Optimal ordering policies for continuous review perishable inventory models

This paper extends the notions of perishable inventory models to the realm of continuous review inventory systems. The traditional perishable inventory costs of ordering, holding, shortage or penalty, disposal and revenue are incorporated into the continuous review framework. The type of policy that is optimal with respect to long run average expected cost is presented for both the backlogging and lost-sales models. In addition, for the lost-sales model the cost function is presented and analyzed.     

A dynamic programming algorithm for dynamic lot size models with piecewise linear costs

We propose a new algorithm for dynamic lot size models (LSM) in which production and inventory cost functions are only assumed to be piecewise linear. In particular, there are no assumptions of convexity, concavity or monotonicity. Arbitrary capacities on both production and inventory may occur, and backlogging is allowed. Thus the algorithm addresses most variants of the LSM appearing in the literature. Computational experience shows it to be very effective on NP-hard versions of the problem. For example, 48 period capacitated problems with production costs defined by eight linear segments are solvable in less than 2.5 minutes of Vax 8600 cpu time.     

A single-item economic lot-sizing problem with a non-uniform resource: Approximation

We study a generalization of the classical single-item capacitated economic lot-sizing problem to the case of a non-uniform resource usage for production. The general problem and several special cases are shown to be non-approximable with any polynomially computable relative error in polynomial time. An optimal dynamic programming algorithm and its approximate modification are presented for the general problem. Fully polynomial time approximation schemes are developed for two NP-hard special cases: (1) cost functions of total production are separable and holding and backlogging cost functions are linear with polynomially related slopes, and (2) all holding costs are equal to zero.     

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.     

需求率依赖于库存水平的离散性库存费的库存模型

传统的库存控制模型都视需求率为固定不变的,放松了这个假定,通过考虑库存费为存储时间的阶梯函数的情形:(1)全单位库存费用,(2)增量库存费用,并且在需求率依赖于库存水平,当库存水平下降到一定程度时,需求率变为常数的形式下,把变化的订购费引入,发展了两个离散性库存费的变质物品的库存控制模型。在模型中允许周期末库存水平不为零,并且提出了最优解的算法。     

On “An EOQ model for perishable items under stock-dependent selling rate and time-dependent partial backlogging” by Dye and Ouyang

In 2005, Dye and Ouyang proposed an EOQ model for perishable items under stock-dependent selling rate and time-dependent partial backlogging, and then established the unique optimal solution to the problem when building up inventory is not profitable. However, they did not provide the optimal solution to the problem when building up inventory is profitable. In this note, we establish an appropriate model in which building up inventory is profitable, and then provide an algorithm to find the optimal solution to the problem. A numerical example is used to illustrate the proposed model.     

An integrated inventory–transportation system with modified periodic policy for multiple products

Efficient management of a distribution system requires an integrated approach towards various logistical functions. In particular, the fundamental areas of inventory control and transportation planning need to be closely coordinated. Our model deals with an inbound material-collection problem. An integrated inventory–transportation system is developed with a modified periodic-review inventory policy and a travelling-salesman component. This is a multi-item joint replenishment problem, in a stochastic setting, with simultaneous decisions made on inventory and transportation policies. We propose a heuristic decomposition method to solve the problem, minimizing the long-run total average costs (major- and minor-ordering, holding, backlogging, stopover and travel). The decomposition algorithm works by using separate calculations for inventory and routing decisions, and then coordinating them appropriately. A lower bound is constructed and computational experience is reported.     

A column generation-based heuristic algorithm for an inventory routing problem with perishable goods

An inventory routing problem is a variation of the vehicle routing problem in which inventory and routing decisions are determined simultaneously over a given time horizon. The objective is to minimize the sum of transportation and inventory costs. In this paper, we study a specific inventory routing problem in which goods are perishable (PIRP). We develop a mathematical model for PIRP and exploit its structure to develop a column generation-based solution approach. Cutting planes are added to improve the formulation. We present computational experiments to demonstrate that our methodology is effective, and that the integration of routing and inventory can yield significant cost savings.     

Optimal selling price and lotsize with time varying deterioration and partial backlogging

The article deals with an EOQ (economic order quantity) model over an infinite time horizon for perishable items where demand is price dependent and partial backorder is permitted. The rate of deterioration is taken to be time proportional and it is assumed that shortage occurs at starting of the inventory cycle. Based on the partial backlogging and lost sale cases, the author develops the criterion for the optimal solution for the replenishment schedule, and proves the optimal ordering policy is unique. Moreover, the article suggests to new functions regarding price-dependent demand and time varying deterioration rate. Finally, numerical examples are illustrated to test the model in various issues.     

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.     

Solving the capacitated lot-sizing problem with backorder consideration

In this research, we formulate and solve a type of the capacitated lot-sizing problem. We present a general model for the lot-sizing problem with backorder options, that can take into consideration various types of production capacities such as regular time, overtime and subcontracting. The objective is to determine lot sizes that will minimize the sum of setup costs, holding cost, backorder cost, regular time production costs, and overtime production costs, subject to resource constraints. Most existing formulations for the problem consider the special case of the problem where a single source of production capacity is considered. However, allowing for the use of alternate capacities such as overtime is quite common in many manufacturing settings. Hence, we provide a formulation that includes consideration of multiple sources of production capacity. We develop a heuristic based on the special structure of fixed charge transportation problem. The performance of our algorithm is evaluated by comparing the heuristic solution value to lower bound value. Extensive computational results are presented.     

Inventory models for perishable items with inventory level dependent demand rate

This paper presents inventory models for perishable items with inventory level dependent demand rate. The models with and without backlogging are studied. In the backlogging model, it is assumed that the backlogging rate is dependent on the waiting time and the amount of products already backlogged simultaneously. Two cases that holding inventory is profitable or not are studied, respectively. The smallest shelf space to ensure shortage not occur when holding inventory is not profitable is obtained. In the model without backlogging, it is assumed that the remaining stock at the end of the inventory cycle is disposed of with salvage value. The necessary and sufficient conditions for the existence and uniqueness of the optimal solution of these models are investigated. At last, some numerical examples are presented to illustrate the effectiveness of the proposed model. The model in this paper is generalization of present ones. In particularly, the model is reduced to Padmanabhan and Vrat’s when δ₁ = 0, and Dye and Ouyang’s when δ₂ = 0. If S = s and δ₂ = 0, it is Chang, Goyal and Teng’s model.     

Optimal production inventory policy for defective items with fuzzy time period

In this paper, an optimal production inventory model with fuzzy time period and fuzzy inventory costs for defective items is formulated and solved under fuzzy space constraint. Here, the rate of production is assumed to be a function of time and considered as a control variable. Also the demand is linearly stock dependent. The defective rate is taken as random, the inventory holding cost and production cost are imprecise. The fuzzy parameters are converted to crisp ones using credibility measure theory. The different items have the different imprecise time periods and the minimization of cost for each item leads to a multi-objective optimization problem. The model is under the single management house and desired inventory level and product cost for each item are prescribed. The multi-objective problem is reduced to a single objective problem using Global Criteria Method (GCM) and solved with the help of Fuzzy Riemann Integral (FRI) method, Kuhn–Tucker condition and Generalised Reduced Gradient (GRG) technique. In optimum results including production functions and corresponding optimum costs for the different models are obtained and then are presented in tabular forms.     

Polynomial cases of the economic lot sizing problem with cost discounts

In this paper we study the economic lot sizing problem with cost discounts. In the economic lot sizing problem a facility faces known demands over a discrete finite horizon. At each period, the ordering cost function and the holding cost function are given and they can be different from period to period. There are no constraints on the quantity ordered in each period and backlogging is not allowed. The objective is to decide when and how much to order so as to minimize the total ordering and holding costs over the finite horizon without any shortages. We study two different cost discount functions. The modified all-unit discount cost function alternates increasing and flat sections, starting with a flat section that indicates a minimum charge for small quantities. While in general the economic lot sizing problem with modified all-unit discount cost function is known to be NP-hard, we assume that the cost functions do not vary from period to period and identify a polynomial case. Then we study the incremental discount cost function which is an increasing piecewise linear function with no flat sections. The efficiency of the solution algorithms follows from properties of the optimal solution. We computationally test the polynomial algorithms against the use of CPLEX.     

Capacitated replenishment and disposal planning for multiple products with resalable returns

We consider a replenishment and disposal planning problem (RDPP) that arises in settings where customer returns are in as-good-as-new condition. These returns can be placed into inventory to satisfy future demand or can be disposed of, in case they lead to excess inventory. Our focus is on a multi-product setting with dynamic demands and returns over a finite planning horizon with explicit replenishment and disposal capacities. The problem is to determine the timing of replenishment and disposal setups, along with the associated quantities for the products, so as to minimize the total costs of replenishment, disposal, and inventory holding throughout the planning horizon. We examine two variants of the RDPP of interest both of which are specifically motivated by a spare part kitting application. In one variant, the replenishment capacity is shared among multiple products while the disposal capacity is product specific. In the other variant, both the replenishment and disposal capacities are shared among the products. We propose a Lagrangian Relaxation approach that relies on the relaxation of the capacity constraints and develop a smoothing heuristic that uses the solution of the Lagrangian problem to obtain near-optimal solutions. Our computational results demonstrate that the proposed approach is very effective in obtaining high-quality solutions with a reasonable computational effort.     

A hybrid tabu search based heuristic for the periodic distribution inventory problem with perishable goods

Most of the research on integrated inventory and routing problems ignores the case when products are perishable. However, considering the integrated problem with perishable goods is crucial since any discrepancy between the routing and inventory cost can double down the risk of higher obsolescence costs due to the limited shelf-life of the products. In this paper, we consider a distribution problem involving a depot, a set of customers and a homogeneous fleet of capacitated vehicles. Perishable goods are transported from the depot to customers in such a way that out-of-stock situations never occur. The objective is to simultaneously determine the inventory and routing decisions over a given time horizon such that total transportation cost is minimized. We present a new “arc-based formulation” for the problem which is deemed more suitable for our new tabu search based approach for solving the problem. We perform a thorough sensitivity analysis for each of the tabu search parameters individually and use the obtained gaps to fine-tune the parameter values that are used in solving larger sized instances of the problem. We solve different sizes of randomly generated instances and compare the results obtained using the tabu search algorithm to those obtained by solving the problem using CPLEX and a recently published column generation algorithm. Our computational experiments demonstrate that the tabu search algorithm is capable of obtaining a near-optimal solution in less computational time than the time required to solve the problem to optimality using CPLEX, and outperforms the column generation algorithm for solving the “path flow formulation” of the problem in terms of solution quality in almost all of the considered instances.     

Dynamic lot-sizing problem with demand time windows and container-based transportation cost

In this paper, we study the dynamic lot-sizing problem with demand time windows and container-based transportation cost. For each particular demand, there are corresponding earliest and latest times, and the duration between such earliest and latest times is the demand time window. If a demand is satisfied by a delivery within demand time window, then there is no holding or backlogging cost incurred. Our purpose is to satisfy demand at a minimum total cost, including setup cost, procurement cost, container cost, and inventory holding cost. This research is supported in part by Hong Kong RGC grant HKUST 6010/02E and NUS ARF grant R-266-000-019-112.     

Production-inventory systems in stochastic environment and stochastic lead times

We consider a production-inventory system where the production and demand rates are modulated by a finite state Continuous Time Markov Chain (CTMC). When the inventory position (inventory on hand – backorders+inventory on order) falls to a reorder point r, we place an order of size q from an external supplier. We consider the case of stochastic leadtimes, where the leadtimes are i.i.d. exponential(μ) random variables, and orders may or may not be allowed to cross. We derive the distribution of the inventory level, and analyze the long run holding, backlogging, and ordering cost rate per unit time. We use simulation to study the sensitivity of the system to the distribution of the lead times.     

An EOQ model for deteriorating items in a declining market with SFI policy

An order-level inventory model fora perishable product with a timedependent demand is developed fora fixed planning period, allowing backlogging in all cycles within the said period. The market demand is assumed todecrease exponentially as time elapses. The average system cost is derived and its optimization procedure is illustrated with a numerical example. Sensitivity of the optimal solution to changes in the values of different parameters of the system is also analysed.     

The two-group heuristic to solve the multi-product,economic lot sizing and scheduling problem in flow shops

This paper presents a new and efficient heuristic to solve the multi-product, economic lot sizing and scheduling problem in flow shops. The problem addressed is that of making sequencing, lot sizing and scheduling decisions for a number of products so as to minimize the sum of setup costs, work-in-process inventory holding costs and final-products inventory holding costs while a given demand is fulfilled without backlogging. The proposed heuristic, called the two-group method (TG), assumes that the cycle time of each product is an integer multiple of a basic period and restricts these multiples to take either the value 1 or K where K is a positive integer. The products to be produced once each K basic period are then partitioned into K sub-groups and each sub-group is assigned to one and only one of the K basic periods of the global cycle. This method first determines a value for K and a feasible partition. Then, a production sequence is determined for each sub-group of products and a non-linear program is solved to determine lot sizes and a feasible schedule. We also show how to adapt our method to the case of batch streaming (transportation of sub-batches from one machine to the next). To evaluate its performance, the TG method was compared to both the common cycle method and a reinforced version of El-Najdawi’s job-splitting heuristic. Numerical results show that the TG method outperforms both of these methods.