A note on the inventory model for deteriorating items with trapezoidal type demand rate

In this paper, we study the inventory model for deteriorating items with trapezoidal type demand rate, that is, the demand rate is a piecewise linearly function. We proposed an inventory replenishment policy for this type of inventory model. The numerical solution of the model is obtained and also examined.     

A Study of the infinite horizon ‘solution’ of inventory lot size models with a linear demand function

The article studies the infinite time horizon inventory lot size model with a linear demand function. The pertinent costs are the carrying and order costs, shortages are prohibited, and replenishments are instantaneous. The problem is to minimize the total cost by exhibiting a schedule of optimal replenishment times.The computerized asymptotic solution suggested here is a generalization of the classical square root law and the more recent cubic root law. Due to its broader usefulness, it should preponderate previously known results.     

Two-warehouse inventory model with ramp-type demand and partially backlogged shortages

In this article, we consider an inventory model for items that are stored in two-warehouses when demand is a general ramp-type function of time. Shortages are allowed and a constant fraction of shortages is backlogged. The existence and uniqueness of optimal solution is proved for both – the single-warehouse and the two-warehouse models. An algorithm is developed to facilitate the choice between the two-warehouse and the single-warehouse systems and hence to obtain the optimal replenishment policy. Numerical examples are presented. Sensitivity analysis with respect to the parameters of the model is performed.     

Optimal replenishment policies with allowable shortages for a product life cycle

In this paper, we investigate replenishment policies with allowable shortages by considering a general, time-varying, continuous, and deterministic demand function for a product life cycle. The objective is to optimally determine the number of inventory replenishments, the inventory replenishment time points, and the beginning time points of shortages within the product life cycle by minimizing the total relevant costs of the inventory replenishment system. The proposed problem is mathematically formulated as a mixed-integer nonlinear programming model. A complete search procedure is developed to find the optimal solution by employing the properties derived in this paper and the well-known Nelder–Mead algorithm. Also, several numerical examples and the corresponding sensitivity analyses are carried out to illustrate the features of our model by utilizing the search procedure developed in this paper.     

A general and optimal approach for three inventory models with a linear trend in demand

This study provides a general and simple algorithm to obtain an optimal solution for three inventory models with a replenishment batching policy, production batching policy, and an integrated replenishment/production batching policy in a manufacturing system, under a finite time horizon and linear trend in demand. This study determines the replenishment or production schedule with one general equation for these three problems and provides fully theoretical proofs for relaxing some of the conjectures in previous studies. A general and explicit procedure to derive the optimal solution for these three inventory models is presented, while considering both linearly increasing and decreasing demands. In addition, demonstrations of applicability are performed.     

Inventory systems for deteriorating items with shortages and a linear trend in demand-taking account of time value

This paper derives an inventory model for deteriorating items with the demand of linear trend and shortages during the finite planning horizon considering the time value of money. A simple solution algorithm using a line search is presented to determine the optimal interval which has positive inventories. Numerical examples are given to explain the solution algorithm. Sensitivity analysis is performed to study the effect of changes in the system parameters.Scope and purpose The traditional inventory model considers the ideal case in which depletion of inventory is caused by a constant demand rate. However, in real-life situations there is inventory loss due to deterioration. In a realistic product life cycle, demand is increasing with time and eventually reaching zero. Most of the classical inventory models did not take into account the effects of inflation and time value of money. But in the past, the economic situation of most of the countries has changed to such an extent due to large scale inflation and consequent sharp decline in the purchasing power of money. So, it has not been possible to ignore the effects of inflation and time value of money any further. The purpose of this article is to present a solution procedure for the inventory problem of deteriorating items with shortages and a linear trend in demand taking account of time value.     

Optimal replenishment policies for the case of a demand function with product-life-cycle shape in a finite planning horizon

In this paper, we explore the demand function follows the product-life-cycle shape for the decision maker to determine the optimal number of inventory replenishments and the corresponding optimal inventory replenishment time points in the finite planning horizon. The objective function of the total relevant costs considered in our model is mathematically formulated as a mixed-integer nonlinear programming problem. A complete search procedure is provided to find the optimal solution by employing the properties derived in this paper and the Nelder–Mead algorithm. Also, based on the search procedure developed in this paper, a decision support system is implemented on a personal computer to solve the proposed problem.     

Situation reactive approach to Vendor Managed Inventory problem

In this research, we deal with VMI (Vendor Managed Inventory) problem where one supplier is responsible for managing a retailer’s inventory under unstable customer demand situation. To cope with the nonstationary demand situation, we develop a retrospective action-reward learning model, a kind of reinforcement learning techniques, which is faster in learning than conventional action-reward learning and more suitable to apply to the control domain where rewards for actions vary over time. The learning model enables the inventory control to become situation reactive in the sense that replenishment quantity for the retailer is automatically adjusted at each period by adapting to the change in customer demand. The replenishment quantity is a function of compensation factor that has an effect of increasing or decreasing the replenishment amount. At each replenishment period, a cost-minimizing compensation factor value is chosen in the candidate set. A simulation based experiment gave us encouraging results for the new approach.     

基于混合果蝇优化算法的选址-库存联合优化策略

针对装备维修保障仓库系统运营费用高、仓库点位布局不合理、备件库存结构不合理等问题,建立以多品种联合补货问题为基础的装备维修备件仓库选址-库存控制决策联合优化模型,模型可用于求解仓库的开设位置、维修活动需求点的指派情况、仓库补货时间以及库存水平等.根据模型的结构特点,利用多种群协同进化的方法改进传统果蝇优化算法的位置更新方式,设计一种内外两层搜索策略的混合果蝇优化算法,外层搜索策略作为算法的主程序用于搜索仓库选址决策变量,内层搜索策略采用改进的RAND算法用于搜索库存控制决策变量.仿真结果表明,混合果蝇优化算法具有良好的求解效率,能够确保库存系统在一定服务水平的基础上有效降低库存运营总成本.     

Coordinating replenishment and pricing policies for non-instantaneous deteriorating items with price-sensitive demand

This article will formulate and solve an inventory system with non-instantaneous deteriorating items and price-sensitive demand. The purpose of this study is to determine the optimal selling price and the length of replenishment cycle such that the total profit per unit time has a maximum value for the retailer. We first establish a proper model for a mathematical formulation. Then we develop several theoretical results and provide the decision-maker with an algorithm to find the optimal solution. Finally, two numerical examples are provided to illustrate the solution procedure, and a sensitivity analysis of the optimal solution with respect to major parameters is carried out.     

An integrated local search method for inventory and routing decisions

The present article studies an inventory routing model which integrates two important components of the supply chain: transportation logistics and inventory control. The distribution system examined consists of customers that face product demand at a deterministic and constant rate. Customer demand is satisfied by a fixed vehicle fleet located at the central depot. The aim of the problem is to determine the timing and size of the replenishment services together with the vehicle routes, so that the total transportation and inventory holding cost of the system is minimized. In methodological terms, we propose a solution approach applying two innovative local search operators for jointly dealing with the inventory and routing aspects of the examined problem, and Tabu Search for further reducing the transportation costs. The proposed algorithmic framework was tested on a set of new benchmark instances of various scales. It produced satisfactory results both in terms of effectiveness and robustness.     

The EPQ model under conditions of two levels of trade credit and limited storage capacity in supply chain management

An inventory problem involves a lot of factors influencing inventory decisions. To understand it, the traditional economic production quantity (EPQ) model plays rather important role for inventory analysis. Although the traditional EPQ models are still widely used in industry, practitioners frequently question validities of assumptions of these models such that their use encounters challenges and difficulties. So, this article tries to present a new inventory model by considering two levels of trade credit, finite replenishment rate and limited storage capacity together to relax the basic assumptions of the traditional EPQ model to improve the environment of the use of it. Keeping in mind cost-minimisation strategy, four easy-to-use theorems are developed to characterise the optimal solution. Finally, the sensitivity analyses are executed to investigate the effects of the various parameters on ordering policies and the annual total relevant costs of the inventory system.     

A stochastic production planning problem with nonlinear cost

Most production planning models are deterministic and often assume a linear relation between production volume and production cost. In this paper, we investigate a production planning problem in a steel production process considering the energy consumption cost which is a nonlinear function of the production quantity. Due to the uncertain environment, the production demands are stochastic. Taking a scenario-based approach to express the stochastic demands according to the knowledge of planners on the demand distributions, we formulate the stochastic production planning problem as a mixed integer nonlinear programming (MINLP) model.Approximated with the piecewise linear functions, the MINLP model is transformed into a mixed integer linear programming model. The approximation error can be improved by adjusting the linearization ranges repeatedly. Based on the piecewise linearization, a stepwise Lagrangian relaxation (SLR) heuristic for the problem is proposed where variable splitting is introduced during Lagrangian relaxation (LR). After decomposition, one subproblem is solved by linear programming and the other is solved by an effective polynomial time algorithm. The SLR heuristic is tested on a large set of problem instances and the results show that the algorithm generates solutions very close to optimums in an acceptable time. The impact of demand uncertainty on the solution is studied by a computational discussion on scenario generation.     

Solving lot-sizing problem with quantity discount and transportation cost

Owing to today's increasingly competitive market and ever-changing manufacturing environment, the inventory problem is becoming more complicated to solve. The incorporation of heuristics methods has become a new trend to tackle the complex problem in the past decade. This article considers a lot-sizing problem, and the objective is to minimise total costs, where the costs include ordering, holding, purchase and transportation costs, under the requirement that no inventory shortage is allowed in the system. We first formulate the lot-sizing problem as a mixed integer programming (MIP) model. Next, an efficient genetic algorithm (GA) model is constructed for solving large-scale lot-sizing problems. An illustrative example with two cases in a touch panel manufacturer is used to illustrate the practicality of these models, and a sensitivity analysis is applied to understand the impact of the changes in parameters to the outcomes. The results demonstrate that both the MIP model and the GA model are effective and relatively accurate tools for determining the replenishment for touch panel manufacturing for multi-periods with quantity discount and batch transportation. The contributions of this article are to construct an MIP model to obtain an optimal solution when the problem is not too complicated itself and to present a GA model to find a near-optimal solution efficiently when the problem is complicated.     

The replenishment policy of agri-products with stochastic demand in integrated agricultural supply chains

Market demand of agri-products is influenced by uncertain factors, such as weather, temperature, and customer preferences. In integrated agricultural supply chains, traditional inventory models are useless because of the stochastic demand and deteriorative characteristic of agri-products. This paper provides a method to determine the optimal replenishment policy of integrated agricultural supply chains with stochastic demand. In these EOQ/EPQ models, shortages are allowed and are backlogged if market demand is stochastic. The objective function is to minimize the total cost of the supply chain in the planning horizon. The total cost includes the ordering cost, the holding cost, the shortage cost and the purchasing cost. Thinking of the nonlinear relationship and dynamic forces in models, a system dynamic (SD) simulation model is constructed to find the optimal lot size and replenishment interval. Finally, an example is given to make a sensitivity analysis of the simulation model. Compared to traditional methods (such as equalize stochastic demand), the total cost decreases by 16.27% if the supply chains adopt the new replenishment policy. The results illustrated that the new replenishment policy (with intelligent method) is beneficial to help supply chain make decision scientifically. Moreover, the intelligent method can simulate stochastic demand perfectly, and it is effectively for solving the complicated and mathematically intractable replenishment problem.     

A simulation-optimization approach for integrated sourcing and inventory decisions

In this paper, we study a generalized vendor selection problem that integrates vendor selection and inventory replenishment decisions of a firm. In addition to vendor-specific procurement and management costs, we consider inventory replenishment, holding, and backorder costs explicitly to meet stationary stochastic demand faced by the firm. Our goal is to select the best set of vendors along with the optimum inventory decisions at each plant of the firm so that we minimize the system-wide total costs and achieve desired service and reliability levels. Due to uncertainties inherent in the problem related to demand observed by the firm, quality provided by vendors, and disruptions observed by vendors, we utilize a simulation-optimization approach to solve the problem. More specifically, we build a discrete-event simulation model to evaluate the objective function of the problem that works in concert with a scatter search-based metaheuristic optimization approach to search the solution space. Computational results not only provide managerial insights and measure the significance of intangible factors in the vendor selection process but also highlight the importance of computational tools such as simulation-optimization for the vendor selection problem.     

Deterministic inventory model with two levels of storage,a linear trend in demand and a fixed time horizon

A deterministic inventory model is developed for a single item having two separate storage facilities (owned and rented warehouses) due to limited capacity of the existing storage (owned warehouse) with linearly time-dependent demand (increasing) over a fixed finite time horizon. The model is formulated by assuming that the rate of replenishment is infinite and the successive replenishment cycle lengths are in arithmetic progression. Shortages are allowed and fully backlogged. As a particular case, the results for the model without shortages are derived. Results are illustrated with two numerical examples.Scope and purposeThroughout the world, the production of food grains is periodical. Normally, in countries where state control is less, the demand of essential food grains is lowest at the time of harvest and goes up to the highest level just before the next harvest. This phenomenon is very common in developing third world countries where most of the people are landless or marginal farmers. At the time of harvest, they share some grain/product with landowners and as soon as the small inventory is exhausted, they are forced to buy food grains from the open market. As a result, demand for food grains increases with time in a period along with the number of the people whose initial stock of food grains gets exhausted.In this paper, a two-storage inventory model with time-dependent demand and fixed time horizon is developed and solved by a mathematical programme based on gradient method. This methodology of model development and its solution are quite general and it can be applied to inventory models of any product whose production is periodical and demand increases linearly with time.     

A local search method for periodic inventory routing problem

Inventory routing problem considers inventory allocation and routing problems simultaneously, in which the replenishment policies and routing arrangement are determined by the supplier under the vendor managed inventory mode. What we consider here is a periodic inventory routing problem that once the delivery time, quantity and routing are determined, they will remain the same in the following periods. The problem is modeled concisely, and then it is decomposed into two subproblems, inventory problem and vehicle routing problem. The inventory problem is solved by proposing a local search method, which is achieved by four operators on delivery quantity and retailer’s demand. Insertion operator aims to insert a new replenishment point of a retailer while removal operator is to remove a replenishment point. Besides, addition operator is adopted as an assistant tool, and crossover operator is proposed for some special cases. We also propose a tabu search method to solve the routing problem. Finally, the computational results show that the method is efficient and stable.     

Inventory and pricing strategies for deteriorating items with shortages: A discounted cash flow approach

In this article, we consider an infinite horizon, single product economic order quantity where demand and deterioration rate are continuous and differentiable function of price and time, respectively. In addition, we allow for shortages and completely backlogged. The objective is to find the optimal inventory and pricing strategies maximizing the net present value of total profit over the infinite horizon. For any given selling price, we first prove that the optimal replenishment schedule not only exists but is unique. Next, we show that the total profit per unit time is a concave function of price when the replenishment schedule is given. We then provide a simple algorithm to find the optimal selling price and replenishment schedule for the proposed model. Finally, we use a couple of numerical examples to illustrate the algorithm.     

An EPQ model for deteriorating items with inventory-level-dependent demand and permissible delay in payments

This article develops an inventory model for exponentially deteriorating items under conditions of permissible delay in payments. Unlike the existing related models, we assume that the items are replenished at a finite rate and the demand rate of the items is dependent on the current inventory level. The objective is to determine the optimal replenishment policies in order to maximise the system's average profit per unit of time. A simple method is shown for finding the optimal solution of the model based on the derived properties of the objective function. In addition, we deduce some previously published results as the special cases of the model. Finally, numerical examples are used to illustrate the proposed model. Some managerial insights are also inferred from the sensitive analysis of model parameters.