Modeling and analysis of single item multi-period procurement lot-sizing problem considering rejections and late deliveries

Integer linear programming approach has been used to solve a multi-period procurement lot-sizing problem for a single product that is procured from a single supplier considering rejections and late deliveries under all-unit quantity discount environment. The intent of proposed model is two fold. First, we aim to establish tradeoffs among cost objectives and determine appropriate lot-size and its timing to minimize total cost over the decision horizon considering quantity discount, economies of scale in transactions and inventory management. Second, the optimization model has been used to analyze the effect of variations in problem parameters such as rejection rate, demand, storage capacity and inventory holding cost for a multi-period procurement lot-sizing problem. This analysis helps the decision maker to figure out opportunities to significantly reduce cost. An illustration is included to demonstrate the effectiveness of the proposed model. The proposed approach provides flexibility to decision maker in multi-period procurement lot-sizing decisions through tradeoff curves and sensitivity analysis.     

New integration of preventive maintenance and production planning with cell formation and group scheduling for dynamic cellular manufacturing systems

This paper proposes a new integration method for cell formation, group scheduling, production, and preventive maintenance (PM) planning problems in a dynamic cellular manufacturing system (CMS). The cell formation sub-problem aims to form part families and machine groups, which minimizes the inter-cell material handling, under-utilization, and relocation costs. The production planning aspect is a multi-item capacitated lot-sizing problem accompanied by sub-contracting decisions, while the group scheduling problem deals with the decisions on the sequential order of the parts and their corresponding completion times. The purpose of the maintenance sub-problem is to determine the availability of the system and the time when the noncyclical perfect PM must be implemented to reduce the number of corrective actions. Numerical examples are generated and solved by Bender’s decomposition pack in GAMS to evaluate the interactions of the proposed model. Statistical analysis, based on a nonparametric method, is also used to study the behavior of the model’s cost components in two different situations. It is shown that by adding the PM planning decisions to the tactical decisions of the dynamic CMS, the optimal configuration and production plans of the system are heavily affected. The results indicate that omitting the PM actions increases the number of sudden failures, which leads to a higher total cost. Finally, it is concluded that the boost in the total availability of the dynamic CMS is one of the main advantages of the proposed integrated method.     

需求损失下两产品联合生产动态批量决策及预测时阈

研究需求损失下两产品联合生产(采购)动态批量决策问题.在各周期成本变动情形下分析多周期动态批量决策的预测时阈和决策时阈,构建包含联合启动成本、两产品的单独启动成本、库存持有成本、变动生产成本和需求损失成本在内的成本最小化模型.在最优解结构特性的基础上,设计出前向动态规划算法求解问题,通过建立两产品生产点的单调性和建立生产集,给出求解预测时阈和决策时阈的充分条件.通过数值算例分析预测时阈求解的具体过程,表明所构建模型的有效性.     

Pricing and production lot-size/scheduling with finite capacity for a deteriorating item over a finite horizon

Although the lately evolved manufacturing technologies such as enterprise resource planning (ERP) provide a unified platform for managing and integrating core business processes within a firm, the decision-making between marketing and production planning still remains rather disjoint. It is due in large parts to the inherent weaknesses of ERP such as the fixed and static parameter settings and uncapacitated assumption. To rectify these drawbacks, we propose a decision model that solves optimally the production lot-size/scheduling problem taking into account the dynamic aspects of customer's demand as well as the restriction of finite capacity in a plant. More specifically, we consider a single product that is subject to continuous decay, faces a price-dependent and time-varying demand, and time-varying deteriorating rate, production rate, and variable production cost, with the objective of maximizing the profit stream over multi-period planning horizon. We propose both coordinated and decentralized decision-making policies that drive the solution of the multivariate maximization problem. Both policies are formulated as dynamic programming models and solved by numerical search techniques. In our numerical experiments, the solution procedure is demonstrated, comparative study is conducted, and sensitivity analysis is carried out with respect to major parameters. The numerical result shows that the solution generated by the coordinated policy outperforms that by the decentralized policy in maximizing net profit and many other quantifiable measures such as minimizing inventory investment and storage capacity.Scope and purposeWe consider a manufacturing firm who produces and sells a single product that is subjected to continuous decay over a lifetime, faces a price-dependent and time-varying demand function, shortages are allowed and a completely backlogged, and has the objective of determining price and production lot-size/scheduling so as to maximize the total profit stream over multi-period planning horizon. We develop a tactical-level decision model that solves the production scheduling problem taking into account the dynamic nature of customer's demand which is partially controllable through pricing schemes. As analogous to the sales and operations planning, the proposed scheme can be used as a coordination center of the APS system within a generic enterprise resource planning framework which integrates and coordinates distinct functions within a firm.This paper differs from the existing works in several ways. First, we propose a dynamic version of the joint pricing and lot-size/scheduling problem taking into account the capacitated constraint. Second, several key factors being considered in the model, such as the demand rate, deteriorating rate, production rate, and variable production cost are assumed time-varying that reflect the dynamic nature of the market and the learning effect of the production system. A third difference between the past research and ours is that the price can be adjusted upward or downward in our model, making the proposed pricing policy more responsive to the structural change in demand or supply.     

Single-Item Dynamic Lot-Sizing Models With Bounded Inventory and Outsourcing

This paper addresses a real-life lot-sizing problem which can be considered a single-item dynamic lot-sizing problem with bounded inventory. The particularity is that the demand of a period can be entirely or partially outsourced with an outsourcing cost. The goal is to minimize the total cost of production, setup, inventory holding, and outsourcing. The cost functions are linear but time-varying. We assume that the unit production cost is constant or nonincreasing over time. The problem is shown to be solvable in a strongly polynomial time with a dynamic-programming approach. The proposed algorithm can solve problems of sizes of up to 400 periods in less than 2 ms on a 1.4-GHz Pentium IV processor.     

Grouping in decomposition method for multi-item capacitated lot-sizing problem with immediate lost sales and joint and item-dependent setup cost

This article examines a dynamic and discrete multi-item capacitated lot-sizing problem in a completely deterministic production or procurement environment with limited production/procurement capacity where lost sales (the loss of customer demand) are permitted. There is no inventory space capacity and the production activity incurs a fixed charge linear cost function. Similarly, the inventory holding cost and the cost of lost demand are both associated with a linear no-fixed charge function. For the sake of simplicity, a unit of each item is assumed to consume one unit of production/procurement capacity. We analyse a different version of setup costs incurred by a production or procurement activity in a given period of the planning horizon. In this version, called the joint and item-dependent setup cost, an additional item-dependent setup cost is incurred separately for each produced or ordered item on top of the joint setup cost.     

An integrated replenishment model with quantity discounts, reentry and downward substitution for control wafers

This paper considers control wafers replenishment problem in wafer fabrication factories. A dynamic lot-sizing replenishment problem with reentry and downward substitution is examined in a pulling control production environment. The objective is to set the inventory level so as to minimize the total cost of control wafers, where the costs include order cost, purchase cost, setup cost, production cost and holding cost, while maintaining the same level of production throughput. In addition, purchase quantity discounts and precise inventory level are considered in the replenishment model. The control wafers replenishment problem is first constructed as a network, and is then transformed into a mixed integer programming model. Lastly, an efficient heuristic algorithm is proposed for solving large-scale problems. A numerical example is given to illustrate the practicality for empirical investigation. The results demonstrate that the proposed mixed integer programming model and the heuristic algorithm are effective tools for determining the inventory level of control wafers for multi-grades in multi-periods.     

Optimal pricing, lot-sizing and marketing planning in a capacitated and imperfect production system

In the classical economic production quantity (EPQ) problem demand is considered to be known in advance. However, in the real-world, demand of a product is a function of factors such as product’s price, its quality, and marketing expenditures for promoting the product. Quality level of the product and specifications of the adopted manufacturing process also affect the unit product’s cost. Therefore, in this paper we consider a profit maximizing firm who wants to jointly determine the optimal lot-sizing, pricing, and marketing decisions along with manufacturing requirements in terms of flexibility and reliability of the process. Geometric programming (GP) technique is proposed to address the resulting nonlinear optimization problem. Using recent advances in optimization techniques we are able to optimally solve the developed, highly nonlinear, mathematical model. Finally, using numerical examples, we illustrate the solution approach and analyze the solution under different conditions.     

Decomposition based heuristic algorithm for lot-sizing and scheduling problem treating time horizon as a continuum

In this paper, we deal with a new lot-sizing and scheduling problem (LSSP) that minimizes the sum of production cost, setup cost, and inventory cost. Incorporating the constraints of setup carry-over and overlapping as well as demand splitting, we develop a mixed integer programming (MIP) formulation. In the formulation, problem size does not increase as we enhance the precision level of a time period; for example, by dividing a time period into a number of microtime periods. Accordingly, in the proposed model, we treat the time horizon as a continuum not as a collection of discrete time periods. Since the problem is theoretically intractable, we develop a simple but efficient heuristic algorithm by devising a decomposition scheme coupled with a local search procedure. Even if in theory the heuristic may not guarantee finding a feasible solution, computational results demonstrate that the proposed algorithm is a viable choice in practice for finding good quality feasible solutions within acceptable time limit.     

Replenishment policy based on modified ant colony optimisation and statistical analysis under the pre-order penetration point

In the supply chain, most businesses in the pre-order penetration point (pre-OPP) operate under the forecast-driven mode, so that the decisions regarding inventory are made in accordance with the forecast and replenishment planning. This paper considers the stochastic dynamic lot-sizing problem of the two-phased transportation cost, service level constraint, and cash flow under a non-deterministic demand. This problem includes a nonlinear integer programming sub-problem. Therefore, this paper proposes an optimisation replenishment policy method based on modified ant colony optimisation (ACO) and response surface methodology. The main differences between the modified ACO and the traditional ACO lie in the modified update of pheromone intensity and the dynamic mutation operator. The experimental result shows that when the demand is normal distribution, the proposed approach, successfully finds the stationary point of minimum response. Besides, in the test of the algorithm solution quality, the modified ACO is better than the traditional ACO in all scenarios.     

Lot-sizing on a single imperfect machine: ILP models and FPTAS extensions

A single-machine multi-product lot-sizing and sequencing problem is studied. In this problem, items of n different products are manufactured in lots. Demands for products as well as per item processing times are known. There are losses of productivity because of non perfect production. There is also a sequence dependent set-up time between lots of different products. Machine yields and product lead times are assumed to be known deterministic functions. The objective is to minimize the cost of the demand dissatisfaction provided that the total processing time does not exceed a given time limit. We propose two integer linear programming (ILP) models for the NP-hard “fraction defective” case of this problem and compare effectiveness of their ILOG CPLEX realizations with a dynamic programming algorithm in a computer experiment. We also show how an earlier developed fully polynomial time approximation scheme (FPTAS) and one of the ILP models can be extended for a more complex case.     

Solution of fuzzy integrated production and marketing planning based on extension principle

The integration of production and marketing planning is crucial in practice for increasing a firm’s profit. However, the conventional inventory models determine the selling price and demand quantity for a retailer’s maximal profit with exactly known parameters. When the demand quantity, unit cost, and production rate are represented as fuzzy numbers, the profit calculated from the model should be fuzzy as well. Unlike previous studies, this paper develops a solution method to find the fuzzy profit of the integrated production and marketing planning problem when the demand quantity, unit cost, and production rate are represented as fuzzy numbers. Based on Zadeh’s extension principle, we transform the problem into a pair of two-level mathematical programming models to calculate the lower bound and upper bound of the fuzzy profit. According to the duality theorem of geometric programming technique, the two-level mathematical program is transformed into the one-level conventional geometric program to solve. At a specific α-level, we can derive the global optimum solutions for the lower and upper bounds of the fuzzy profit by applying well-developed theories of geometric programming. Examples are given to illustrate the whole idea proposed in this paper.     

列生成与GUB相结合求解钢铁原料采购批量问题

钢铁原料采购批量问题的目标是确定各种原料在一定时期(通常为一年)的各个时段 (一个月)内的采购量,在满足生产需求的前提下,使总的采购成本和库存费用之和最小.一般批 量问题是NP-hard,目前只存在有限的、启发式的方法求解小规模问题.建立了原料采购批量模 型并提出一种新的方法--列生成与GUB(广义上界)相结合方法求解该模型.一组实际问题测 试结果证明了该方法的有效性,同时也表明了该方法的潜在应用价值.     

A goal programming model for joint decision making of inventory lot-size,supplier selection and carrier selection

In this paper, we address a problem in which a storage space constrained buyer procures a single product in multiple periods from multiple suppliers. The production capacity constrained suppliers offer all-unit quantity discounts. The late deliveries and rejections are also incorporated in sourcing. In addition, we consider transportation cost explicitly in decision making which may vary because of freight quantity and distance of shipment between the buyer and a supplier. We propose a multi-objective integer linear programming model for joint decision making of inventory lot-sizing, supplier selection and carrier selection problem. In the multi-objective formulation, net rejected items, net costs and net late delivered items are considered as three objectives that have to be minimized simultaneously over the decision horizon. The intent of the model is to determine the timings, lot-size to be procured, and supplier and carrier to be chosen in each replenishment period. We solve the multi-objective optimization problem using three variants of goal programming (GP) approaches: preemptive GP, non-preemptive GP and weighted max–min fuzzy GP. The solution of these models is compared at different service-level requirements using value path approach.     

A stochastic fleet composition problem

In this paper, we consider the problem of forming a new vehicle fleet, consisting of multiple vehicle types, to cater for uncertain future requirements. The problem is to choose the number of vehicles of each type to purchase so that the total expected cost of operating the fleet is minimized. The total expected cost includes fixed and variable costs associated with the fleet, as well as hiring costs that are incurred whenever vehicle requirements exceed fleet capacity. We develop a novel algorithm, which combines dynamic programming and the golden section method, for determining the optimal fleet composition. Numerical results show that this algorithm is highly effective, and takes just seconds to solve large-scale problems involving hundreds of different vehicle types.     

修改D-W分解求具有需求时间窗和投机性成本的批量问题

研究多产品具有能力约束、需求时间窗、允许延期交货和投机性成本的批量问题.分析无能力约束凸包极点的特征,采用修正的Dantzig-Wolfe分解对原问题进行等价变换.使用列生成获得下界,同时采用启发式分支定界寻找近优解.对随机算例进行了测试与比较,计算结果表明上界与下界之间的间隙非常小;另外分析了当能力参数和订单规模变化时解的质量和计算时间.     

Optimal production run length in deteriorating production processes with fuzzy elapsed time

This paper investigates the optimal production run length in deteriorating production processes, where the elapsed time until the production process shifts is characterized as a fuzzy variable, also the setup cost and the holding cost are characterized as fuzzy variables, respectively. A mathematical formula representing the expected average cost per unit time is derived, and some properties are obtained to establish an efficient solution procedure. Since there is no closed-form expression for the optimal production run length, an approximate solving approach is presented. Finally, two numerical examples are given to illustrate the procedure of searching the optimal solutions.     

Optimal lot size and offline inspection policy

In this study, we investigate the effects of using process status at the end of the production lot (PSPL), on determining the optimal policies for products inspection and production lot size. First, we obtain the optimal product inspection policies for different PSPL for a given lot in the in-control or out-of-control state. Properties for the inspection policy are explored. Then, the expected total cost function, which includes setup cost, process maintenance cost and quality-related control cost, is obtained. The optimal production lot size that minimizes the expected total cost per item is determined. Our proposed inspection policy is compared with the three policies of no inspection, full inspection, and disregarding the first s items policy, in which only items from s+1 until the end of the production lot are inspected. Differences in the minimum expected total cost per item between our proposed inspection policy and the other three policies are investigated with a numerical example.     

Dynamic brand-image-based production location decisions

In this paper, we study the dynamic production location decisions of a manufacturer of a certain branded product. Considering brand-image as a form of goodwill, we extend the well-known Nerlove-Arrow dynamic model by adding both country-image and price. Formulating an optimal control problem for a group of countries in which the cost of production is convexly increasing with country-image, we are able to develop optimal decision rules for a manufacturer regarding the location of production and pricing over time. The resulted optimal policy has a very interesting pattern. Assuming that the demand rises by more than the value of the new brand-image in percentage terms, then, if brand-image is increasing toward a stationary value level, the optimal policy should be to initially locate production in countries with high image and set a high price that signals high quality. Later, the production should gradually shift to countries with lower production costs and lower image and the price lowered until the stationary value level is reached. For brand-images beyond the stationary value level, the location of production should start in a country with low costs and country-image while setting prices that signal relatively low quality. Over time, production should be shifted to countries with gradually higher costs and images while setting higher prices until the brand-image approaches the level of stationary value.     

Dynamic selling of quality-graded products under demand uncertainties

This paper examines dynamic selling (DS) problems under demand uncertainties. Quality-graded products with fully downward substitutable demands are considered. Downward demand substitution indicates that demands for lower quality grade products can be fulfilled by either designated or higher quality grade products. In this dynamic selling problem, decision makers need to choose an optimal selling policy in each decision epoch. The objective is to identify an optimal policy for the dynamic selling of quality-graded inventory.DS problems are formulated as a discrete-time Markov decision process (MDP) model. In the MDP model, demand type and inventory levels are state variables. The objective is to maximize expected profits. In such a multi-dimensional dynamic decision problem, computational complexity is a chief concern. This study proves the structure of optimal policies that significantly reduce computational complexity. Performance of optimal dynamic selling policies is evaluated in detailed numerical studies.