Profit maximization in an inventory system with time-varying demand,partial backordering and discrete inventory cycle

In this paper, an inventory problem where the inventory cycle must be an integer multiple of a known basic period is considered. Furthermore, the demand rate in each basic period is a power time-dependent function. Shortages are allowed but, taking necessities or interests of the customers into account, only a fixed proportion of the demand during the stock-out period is satisfied with the arrival of the next replenishment. The costs related to the management of the inventory system are the ordering cost, the purchasing cost, the holding cost, the backordering cost and the lost sale cost. The problem is to determine the best inventory policy that maximizes the profit per unit time, which is the difference between the income obtained from the sales of the product and the sum of the previous costs. The modeling of the inventory problem leads to an integer nonlinear mathematical programming problem. To solve this problem, a new and efficient algorithm to calculate the optimal inventory cycle and the economic order quantity is proposed. Numerical examples are presented to illustrate how the algorithm works to determine the best inventory policies. A sensitivity analysis of the optimal policy with respect to some parameters of the inventory system is developed. Finally, conclusions and suggestions for future research lines are given.

     

Revised multi-choice goal programming for multi-period,multi-stage inventory controlled supply chain model with popup stores in Guerrilla marketing

In this paper, we consider a supply chain network design problem with popup stores which can be opened for a few weeks or months before closing seasonally in a marketplace. The proposed model is multi-period and multi-stage with multi-choice goals under inventory management constraints and formulated by 0–1 mixed integer linear programming. The design tasks of the problem involve the choice of the popup stores to be opened and the distribution network design to satisfy the demand with three multi-choice goals. The first goal is minimization of the sum of transportation costs in all stages; the second is to minimization of set up costs of popup stores; and the third goal is minimization of inventory holding and backordering costs. Revised multi-choice goal programming approach is applied to solve this mixed integer linear programming model. Also, we provide a real-world industrial case to demonstrate how the proposed model works.     

A piecewise linearization framework for retail shelf space management models

Managing shelf space is critical for retailers to attract customers and optimize profits. This article develops a shelf-space allocation optimization model that explicitly incorporates essential in-store costs and considers space- and cross-elasticities. A piecewise linearization technique is used to approximate the complicated nonlinear space-allocation model. The approximation reformulates the non-convex optimization problem into a linear mixed integer programming (MIP) problem. The MIP solution not only generates near-optimal solutions for large scale optimization problems, but also provides an error bound to evaluate the solution quality. Consequently, the proposed approach can solve single category-shelf space management problems with as many products as are typically encountered in practice and with more complicated cost and profit structures than currently possible by existing methods. Numerical experiments show the competitive accuracy of the proposed method compared with the mixed integer nonlinear programming shelf-space model. Several extensions of the main model are discussed to illustrate the flexibility of the proposed methodology.     

Optimizing inventory decisions in a two-level supply chain with order quantity constraints

A fundamental assumption in traditional inventory models is that all of the ordered items are of perfect quality. A two-level supply chain is considered consists of one retailer and a collection of suppliers that operate within a finite planning horizon, including multiple periods, and a model is formulated that simultaneously determines both supplier selection and inventory allocation problems in the supply chain. It is supposed that the ordered products dependent on the suppliers include a certain percentage of imperfect quality products and have different prices. In this paper, we study the impact of the retailer’s financial constraint. On the other hand, suppliers have restricted capacities and set minimum order quantity (MOQ) policy for the retailer’s order amount happened in each period. So, the problem is modeled as a mixed integer nonlinear programming. The purpose of this model is to maximize the total profit. The nutrients, fishery and fruitage industries give good examples for the proposed model. A numerical example is presented to indicate the efficiency of the proposed model. Considering the complexity of the model, a genetic algorithm (GA) is presented to solve the model. We demonstrate analytically that the proposed genetic algorithm is suitable in the feasible situations.     

具有非线性采购成本库存控制问题的研究现状与挑战

库存管理是基于运筹学而发展起来的一门学科,并成为近几十年来运筹学和管理科学重要的研究领域之一。在库存系统中,采购成本是必不可少的成本之一,主要包含产品成本、运输成本、装卸成本等。现实中,采购成本依赖于采购量,且往往是采购量的非线性函数。介绍了几类常见的采购成本函数：依赖于采购量的固定成本、增量折扣、全量折扣、车载容量折扣和凸采购成本等。基于周期盘点库存模型和连续盘点库存模型,综述了带有这些非线性采购成本函数的库存模型研究进展。虽然经过了几十年的研究,但很多带有非线性采购成本的库存模型的最优采购策略因为其复杂性至今未能被完整刻画。通过综述来简单讨论该类问题的挑战和机会。     

Analytical models for supplier selection and order quantity allocation

In the literature, decision models and techniques for supplier selection do not often consider inventory management of the items being purchased as part of the analysis. In this article, two mixed integer nonlinear programming models are proposed to select the best set of suppliers and determine the proper allocation of order quantities while minimizing the annual ordering, inventory holding, and purchasing costs under suppliers’ capacity and quality constraints. The first model allows independent order quantities for each supplier while the second model restricts all order quantities to be of equal size, as it would be required in a multi-stage (supply chain) inventory model. Illustrative examples are used to highlight the advantages of the proposed models over a previous model introduced in the literature.     

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.     

A dynamic inventory model with supplier selection in a serial supply chain structure

Considering the inherent connection between supplier selection and inventory management in supply chain networks, this article presents a multi-period inventory lot-sizing model for a single product in a serial supply chain, where raw materials are purchased from multiple suppliers at the first stage and external demand occurs at the last stage. The demand is known and may change from period to period. The stages of this production–distribution serial structure correspond to inventory locations. The first two stages stand for storage areas for raw materials and finished products in a manufacturing facility, and the remaining stages symbolize distribution centers or warehouses that take the product closer to customers. The problem is modeled as a time-expanded transshipment network, which is defined by the nodes and arcs that can be reached by feasible material flows. A mixed integer nonlinear programming model is developed to determine an optimal inventory policy that coordinates the transfer of materials between consecutive stages of the supply chain from period to period while properly placing purchasing orders to selected suppliers and satisfying customer demand on time. The proposed model minimizes the total variable cost, including purchasing, production, inventory, and transportation costs. The model can be linearized for certain types of cost structures. In addition, two continuous and concave approximations of the transportation cost function are provided to simplify the model and reduce its computational time.     

A stochastic aggregate production planning model in a green supply chain: Considering flexible lead times,nonlinear purchase and shortage cost functions

In this paper we develop a stochastic programming approach to solve a multi-period multi-product multi-site aggregate production planning problem in a green supply chain for a medium-term planning horizon under the assumption of demand uncertainty. The proposed model has the following features: (i) the majority of supply chain cost parameters are considered; (ii) quantity discounts to encourage the producer to order more from the suppliers in one period, instead of splitting the order into periodical small quantities, are considered; (iii) the interrelationship between lead time and transportation cost is considered, as well as that between lead time and greenhouse gas emission level; (iv) demand uncertainty is assumed to follow a pre-specified distribution function; (v) shortages are penalized by a general multiple breakpoint function, to persuade producers to reduce backorders as much as possible; (vi) some indicators of a green supply chain, such as greenhouse gas emissions and waste management are also incorporated into the model. The proposed model is first a nonlinear mixed integer programming which is converted into a linear one by applying some theoretical and numerical techniques. Due to the convexity of the model, the local solution obtained from linear programming solvers is also the global solution. Finally, a numerical example is presented to demonstrate the validity of the proposed model.     

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.     

Optimizing a multi-stage production/inventory system by DC programming based approaches

This paper deals with optimizing the cost of set up, transportation and inventory of a multi-stage production system in presence of bottleneck. The considered optimization model is a mixed integer nonlinear program. We propose two methods based on DC (Difference of Convex) programming and DCA (DC Algorithm)—an innovative approach in nonconvex programming framework. The mixed integer nonlinear problem is first reformulated as a DC program and then DCA is developed to solve the resulting problem. In order to globally solve the problem, we combine DCA with a Branch and Bound algorithm (BB-DCA). A convex minorant of the objective function is introduced. DCA is used to compute upper bounds while lower bounds are calculated from a convex relaxation problem. The numerical results compared with those of COUENNE (http://www.coin-or.org/download/binary/Couenne/), a solver for mixed integer nonconvex programming, show the rapidity and the ?-globality of DCA in almost cases, as well as the efficiency of the combined DCA-Branch and Bound algorithm. We also propose a simple heuristic algorithm which is proved by experimental results to be better than an existing heuristic in the literature for this problem.     

A linear programming embedded genetic algorithm for an integrated cell formation and lot sizing considering product quality

Production lot sizing models are often used to decide the best lot size to minimize operation cost, inventory cost, and setup cost. Cellular manufacturing analyses mainly address how machines should be grouped and parts be produced. In this paper, a mathematical programming model is developed following an integrated approach for cell configuration and lot sizing in a dynamic manufacturing environment. The model development also considers the impact of lot sizes on product quality. Solution of the mathematical model is to minimize both production and quality related costs. The proposed model, with nonlinear terms and integer variables, cannot be solved for real size problems efficiently due to its NP-complexity. To solve the model for practical purposes, a linear programming embedded genetic algorithm was developed. The algorithm searches over the integer variables and for each integer solution visited the corresponding values of the continuous variables are determined by solving a linear programming subproblem using the simplex algorithm. Numerical examples showed that the proposed method is efficient and effective in searching for near optimal solutions.     

The multi-product newsboy problem with supplier quantity discounts and a budget constraint

This paper considers the multi-product newsboy problem with both supplier quantity discounts and a budget constraint, while each feature has been addressed separately in the literature. Different from most previous nonlinear optimization models on the topic, the problem is formulated as a mixed integer nonlinear programming model due to price discounts. A Lagrangian relaxation approach is presented to solve the problem. Computational results on both small and large-scale test instances indicate that the proposed algorithm is extremely effective for the problem. An extension to multiple constraints and preliminary computational results are also reported.     

Optimizing multi-item multi-period inventory control system with discounted cash flow and inflation: Two calibrated meta-heuristic algorithms

A mixed binary integer mathematical programming model is developed in this paper for ordering items in multi-item multi-period inventory control systems, in which unit and incremental quantity discounts as well as interest and inflation factors are considered. Although the demand rates are assumed deterministic, they may vary in different periods. The situation considered for the problem at hand is similar to a seasonal inventory control model in which orders and sales happen in a given season. To make the model more realistic, three types of constraints including storage space, budget, and order quantity are simultaneously considered. The goal is to find optimal order quantities of the products so that the net present value of total system cost over a finite planning horizon is minimized. Since the model is NP-hard, a genetic algorithm (GA) is presented to solve the proposed mathematical problem. Further, since no benchmarks can be found in the literature to assess the performance of the proposed algorithm, a branch and bound and a simulated annealing (SA) algorithm are employed to solve the problem as well. In addition, to make the algorithms more effective, the Taguchi method is utilized to tune different parameters of GA and SA algorithms. At the end, some numerical examples are generated to analyze and to statistically and graphically compare the performances of the proposed solving algorithms.     

An application of Genetic Algorithm in solving an inventory model with advance payment and interval valued inventory costs

The purpose of this research is to solve the mixed integer constrained optimization problem with interval coefficient by a real-coded genetic algorithm (RCGA) with ranking selection, whole arithmetical crossover and non-uniform mutation for non-integer decision variables. In the ranking selection, as well as in finding the best solution in each generation of RCGA, recently developed modified definitions of order relations between interval numbers with respect to decision-making are used. Also, for integer decision variables, new types of crossover and mutation are introduced. This methodology is applied to solve a finite time horizon inventory model with constant lead-time, uniform demand rate and a discount by paying an amount of money in advance. Moreover, different inventory costs are considered to be interval valued. According to the consumption of items during lead-time and reorder level, two cases may arise. For each case, the mathematical model becomes a constrained nonlinear mixed integer problem with interval objective. Our objective is to determine the optimal number of cycles in the finite time horizon, lot-size in each cycle and optimal profit. The model is illustrated with some numerical examples and sensitivity analysis has been done graphically with the variation of different inventory parameters.     

Multi-source facility location–allocation and inventory problem

We consider a joint facility location–allocation and inventory problem that incorporates multiple sources of warehouses. The problem is motivated by a real situation faced by a multinational applied chemistry company. In this problem, multiple products are produced in several plants. Warehouse can be replenished by several plants together because of capabilities and capacities of plants. Each customer in this problem has stochastic demand and certain amount of safety stock must be maintained in warehouses so as to achieve certain customer service level. The problem is to determine number and locations of warehouses, allocation of customers demand and inventory levels of warehouses. The objective is to minimize the expected total cost with the satisfaction of desired demand weighted average customer lead time and desired cycle service level. The problem is formulated as a mixed integer nonlinear programming model. Utilizing approximation and transformation techniques, we develop an iterative heuristic method for the problem. An experiment study shows that the proposed procedure performs well in comparison with a lower bound.     

Genetic algorithm-based multi-criteria project portfolio selection

Project portfolio selection is one of the most important decision-making problems for most organizations in project management and engineering management. Usually project portfolio decisions are very complicated when project interactions in terms of multiple selection criteria and preference information of decision makers (DMs) in terms of the criteria importance are taken into consideration simultaneously. In order to solve this complex decision-making problem, a multi-criteria project portfolio selection problem considering project interactions in terms of multiple selection criteria and DMs?? preferences is first formulated. Then a genetic algorithm (GA)-based nonlinear integer programming (NIP) approach is used to solve the multi-criteria project portfolio selection problem. Finally, two illustrative examples are presented for demonstration and verification purposes. Experimental results obtained indicate that the GA-based NIP approach can be used as a feasible and effective solution to multi-criteria project portfolio selection problems.     

物流服务供应链订单分配优化及其遗传算法

针对物流服务供应链订单分配问题中,物流服务集成商通常会按照所分配的订单价值向分包商收取一定比例交易费用的特点,设定交易费用为交易额的线性函数,构建了新的物流服务供应链订单分配优化混合整数规划模型,其优化目标为最小化交易费用、采购费用、短缺服务与延迟供给的物流能力数量。鉴于问题的NP-hard特性,设计了相应的遗传算法,并结合基于优先权的启发式规则避免了大量非法初始解的出现。实验算例表明所建立的模型能够反映物流服务供应链订单分配过程中的线性交易费用因素,其所设计的算法能够在可接受的时间内获得质量较高的满意解,并且对于大规模订单分配优化问题,遗传算法的求解时间与求解结果要优于LINGO软件。     

基于时间的供应链物流运作能力计划模型研究

本文考察了在基于时间的多阶供应链环境下的物流运作能力计划,主要的系统成本是物流要素能力固定成本和库存成本.为了决定最优的稳定周期性的物流运作能力计划,提出了一般性的整数非线性规划公式.强调了在这样的系统运作中的开始时间的关键作用,报告了在确保在任何阶段没有缺货条件下的物流能力决策的一些有价值的结论.