模糊环境下动态供应商选择的混合整数非线性规划模型

基于供应商选择问题的动态性和模糊性,考虑在每个周期内生产商的需求能力及供应商的供应能力为模糊变量,本文将一个多阶段多商品多渠道的供应商选择问题视为一个0-1混合整数模糊动态非线性规划问题,目标函数为总成本最小化。然后建立了0-1混合整数模糊动态非线性规划模型。为了求解该模型,通过可信性理论把模型中模糊机会约束清晰化,将该模型转化为一个确定型的0-1混合整数动态非线性规划模型。最后给出了一个数值算例验证了模型的可行性。     

A stochastic periodic review integrated inventory model involving defective items, backorder price discount, and variable lead time

The purpose of this article is to investigate a stochastic integrated supplier-retailer inventory problem. The model analyzed in this article explores the problem of the protection interval, the backorder price discount, the lead time, and the numbers of shipments from the supplier to the retailer in one production run as control variables to widen applications for an integrated periodic review inventory model. We consider the situation in which the supplier and the retailer establish a long-term strategic partnership and contract to jointly determine the best strategy. We assume that the protection interval demand follows a normal distribution. Our objective is to determine the optimal review period, the optimal backorder price discount, the optimal lead time, and the optimal number of shipments from the supplier to the retailer in one production run, so that the joint expected annual total cost incurred has the minimum value. Furthermore, an algorithm of finding the optimal solution is developed. Also, the sensitivity analysis included and a numerical example is given to illustrate the results of the proposed model.     

A fuzzy optimization approach for procurement transport operational planning in an automobile supply chain

We consider a real-world automobile supply chain in which a first-tier supplier serves an assembler and determines its procurement transport planning for a second-tier supplier by using the automobile assembler’s demand information, the available capacity of trucks and inventory levels. The proposed fuzzy multi-objective integer linear programming model (FMOILP) improves the transport planning process for material procurement at the first-tier supplier level, which is subject to product groups composed of items that must be ordered together, order lot sizes, fuzzy aspiration levels for inventory and used trucks and uncertain truck maximum available capacities and minimum percentages of demand in stock. Regarding the defuzzification process, we apply two existing methods based on the weighted average method to convert the FMOILP into a crisp MOILP to then apply two different aggregation functions, which we compare, to transform this crisp MOILP into a single objective MILP model. A sensitivity analysis is included to show the impact of the objectives weight vector on the final solutions. The model, based on the full truck load material pick method, provides the quantity of products and number of containers to be loaded per truck and period. An industrial automobile supply chain case study demonstrates the feasibility of applying the proposed model and the solution methodology to a realistic procurement transport planning problem. The results provide lower stock levels and higher occupation of the trucks used to fulfill both demand and minimum inventory requirements than those obtained by the manual spreadsheet-based method.     

On solving a large-scale problem on facility location and customer assignment with interaction costs along a time horizon

A 0-1 quadratic programming model is presented for solving the strategic problem of timing the location of facilities and the assignment of customers to facilities in a multi-period setting. It is assumed that all parameters are known and, on the other hand, the quadratic character of the objective function is due to considering the interaction cost incurred by the joint assignment of customers belonging to different categories to a facility at a period. The plain use of a state-of-the-art MILP engine with capabilities for dealing with quadratic terms does not give any advantage over the matheuristic algorithm proposed in this work. In fact, the MILP engine was frequently running out of memory before reaching optimality for the equivalent mixed 0-1 linear formulation, being its best lower bound at that time instant too far from the incumbent solution for the large-sized instances which we have worked with. As an alternative, a fix-and-relax algorithm is presented. A deep computational comparison between MILP alternatives is performed, such that fix-and-relax provides a solution value very close to (and, frequently, a better than) the one provided by the MILP engine. The time required by fix-and-relax is very affordable, being frequently two times smaller than the time required by the MILP engine.     

Solving a two-dimensional trim-loss problem with MILP

In this paper a two-dimensional trim-loss problem connected to the paper-converting industry is considered. The problem is to produce a set of product paper rolls from larger raw paper rolls such that the cost for waste and the cutting time is minimized. The problem is generally non-convex due to a bilinear objective function and some bilinear constraints, which give rise to difficulties in finding efficient numerical procedures for the solution. The problem can, however, be solved as a two-step procedure, where the latter step is a mixed integer linear programming (MILP) problem. In the present formulation, both the width and length of the raw paper rolls as well as the lengths of the product paper rolls are considered variables. All feasible cutting patterns are included in the problem and global optimal cutting patterns are obtained as the solution from the corresponding MILP problem. A numerical example is included to illustrate the proposed procedure.     

顾客部分延期付款下两级商业信用易腐品订货策略

针对易腐品的经济订货批量问题展开研究。在供应商给零售商提供延迟还款的同时,零售商也给顾客提供部分延期还款条件。分五种情况分别讨论零售商的成本构成,并由此建立数学模型以求解最优订货周期使得零售商单位时间总成本最小化。通过数学证明得到了目标函数的解析性质,结果显示每种情况下在可行域范围内至多存在一个极小值点。以此为基础给出了相应的命题以有效地确定零售商的最优决策。最后通过数值算例说明了模型的有效性。     

A serial inventory system with supplier selection and order quantity allocation

Given the prevalence of both supplier selection and inventory control problems in supply chain management, this article addresses these problems simultaneously by developing a mathematical model for a serial system. This model determines an optimal inventory policy that coordinates the transfer of items between consecutive stages of the system while properly allocating orders to selected suppliers in stage 1. In addition, a lower bound on the minimum total cost per time unit is obtained and a 98% effective power-of-two (POT) inventory policy is derived for the system under consideration. This POT algorithm is advantageous since it is simple to compute and yields near optimal solutions.     

Optimizing product assortment under customer-driven demand substitution

The problem of product assortment and inventory planning under customer-driven demand substitution is analyzed and a mathematical model for this problem is provided in this paper. Realistic issues in a retail context such as supplier selection, shelf space constraints, and poor quality procurement are also taken into account. The performance of three modified models, one that neglects customers’ substitution behavior, another that excludes supplier selection decision, and one that ignores shelf space limitations, are analyzed separately with computational experiments. The results of the analysis demonstrate that neglecting customer-driven substitution or excluding supplier selection or ignoring shelf space limitations may lead to significantly inefficient assortments. The effects of demand variability and substitution cost on optimal assortment and supplier selection decisions as well as on the optimal revenue are also investigated. The main contribution of this paper is the development of a practical and flexible model to aid retailers in finding optimal assortments to maximize the expected profit.     

Optimal policy for a dynamic,non-stationary,stochastic inventory problem with capacity commitment

This paper studies a single-product, dynamic, non-stationary, stochastic inventory problem with capacity commitment, in which a buyer purchases a fixed capacity from a supplier at the beginning of a planning horizon and the buyer’s total cumulative order quantity over the planning horizon is constrained with the capacity. The objective of the buyer is to choose the capacity at the beginning of the planning horizon and the order quantity in each period to minimize the expected total cost over the planning horizon. We characterize the structure of the minimum sum of the expected ordering, storage and shortage costs in a period and thereafter and the optimal ordering policy for a given capacity. Based on the structure, we identify conditions under which a myopic ordering policy is optimal and derive an equation for the optimal capacity commitment. We then use the optimal capacity and the myopic ordering policy to evaluate the effect of the various parameters on the minimum expected total cost over the planning horizon.     

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 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.     

Cost Comparisons for Out-of-Phase Inventory Models

Inventory costs for a fixed time period have traditionally been determined by allocating total costs per cycle uniformly throughout that cycle as well as any partial cycles. This procedure for cost allocation has led to the solution of numerous inventory problems, most notable of which is the anticipated price-increase model. When comparing two out-of-phase inventory models, if costs are accounted for when they occur over a fixed planning horizon, inventory policies should be changed to reflect the impact of this different cost-allocation procedure. For the anticipated price-increase model, the ‘optimal’ order quantity as well as the implied savings in inventory costs will be different when cost models are developed based on these different cost-allocation methods. If the objective is to maximize over a fixed planning horizon the actual savings in inventory costs as they occur, the cost models presented here should be used.     

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 impact of price skimming on supply and exit decisions

Stochastic inventory control theory has focused on the order and/or pricing policy when the length of the selling period is known. In contrast to this focus, we examine the optimal length of the selling period—which we refer to as market exit time—in the context of a novel inventory replenishment problem faced by a supplier of a new, trendy, and relatively expensive product with a short life cycle. An important characteristic of the problem is that the supplier applies a price skimming strategy over time and the demand is modeled as a nonhomogeneous Poisson process with an intensity that is dependent on time. The supplier's problems of finding the optimal order quantity and market exit time, with the objective of maximizing expected profit, is studied. Procedures are proposed for joint optimization of the objective function with respect to the order quantity and the market exit time. Then, the effects of the order quantity and market exit time on the supplier's profitability are explored on the basis of a quantitative investigation. Copyright © 2014 John Wiley & Sons, Ltd.     

A capacity constrained single-facility dynamic lot-size model

A computationally efficient algorithm for a multi-period single commodity production planning problem with capacity constraints is developed. The model differs from earlier well-known studies involving concave cost functions in the introduction of production capacity constraints which need not be equal in every period. The objective is to find an optimal production schedule that minimizes the total production and inventory costs. Backlogging is not allowed. The structure of the optimal solution is characterized and then used in an efficient algorithm.     

Two-echelon inventory model for deteriorating items with credit period dependent demand including shortages under trade credit

In real life situation, it is observed that demand of an item depends on the length of the credit period offered by the retailer to his customers which has a positive impact on demand of an item. But the impact of credit period on demand has received a very little attention by researchers. Furthermore, by allowing shortages as backlogging, the impact on the cost from the decay of the products can be balanced out. A profitable decision policy between a supplier and the retailers can be characterized by an agreement on the permissible delay in payments. Recently, Jaggi et al. (Eur J Oper Res 190:130–135, 2008) have investigated the impact of credit linked demand on the retailer’s optimal replenishment policy. The objective of this study is to extend Jaggi et al. (Eur J Oper Res 190:130–135, 2008) model by incorporating deterioration and backlogging. That is, we formulate a two-echelon inventory model for deteriorating items with credit period dependent demand including shortages under two-level trade credit financing and determine the retailer’s optimal replenishment policy when both the supplier as well as the retailer offers the credit period to stimulate customer demand. Furthermore, we establish some useful theorems to characterize the optimal solution and provide an easy and useful computational algorithm with the help of computer code using the software Matlab 7.0 to determine the optimal shortage point, cycle length, ordering quantity and credit period. A numerical example is included to illustrate the solution procedure for the mathematical model developed. Finally, we implement sensitivity analysis of the optimal solution with respect to the major parameters of the system and obtain some important managerial insights.     

An integrated inventory model for a deteriorating item with allowable shortages and credit linked wholesale price

This study proposes a single manufacturer, single retailer integrated inventory model that includes deterioration and shortages in the retailer’s inventory. The manufacturer’s production process is assumed to be imperfect as it produces a certain percentage of defective items. The retailer performs a 100  % screening process immediately on receiving a lot from the manufacturer and returns the detected defective items to the manufacturer in the next delivery. The manufacturer disposes the defective items and incurs a disposal cost. To increase sales, (s)he offers a trade credit to the retailer. The retailer’s wholesale price varies linearly with the credit period. The objective is to determine the optimal replenishment cycle time, the time of running out of stock, the length of the credit period and the number of lots from the manufacturer to the retailer so as to maximize the total profit of the integrated system. A solution algorithm is designed and illustrated through numerical examples. Furthermore, a sensitivity analysis is carried out to study the influence of the model-parameters on the optimal solution.     

An inventory control problem for deteriorating items with back-ordering and financial considerations

This paper investigates the effects of time value of money and inflation on the optimal ordering policy in an inventory control system. We proposed an economic order quantity model to manage a perishable item over the finite horizon planning under which back-ordering and delayed payment are assumed. The demand and deterioration rates are constant. The present value of total cost during the planning horizon in this inventory system is modeled first, then a three phases solution procedure is proposed to derive the optimal order and shortage quantities, and the number of replenishment during the planning horizon. Finally, the proposed model is illustrated through numerical examples and the sensitivity analysis is reported to find some managerial insights.     

带时变生产成本的易变质经济批量模型的最优策略分析

考虑了具有时变生产成本的易变质产品经济批量模型．有限计划期内,单位生产成本、生产率以及需求率假定为时间的连续函数,生产固定成本则具有遗忘效应现象．当不允许缺货时,建立了以总成本最小为目标的混合整数优化模型并证明了此问题最优解的相关性质．对于此问题的特殊情形,将成本函数中的离散型变量松弛为连续型变量,通过分析其最优解的存在性及唯一性,求解了此最优解,将其作为初始值设计了求取一般情形最优解的有效算法．最后通过算例验证了理论结果的有效性．