Interactive fuzzy programming for two-level linear and linear fractional production and assignment problems: A case study

In this paper, we deal with actual problems on production and work force assignment in a housing material manufacturer and a subcontract firm. We formulate two kinds of two-level programming problems: one is a profit maximization problem of both the housing material manufacturer and the subcontract firm, and the other is a profitability maximization problem of them. Applying the interactive fuzzy programming for two-level linear and linear fractional programming problems, we derive satisfactory solutions to the problems. After comparing the two problems, we discuss the results of the applications and examine actual planning of the production and the work force assignment of the two firms to be implemented.     

Fuzzy programming and profit and cost allocation for a production and transportation problem

In this paper, we deal with a real problem on production and transportation in a housing material manufacturer, and consider a production and transportation planning under the assumption that the manufacturer makes multiple products at factories in multiple regions and the products are in demand in each of the regions. First, we formulate mixed zero–one programming problems such that the cost of production and transportation is minimized subject to capacities of factories and demands of regions. Second, to realize stable production and satisfactory supply of the products in fuzzy environments, fuzzy programming for the production and transportation problem is incorporated. Finally, under the optimal planning of production and transportation, we show a profit and cost allocation by applying a solution concept from game theory. Using actual data, we show usefulness of the fuzzy programming and a rational allocation scheme of the profit and cost.     

具有两种运输方式选择的物流排序问题

本文研究了单机环境下,有两种运输方式可供选择的集成生产和运输的排序问题。有多个工件需要在一台机器上进行加工,工件生产完后需要分批运到客户处。有两种运输方式,普通运输和特快运输可供选择。制造商需要安排工件的加工顺序,选择合适的运输方式和出发时间,以极小化相应的时间目标与运输费用的加权和。研究了排序理论中主要的两个目标函数,分析了问题的复杂性,对于这些问题给出了它们的最优算法。     

Optimization of chance constraint programming with sum-of-fractional objectives – An application to assembled printed circuit board problem

This paper presents a procedure to solve a chance constraint programming problem with sum-of-fractional objectives. The problem and the solution procedure are described with the help of a practical problem – assembled printed circuit boards (PCBs). Errors occurring during assembling PCBs are in general classified into three kinds, viz. machine errors, manual errors and other errors. These errors may lead to the rejection of the major portion of the production and hence result the manufacturer a huge loss. The problem is decomposed to have two objective functions; one is a sum-of-fractional objectives and the other is a non-linear objective with bounded constraints. The interest is to maximize the sum-of-fractional objectives and to minimize the non-linear objective, subject to stochastic and non-stochastic bounded environment. The first problem deals with the maximization of the profit (maximizing sum-of-fractional objectives) and the second deals with the minimization of the loss (errors), so as to obtain the maximum profit after removing the number of defective PCBs.     

Pareto optimal solutions of a cost–time trade-off bulk transportation problem

A cost–time trade-off bulk transportation problem with the objectives to minimize the total cost and duration of bulk transportation without according priorities to them is considered. The entire requirement of each destination is to be met from one source only; however a source can supply to any number of destinations subject to the availability of the commodity at it. Two new algorithms are provided to obtain the set of Pareto optimal solutions of this problem. This work extends and generalizes the work related to single-objective and prioritized two-objective bulk transportation problems done in the past while providing flexibility in decision making.     

Optimizing the periodic pick-up of raw materials for a manufacturer of auto parts

We describe a solution procedure for a special case of the periodic vehicle routing problem (PVRP). Operation managers at an auto parts manufacturer in the north of Spain described the optimization problem to the authors. The manufacturer must pick up parts (raw material) from geographically dispersed locations. The parts are picked up periodically at scheduled times. The problem consists of assigning a pickup schedule to each of its supplier’s locations and also establishing daily routes in order to minimize total transportation costs. The time horizon under consideration may be as long as 90 days. The resulting PVRP is such that the critical decision is the assignment of locations to schedules, because once this is done, the daily routing of vehicles is relatively straightforward. Through extensive computational experiments, we show that the metaheuristic procedure described in this paper is capable of finding high-quality solutions within a reasonable amount of computer time. Our main contribution is the development of a procedure that is more effective at handling PVRP instances with long planning horizons when compared to those proposed in the literature.     

New mathematical model for the bi-objective inventory routing problem with a step cost function: A multi-objective particle swarm optimization solution approach

Inventory management and satisfactory distribution are among the most important issues considered by distribution companies. One of the key objectives is the simultaneous optimization of the inventory costs and distribution expenses, which can be addressed according to the inventory routing problem (IRP). In this study, we present a new transport cost calculation pattern for the IRP based on some real cases. In this pattern, the transportation cost is calculated as a function of the load carried and the distance traveled by the vehicle based on a step cost function. Furthermore, previous methods usually aggregate the inventory and transportation costs to formulate them as a single objective function, but in non-cooperative real-life cases, the inventory-holding costs are paid by retailers whereas the transportation-related costs are paid by the distributor. In this study, we separate these two cost elements and introduce a bi-objective IRP formulation where the first objective is to minimize the inventory-holding cost and the second is minimizing the transportation cost. We also propose an efficient particle representation and employ a multi-objective particle swarm optimization algorithm to generate the non-dominated solutions for the inventory allocation and vehicle routing decisions. Finally, in order to evaluate the performance of the proposed algorithm, the results obtained were compared with those produced using the augmented ε-constraint method, thereby demonstrating the practical utility of the proposed multi-objective model and the proposed solution algorithm.     

Integrated markdown pricing and aggregate production planning in a two echelon supply chain: A hybrid fuzzy multiple objective approach

Given high variability of demands for short life cycle products, a retailer has to decide about the products’ prices and order quantities from a manufacturer. In the meantime, the manufacturer has to determine an aggregate production plan involving for example, production, inventory and work force levels in a multi period, multi product environment. Due to imprecise and fuzzy nature of products’ parameters such as unit production and replenishment costs, a hybrid fuzzy multi-objective programming model including both quantative and qualitative constraints and objectives is proposed to determine the optimalprice markdown policy and aggregate production planning in a two echelon supply chain. The model aims to maximize the total profit of manufacturer, the total profit of retailer and improving service aspects of retailing simultaneously. After applying appropriate strategies to defuzzify the original model, the equivalent multi-objective crisp model is then solved by a fuzzy goal programming method. An illustrative example is also provided to show the applicability and usefulness of the proposed model and solution method.     

Optimal production-inventory policy for the multi-period fixed proportions co-production system

Consider the multi-period production-inventory problem where a manufacturer purchases and processes a raw material into two products in fixed proportions when facing uncertain demands. In each period, the manufacturer first reviews the on-hand inventories of the products and then decides the purchase/processing quantity of the raw material. After processing the raw material into the end products, the demands of the two products are realized and satisfied by the available inventories. Any leftover inventories are carried to the next period while the unsatisfied demands are backordered. By proving the concavity and submodularity of the expected profit-to-go function, we establish that the one-dimensional produce-up-to policy is optimal. We also study the case where the raw material is seasonal and the manufacturer has only one chance to purchase. Modeling it as a dynamic program, we establish that the one-dimensional produce-down-to policy is optimal. Finally, we conduct numerical studies to examine the impacts of supply-demand balance and price fluctuation on the optimal policy, and derive managerial insights from the analytical findings.     

Interactive solutions for the linear multiobjective transportation problem

Mathematical programming problems that exhibit the mathematical structure of a transportation problem often arise in settings with multiple conflicting objectives. Existing procedures for analyzing these problems fall into two general categories. These methods either generate all nondominated solutions or they construct a single compromise solution. This paper presents two interactive algorithms which take advantage of the special form of the multiple objective transportation problem. Two examples are included to illustrate these algorithms and to demonstrate their viability.     

Optimal production planning and cap setting under cap-and-trade regulation

This paper investigates the optimal production decisions of a self-pricing manufacturer and the optimal cap-setting decisions of a regulator under the cap-and-trade regulation. The objectives of the manufacturer and the regulator are to maximize profit and to maximize social welfare, respectively. We first derive the optimal production decisions and the corresponding total emissions of the manufacturer, with given parameters of the cap-and-trade regulation. Based on these results, we then solve the optimal cap-setting problem of the regulator. Furthermore, through sensitive analyses, we show that as the emissions intensity (i.e., the emissions generated from one unit of product) increases, both the optimal total emissions and the optimal cap first increase and then decrease.     

装备维修器材生产路径双目标问题优化决策方法研究

在装备维修器材供应保障中,针对精确保障背景下部队用户对器材保障精度的要求,构建了最小化总成本和最大化订单精准执行率的双目标优化决策模型。在ε-约束法框架内,开发可生成近似Pareto前沿的两阶迭代启发式算法,并采用模糊逻辑决策法选择符合决策者偏好的折中最优解。随机实例测试结果表明所提出的模型和算法可以很好地应用在双目标优化问题的研究中,并在求解不同规模实例时表现出优异的性能。     

Optimal material ordering policy and allocation rule for a manufacturer making multiple products

Material ordering and allocation are important decisions for manufactures making multiple products, because those firms usually possess flexible production systems which can produce different products based on the same raw material. In this paper, we investigate the ordering policy (OP) and allocation rule (AR) of the raw materials for a manufacturer selling multiple products. The manufacturer’s decision-making problem is analyzed under three scenarios: (1) joint decisions on OP and AR, (2) fixed AR, and (3) predetermined OP. We show that the latter two are not special cases of the first scenario, and they require different solution methods. Our objective is to derive the optimal solutions analytically. For the first scenario, we obtain the closed-form solution that is indeed optimal for the nonconcave profit function. For the fixed AR scenario, the products with twice-differentiable demands are studied, and the exact optimal OP for the raw material is achieved. Finally, if the OP is predetermined, we prove that the profit function is concave in AR and provide the associated optimality conditions, for which the optimal AR can be reached numerically. Different from the pervious heuristic approaches, these mathematically tractable solutions are easy to be applied by the practitioners.     

A Branch and Bound Algorithm to Minimize the Total Weighed Number of Tardy Jobs and Delivery Costs

This paper addresses the production and delivery scheduling integration problem; a manufacturer receives ₋ orders from one customer while the orders need to be processed on one or two machines and be sent to the customer in batches. Sending several jobs in batches will reduce the transportation cost but it may increase the number of tardy jobs. The objective is to minimize the sum of the total weighted number of tardy jobs and the delivery costs. The structural properties of the problem for a single machine and special cases of the two-machine flow shop problem are investigated and used to set up a new branch and bound algorithm. A heuristic algorithm for upper bound calculation and two approaches for lower bound calculation are also introduced. Results of computational tests show significant improvement over an existing dynamic programming method.     

An agent-based negotiation model on price and delivery date in a fashion supply chain

Price and due-date negotiation between supply chain members is a critical issue. Motivated by industrial practice, we consider in this paper a make-to-order fashion supply chain in which the downstream manufacturer and the upstream supplier are cooperative on due-date and competitive on price. We propose a two-phase negotiation agenda based on such characteristics, and aim to find an optimal solution to deal with the negotiation problem considering production cost and mutual benefit. We build an analytical negotiation model for a manufacturer-supplier pair, discuss their utilities, and examine the Pareto efficiency frontier from the theoretical perspective. After that, from an application perspective, we build an agent-based two-phase negotiation system where agents are used to represent the two parties to enhance communication. In the cooperative phase, a simulated annealing based intelligent algorithm is employed to help the manufacturer agent and the supplier agent search tentative agreement on due dates which can minimize the total supply chain cost. In the competitive phase, the two parties bargain on the pricing issue using concession based methods. They adjust the reservation value and aspiration value for pricing accordingly based on the integrated utility and the result of the previous phase. Simulation results show that, the proposed negotiation approach can achieve optimal utility of agents and reach a win-win situation for the bilateral parties. Sensitivity analysis is conducted to further generate insights on how different parameters affect the performance of the proposed system.     

An iterative sequential heuristic procedure to a real-life 1.5-dimensional cutting stock problem

This paper addresses a real-life 1.5D cutting stock problem, which arises in a make-to-order plastic company. The problem is to choose a subset from the set of stock rectangles to be used for cutting into a number of smaller rectangular pieces so as to minimize total production cost and meet orders. The total production cost includes not only material wastage, as in traditional cutting stock problems, but also production time. A variety of factors are taken into account, like cutter knife changes, machine restrictions, due dates and other work in progress limitations. These restrictions make the combinatorial structure of the problem more complex. As a result, existing algorithms and mathematical models are no longer appropriate. Thus we developed a new 1.5D cutting stock model with multiple objectives and multi-constraints and solve this problem in an incomplete enumerative way. The computational results show that the solution procedure is easy to implement and works very well.     

Multi-objective multi-item solid transportation problem in fuzzy environment

A multi-objective multi-item solid transportation problem with fuzzy coefficients for the objectives and constraints, is modeled and then solved by two different methods. A defuzzification method based on fuzzy linear programming is applied for fuzzy supplies, demands and conveyance capacities, including the condition that both total supply and conveyance capacity must not fall below the total demand. First, expected values of the fuzzy objective functions are considered to derive crisp values. Another method based on the concept of “minimum of fuzzy number” is applied for the objective functions that yields fuzzy values instead of particular crisp values for the fuzzy objectives. Fuzzy programming technique and global criterion method are applied to derive optimal compromise solutions of multi-objectives. A numerical example is solved using above mentioned methods and corresponding results are compared.     

Synchronizing production and air transportation scheduling using mathematical programming models

Traditional scheduling problems assume that there are always infinitely many resources for delivering finished jobs to their destinations, and no time is needed for their transportation, so that finished products can be transported to customers without delay. So, for coordination of these two different activities in the implementation of a supply chain solution, we studied the problem of synchronizing production and air transportation scheduling using mathematical programming models. The overall problem is decomposed into two sub-problems, which consists of air transportation allocation problem and a single machine scheduling problem which they are considered together. We have taken into consideration different constraints and assumptions in our modeling such as special flights, delivery tardiness and no delivery tardiness. For these purposes, a variety of models have been proposed to minimize supply chain total cost which encompass transportation, makespan, delivery earliness tardiness and departure time earliness tardiness costs.     

Extremization of multi-objective stochastic fractional programming problem

This paper addresses classes of assembled printed circuit boards, which faces certain kinds of errors during its process of manufacturing. Occurrence of errors may lead the manufacturer to be in loss. The encountered problem has two objective functions, one is fractional and the other is a non-linear objective. The manufacturers are confined to maximize the fractional objective and to minimize the non-linear objective subject to stochastic and non-stochastic environment. This problem is decomposed into two problems. A solution approach to this model has been developed in this paper. Results of some test problems are provided.     

Use of the Transportation Method of Linear Programming in Production Planning: A Case Study<Superscript>†</Superscript>

An application of the transportation method of linear programming to a production planning problem of a large footwear manufacturer is described. Factors affecting the choice of approach are discussed along with details of data requirements and their implications. Results, including a comparison of the analytic solution with that obtained by management judgement, and related sensitivity analyses are presented. The model used in one sub-problem is detailed in the appendix and incorporates spare capacity constraints and conditions on product mix but preserves the standard transportation format. An additional ratio constraint is satisfied algebraically after obtaining an original optimal solution.