策问题最优解的性质进行了讨论,在此基础上利用动态规划的思想给出了时间复杂性为O(T4)的算法,并用C++实现了本算法,实验表明本算法是有效的。     

Multi-item inventory control with full truckloads: A comparison of aggregate and individual order triggering

In this paper, we consider the stochastic joint replenishment problem in an environment where transportation costs are dominant and full truckloads or full container loads are required. One replenishment policy, taking into account capacity restrictions of the total order volume, is the so-called QS policy, where replenishment orders are placed to raise the individual inventory positions of all items to their order-up-to levels, whenever the aggregate inventory position drops below the reorder level. We first provide a method to compute the policy parameters of a QS policy such that item target service levels can be met, under the assumption that demand can be modeled as a compound renewal process. The approximation formulas are based on renewal theory and are tested in a simulation study which reveals good performance. Second, we compare the QS policy with a simple allocation policy where replenishment orders are triggered by the individual inventory positions of the items. At the moment when an individual inventory position drops below its item reorder level, a replenishment order is triggered and the total vehicle capacity is allocated to all items such that the expected elapsed time before the next replenishment order is maximized. In an extensive simulation study it is illustrated that the QS policy outperforms this allocation policy since it results in lower inventory levels for the same service level. Although both policies lead to similar performance if items are identical, it can differ substantially if the item characteristics vary.     

Inventory rationing in an (<Emphasis Type="Italic">s,Q</Emphasis>) inventory model with lost sales and two demand classes

Whenever demand for a single item can be categorised into classes of different priority, an inventory rationing policy should be considered. In this paper we analyse a continuous review (s, Q) model with lost sales and two demand classes. A so-called critical level policy is applied to ration the inventory among the two demand classes. With this policy, low-priority demand is rejected in anticipation of future high-priority demand whenever the inventory level is at or below a prespecified critical level. For Poisson demand and deterministic lead times, we present an exact formulation of the average inventory cost. A simple optimisation procedure is presented, and in a numerical study we compare the optimal rationing policy with a policy where no distinction between the demand classes is made. The benefit of the rationing policy is investigated for various cases and the results show that significant cost reductions can be obtained.     

The stochastic transportation problem with single sourcing

We propose a branch-and-price algorithm for solving a class of stochastic transportation problems with single-sourcing constraints. Our approach allows for general demand distributions, nonlinear cost structures, and capacity expansion opportunities. The pricing problem is a knapsack problem with variable item sizes and concave costs that is interesting in its own right. We perform an extensive set of computational experiments illustrating the efficacy of our approach. In addition, we study the cost of the single-sourcing constraints.     

The multi-item single period problem with an initial stock of convertible units

This paper deals with the situation of a number of end items, each facing uncertain demand in a single period of interest. Besides being able to purchase units of the end items there is also available a stock of units that can be converted into end items but at unit costs that depend on the specific end item. Efficient solution procedures are presented for two situations: (i) where the end item demand distributions are assumed known (illustrated for the case of normally distributed demand) and (ii) a distribution free approach where only the first two moments of the distributions are assumed known. Computational results for a set of problems are presented.     

Augmenting the lot sizing order quantity when demand is probabilistic

In this paper we consider a single item, discrete time, lot sizing situation where demand is random and its parameters (e.g., mean and standard deviation) can change with time. For the appealing criterion of minimizing expected total relevant costs per unit time until the moment of the next replenishment we develop two heuristic ways of selecting an appropriate augmentation quantity beyond the expected total demand through to the planned (deterministic) time of the next replenishment. The results of a set of numerical experiments show that augmentation is important, particularly when orders occur frequently (i.e., the fixed cost of a replenishment is low relative to the costs of carrying one period of demand in stock) and the coefficient of variability of demand is relatively low, but also under other specified circumstances. The heuristic procedures are also shown to perform very favourably against a hindsight, baseline (s, S) policy, especially for larger levels of non-stationarity.     

Evaluating shortfall distributions in periodic inventory systems with stochastic endogenous demands and lead-times

This paper addresses a multi-period production/inventory problem with two suppliers, where demand sizes and supplier lead time are stochastic and correlated. A discrete time, single item inventory system is considered, where inventory levels are reviewed periodically and managed using a base-stock policy. At the end of each period, a replenishment order is placed, which enters a queue at the buffer stage and is consequently forwarded to the first available supplier. We present a mathematical model of this inventory system and determine optimal safety stock levels for it, in closed form, using matrix analytic techniques and the properties of phase type distributions. To account for the effect of order crossovers, which occur whenever replenishment orders do not arrive in the sequence in which they were placed, the inventory shortfall distribution is analyzed. Finally, a set of numerical experiments with a system with two suppliers is presented, where the proposed model is compared to other existing models.     

An extension of partial backordering EOQ with correlated demand caused by cross-selling considering multiple minor items

Zhang et al. (2011) proposed the partial backordering EOQ with correlated demand caused by cross-selling, where a portion of the sales of a minor item is associated with those of a major item. In this paper, we extend their model to make it more applicable to dealing with the inventory replenishment problem for multiple associated items. We formulate the model as a mixed integer nonlinear programming (MINLP) problem and develop a global optimum search procedure with the fill rate given. We further employ a one-dimensional search on the fill rate to obtain the minimum total inventory cost within a predetermined precision, which enjoys polynomial computational complexity.     

Optimal control of an inventory-production system with state-dependent random yield

In this paper we study a single stage, periodic-review inventory problem for a single item with stochastic demand. The inventory manager determines order sizes according to an order-up-to logic and observes a random yield due to quality problems in the production. We distinguish between two different states of the production process combined with different probabilities to produce a defective unit. In order to improve the production process, periodic inspections are conducted and in case of a failure the machine is repaired. Approximations are developed to evaluate the average cost for a given order-up-to level and a given inspection interval and we illustrate the existence of optimal policy parameters. The approximations are tested in a simulation study and reveal an excellent performance as they lead to near optimal policy parameters. Moreover, we decompose the problem and test different methods to compute the policy parameters either sequentially or separately. Our results show that a joint optimization of the inventory and maintenance policy leads to a better system performance and reduced costs.     

Algorithms for Capacitated,Multi-Item Lot-Sizing without Set-Ups

The multi-item lot-sizing problem considered here is concerned with finding the lot sizes over a horizon of discrete time periods to meet known future demand without incurring backlogs, such that the total cost of production and inventory holding is minimized. The capacity constraints arise because the production of each item consumes capacitated production resources at a given rate. Production is assumed to occur without set-ups. The problem is formulated as a capacitated trans-shipment problem. Use of modern, minimum-cost network flow algorithms, coupled with appropriate starting procedures, allows realistically large problem instances to be solved efficiently; thus obviating the need for specialized algorithms based on restrictive assumptions regarding cost structures.     

The multi-item newsvendor problem with a budget constraint and fixed ordering costs

This paper deals with a multi-item newsvendor problem subject to a budget constraint on the total value of the replenishment quantities. Fixed costs for non-zero replenishments have been explicitly considered. Dynamic programming procedures are presented for two situations: (i) where the end item demand distributions are assumed known (illustrated for the case of normally distributed demand) and (ii) a distribution free approach where only the first two moments of the distributions are assumed known. In addition, simple and efficient heuristic algorithms have been developed. Computational experiments show that the performance of the heuristics are excellent based on a set of test problems.     

Deterministic models of perishable inventory with stock-dependent demand rate and nonlinear holding cost

This paper deals with an extended EOQ-type inventory model for a perishable product where the demand rate is a function of the on-hand inventory. The traditional parameters of unit item cost and ordering cost are kept constant; but the holding cost is treated as (i) a nonlinear function of the length of time for which the item is held in stock, and (ii) a functional form of the amount of the on-hand inventory. The approximate optimal solution in both the cases are derived. Computational results are presented indicating the effects of nonlinearity in holding costs.     

A pricing problem under Monge property

We study a pricing problem where buyers with non-uniform demand purchase one of many items. Each buyer has a known benefit for each item and purchases the item that gives the largest utility, which is defined to be the difference between the benefit and the price of the item. The optimization problem is to decide on the prices that maximize total revenue of the seller. This problem is also called the optimal product line design problem in the absence of competition.

Even though the general problem is known to be NP-hard, it can be solved efficiently under some natural assumptions on customer benefits. In this paper we study properties of optimal solutions and present a dynamic programming algorithm when customer benefits satisfy the Monge property. The same algorithm can also be used to solve the problem under the additional requirement that all buyers should be served.     

A production-repairing inventory model with fuzzy rough coefficients under inflation and time value of money

In this paper, a production-repairing inventory model in fuzzy rough environment is proposed incorporating inflationary effects where a part of the produced defective units are repaired and sold as fresh units. Here, production and repairing rates are assumed as dynamic control variables. Due to complexity of environment, different costs and coefficients are considered as fuzzy rough type and these are reduced to crisp ones using fuzzy rough expectation. Here production cost is production rate dependent, repairing cost is repairing rate dependent and demand of the item is stock-dependent. Goal of the research work is to find decisions for the decision maker (DM) who likes to maximize the total profit from the above system for a finite time horizon. The model is formulated as an optimal control problem and solved using a gradient based non-linear optimization method. Some particular cases of the general model are derived. The results of the models are illustrated with some numerical examples.     

Determining optimal order splitting and reorder level for N-supplier inventory systems

This paper considers multiple-supplier single-item inventory systems, where the item acquisition lead times of suppliers and demand arrival are random, and backorder is allowed. The acquisition takes place when the inventory level depletes to a reorder level, and the order is split among multiple suppliers. The acquisition lead times may have different distributions, the unit purchasing prices from suppliers may be different, and thus the order quantities for different suppliers may be different. The problem is to determine the reorder level and order quantity for each supplier so that the expected total cost per unit time, consisting of the fixed ordering cost, procurement cost, inventory holding cost and shortage cost, is minimized. We develop a mathematical model describing the system in detail. We also conduct extensive numerical experiments to analyze the advantages and distinct characteristics of multiple-supplier systems.