Duality in infinite dimensional linear programming

We consider the class of linear programs with infinitely many variables and constraints having the property that every constraint contains at most finitely many variables while every variable appears in at most finitely many constraints. Examples include production planning and equipment replacement over an infinite horizon. We form the natural dual linear programming problem and prove strong duality under a transversality condition that dual prices are asymptotically zero. That is, we show, under this transversality condition, that optimal solutions are attained in both primal and dual problems and their optimal values are equal. The transversality condition, and hence strong duality, is established for an infinite horizon production planning problem.This material is based on work supported by the National Science Foundation under Grant No. ECS-8700836.     

Goal Programming and Agricultural Planning

Goal Programming is similar in structure to linear programming, but offers a more flexible approach to planning problems by allowing a number of goals which are not necessarily compatible to be taken into account, simultaneously. The use of linear programming in farm planning is reviewed briefly. Consideration is given to published evidence of the goals of farmers, and ways in which these goals can be represented. A goal programming model of a 600 acre mixed farm is described and evaluated. Advantages and shortcomings of goal programming in relation to linear programming are considered. It is found that goal programming can be used as a means of generating a range of possible solutions to the planning problem.     

基于模糊需求的多产品供应商选择的多目标规划模型

由决策于环境的不确定性,供应商选择问题存在大量的模糊信息,传统的确定性规划模型已经不能够很好地处理此类问题。本文基于模糊需求量信息,对于多产品供应商问题建立了模糊多目标规划模型。同时考虑到各目标及约束的重要性程度不同的影响,通过引进适当的权重对多目标规划模型进行求解。文中结合实际算例验证模型的可行性和有效性。     

An interactive fuzzy goal programming approach for multi-period multi-product production planning problem

This study addresses an interactive multiple fuzzy goal programming (FGP) approach to the multi-period multi-product (MPMP) production planning problem in an imprecise environment. The proposed model attempts to simultaneously minimize total production costs, rates of changes in labor levels, and maximizing machine utilization, while considering individual production routes of parts, inventory levels, labor levels, machine capacity, warehouse space, and the time value of money. Piecewise linear membership functions are utilized to represent decision maker’s (DM’s) overall satisfaction levels. A numerical example demonstrates the feasibility of applying the proposed model to the MPMP problem. Furthermore, the proposed interactive approach facilitates the DM with a systematic framework of decision making process which enables DM to modify the search direction to reach the most satisfactory results during solving process.     

A scenario decomposition approach for stochastic production planning in sawmills

This study considers a real world stochastic multi-period, multi-product production planning problem. Motivated by the challenges encountered in sawmill production planning, the proposed model takes into account two important aspects: (i) randomness in yield and in demand; and (ii) set-up constraints. Rather than considering a single source of randomness, or ignoring set-up constraints as is typically the case in the literature, we retain all these characteristics while addressing real life-size instances of the problem. Uncertainties are modelled by a scenario tree in a multi-stage environment. In the case study, the resulting large-scale multi-stage stochastic mixed-integer model cannot be solved by using the mixed-integer solver of a commercial optimization package, such as CPLEX. Moreover, as the production planning model under discussion is a mixed-integer programming model lacking any special structure, the development of decomposition and cutting plane algorithms to obtain good solutions in a reasonable time-frame is not straightforward. We develop a scenario decomposition approach based on the progressive hedging algorithm, which iteratively solves the scenarios separately. CPLEX is then used for solving the sub-problems generated for each scenario. The proposed approach attempts to gradually steer the solutions of the sub-problems towards an implementable solution by adding some penalty terms in the objective function used when solving each scenario. Computational experiments for a real-world large-scale sawmill production planning model show the effectiveness of the proposed solution approach in finding good approximate solutions.     

DEA models for resource reallocation and production input/output estimation

This paper discusses the “inverse” data envelopment analysis (DEA) problem with preference cone constraints. An inverse DEA model can be used for a decision making unit (DMU) to estimate its input/output levels when some or all of its input/output entities are revised, given its current DEA efficiency level. The extension of introducing additional preference cones to the previously developed inverse DEA model allows the decision makers to incorporate their preferences or important policies over inputs/outputs into the production analysis and resource allocation process. We provide the properties of the inverse DEA problem through a discussion of its related multi-objective and weighted sum single-objective programming problems. Numerical examples are presented to illustrate the application procedure of our extended inverse DEA model. In particular, we demonstrate how to apply the model to the case of a local home electrical appliance group company for its resource reallocation decisions.     

Semi-Infinite Programming Approach to Continuously-Constrained Linear-Quadratic Optimal Control Problems

Consider the class of linear-quadratic (LQ) optimal control problems with continuous linear state constraints, that is, constraints imposed on every instant of the time horizon. This class of problems is known to be difficult to solve numerically. In this paper, a computational method based on a semi-infinite programming approach is given. The LQ optimal control problem is formulated as a positive-quadratic infinite programming problem. This can be done by considering the control as the decision variable, while taking the state as a function of the control. After parametrizing the decision variable, an approximate quadratic semi-infinite programming problem is obtained. It is shown that, as we refine the parametrization, the solution sequence of the approximate problems converges to the solution of the infinite programming problem (hence, to the solution of the original optimal control problem). Numerically, the semi-infinite programming problems obtained above can be solved efficiently using an algorithm based on a dual parametrization method.     

Production planning via scenario modelling

Several Linear Programming (LP) and Mixed Integer Programming (MIP) models for the production and capacity planning problems with uncertainty in demand are proposed. In contrast to traditional mathematical programming approaches, we use scenarios to characterize the uncertainty in demand. Solutions are obtained for each scenario and then these individual scenario solutions are aggregated to yield a nonanticipative or implementable policy. Such an approach makes it possible to model nonstationarity in demand as well as a variety of recourse decision types. Two scenario-based models for formalizing implementable policies are presented. The first model is a LP model for multi-product, multi-period, single-level production planning to determine the production volume and product inventory for each period, such that the expected cost of holding inventory and lost demand is minimized. The second model is a MIP model for multi-product, multi-period, single-level production planning to help in sourcing decisions for raw materials supply. Although these formulations lead to very large scale mathematical programming problems, our computational experience with LP models for real-life instances is very encouraging.     

An interval approach based on expectation optimization for fuzzy random bilevel linear programming problems

This paper considers a class of bilevel linear programming problems in which the coefficients of both objective functions are fuzzy random variables. The main idea of this paper is to introduce the Pareto optimal solution in a multi-objective bilevel programming problem as a solution for a fuzzy random bilevel programming problem. To this end, a stochastic interval bilevel linear programming problem is first introduced in terms of α-cuts of fuzzy random variables. On the basis of an order relation of interval numbers and the expectation optimization model, the stochastic interval bilevel linear programming problem can be transformed into a multi-objective bilevel programming problem which is solved by means of weighted linear combination technique. In order to compare different optimal solutions depending on different cuts, two criterions are given to provide the preferable optimal solutions for the upper and lower level decision makers respectively. Finally, a production planning problem is given to demonstrate the feasibility of the proposed approach.     

An improved solution methodology for the arsenal exchange model (AEM)

We develop an iterative approach for solving a linear programming problem with prioritized goals. We tailor our approach to preemptive goal programming problems and take advantage of the fact that at optimality, most constraints are not binding. To overcome the problems posed by redundant constraints, our procedure ensures redundant constraints are not present in the problems we solve. We apply our approach to the arsenal exchange model (AEM). AEM allocates weapons to targets using linear programs (LPs) formulated by the model. Our methodology solves a subproblem using a specific subset of the constraints generated by AEM. Violated constraints are added to the original subproblem and redundant constraints are not included in any of the subproblems. Our methodology was used to solve five test cases. In four of the five test cases, our methodology produced an optimal integer solution. In all five test cases, solution quality was maintained or improved.     

Using a MILP model to establish a framework for an annualised hours agreement

Production flexibility is essential for industrial companies that have to deal with seasonal demand. Human resources are one of the main sources of flexibility. Annualising working hours (i.e., the possibility of irregularly distributing the total number of working hours over the course of a year) is a tool that provides flexibility to organizations; it enables a firm to adapt production capacity to fluctuations in demand. However, it can imply a worsening of the staff’s working conditions. To take the human aspect into account, the planning and scheduling of working time should comply with constraints derived from the law or from a collective bargaining agreement. Furthermore, new and more difficult working-time planning and scheduling problems are arising. This paper proposes a mixed-integer linear program model to solve the problem of planning the production and the working hours of a human team that operates in a multi-product process. Solving the model for different settings provides the essential quantitative information to negotiate the best conditions of the annualised hours system (the elements to establish the trade-off between weekly flexibility and economic or working-time reduction compensation can be obtained). The results achieved in a computational experiment were very satisfactory.     

Resource allocation in a large decentralized enterprise

We present a model of an enterprise comprising several operating units pursuing their production goals with a fair degree of autonomy, but under resource constraints imposed by a headquarters function. Specifically, each operating unit is assumed to seek a maximization of a perceived market value of its product output, subject to constraints on resources such as capital for plants and equipment, headcount, etc. imposed by headquarters. The headquarters function pursues a global optimization problem which takes into account the market values of all the products of the operating units, but also the cost of the resources and their regulation. Under suitable assumptions of linearity, the operation of the enterprise is formulated as a novel hierarchical structure of linear programming problems. An algorithm is presented for the solution of a class of such problems.     

A polyhedral approach to a production planning problem

Production planning in manufacturing industries is concerned with the determination of the production quantities (lot sizes) of some items over a time horizon, in order to satisfy the demand with minimum cost, subject to some production constraints. In general, production planning problems become harder when different types of constraints are present, such as capacity constraints, minimum lot sizes, changeover times, among others. Models incorporating some of these constraints yield, in general, NP-hard problems. We consider a single-machine, multi-item lot-sizing problem, with those difficult characteristics. There is a natural mixed integer programming formulation for this problem. However, the bounds given by linear relaxation are in general weak, so solving this problem by LP based branch and bound is inefficient. In order to improve the LP bounds, we strengthen the formulation by adding cutting planes. Several families of valid inequalities for the set of feasible solutions are derived, and the corresponding separation problems are addressed. The result is a branch and cut algorithm, which is able to solve some real life instances with 5 items and up to 36 periods. This revised version was published online in June 2006 with corrections to the Cover Date.     

Development of a new approach for deterministic supply chain network design

This paper proposes a mixed integer linear programming model and solution algorithm for solving supply chain network design problems in deterministic, multi-commodity, single-period contexts. The strategic level of supply chain planning and tactical level planning of supply chain are aggregated to propose an integrated model. The model integrates location and capacity choices for suppliers, plants and warehouses selection, product range assignment and production flows. The open-or-close decisions for the facilities are binary decision variables and the production and transportation flow decisions are continuous decision variables. Consequently, this problem is a binary mixed integer linear programming problem. In this paper, a modified version of Benders’ decomposition is proposed to solve the model. The most difficulty associated with the Benders’ decomposition is the solution of master problem, as in many real-life problems the model will be NP-hard and very time consuming. In the proposed procedure, the master problem will be developed using the surrogate constraints. We show that the main constraints of the master problem can be replaced by the strongest surrogate constraint. The generated problem with the strongest surrogate constraint is a valid relaxation of the main problem. Furthermore, a near-optimal initial solution is generated for a reduction in the number of iterations.     

A multi-objective production smoothing model with compressible operating times

Real life multi-product multi-period production planning often deals with several conflicting objectives while considering a set of technological constraints. The solutions of these problems can provide deeper insights to the decision makers/managers than those of single-objective problems. Some managers want to use from a production plan that is corresponding to minimum change in production policy along with minimum total cost simultaneously as possible. On the other hand, these two objectives have intrinsic conflicts such that producing in a fixed rate will cause huge costs than producing economically or according to JIT. So this paper presents a novel multi-objective model for the production smoothing problem on a single stage facility that some of the operating times could be determined in a time interval for. The model is to: (a) smooth the variations of production volume, and (b) minimize total cost of the corresponding production plan, while satisfying a set of technological constraints such as limited available time. The proposed model is developed in a real case study and is solved by a new genetic algorithm. The proposed genetic algorithm uses a novel achievement function for exploring the solution space, based on LP-metric concepts. Computational experiences reveal the sufficiency and efficiency of the proposed approach in contrast to previous researches.     

A mathematical programming model and solution for scheduling production orders in Shanghai Baoshan Iron and Steel Complex

In this paper, we investigate the production order scheduling problem derived from the production of steel sheets in Shanghai Baoshan Iron and Steel Complex (Baosteel). A deterministic mixed integer programming (MIP) model for scheduling production orders on some critical and bottleneck operations in Baosteel is presented in which practical technological constraints have been considered. The objective is to determine the starting and ending times of production orders on corresponding operations under capacity constraints for minimizing the sum of weighted completion times of all orders. Due to large numbers of variables and constraints in the model, a decomposition solution methodology based on a synergistic combination of Lagrangian relaxation, linear programming and heuristics is developed. Unlike the commonly used method of relaxing capacity constraints, this methodology alternatively relaxes constraints coupling integer variables with continuous variables which are introduced to the objective function by Lagrangian multipliers. The Lagrangian relaxed problem can be decomposed into two sub-problems by separating continuous variables from integer ones. The sub-problem that relates to continuous variables is a linear programming problem which can be solved using standard software package OSL, while the other sub-problem is an integer programming problem which can be solved optimally by further decomposition. The subgradient optimization method is used to update Lagrangian multipliers. A production order scheduling simulation system for Baosteel is developed by embedding the above Lagrangian heuristics. Computational results for problems with up to 100 orders show that the proposed Lagrangian relaxation method is stable and can find good solutions within a reasonable time.     

LP modelling of the finishing hot rolling programme for low carbon steel coils

In this paper a mathematical model was developed to optimize the finishing rolling of hot rolled coils by increasing the productivity of the rolling programme, to help achieve the required level of quality assurance and to facilitate production planning and control in the hot rolling mill. A brief account of the technological and planning aspects of the hot rolling processes and mills relevant to strip steel is given. Linear (mixed integer) programming is used to formulate the objective function and the various types of constraints of the model. The model takes into consideration, the general aspects pertinent to hot rolling of low carbon steel and the characteristics of the hot rolling mills as stipulated by the operational codes and guidelines of the relevant establishments. Owing to the flexibility offered by linear programming the model can incorporate any modifications and/or additional requirements, if any, in case of other types of steel and/or other types of mills. The full modelling of the problem required the incorporation of some zero/one variable constraints. Owing to the complexity involved and the need to keep the model as simple as possible, it was decided to exclude these constraints and deal with them externally. HYPER LINDO PC was used to solve the programme. Using available data, in the case under consideration the model showed astonishing results in achieving the objectives. Taking into account the effect on the overall productivity as well as quality improvement, the investigation showed that a net improvement in conforming output to the effect of around 43%, could have been obtained had the model been used in the case under consideration.     

Using separable programming to solve the multi-product multiple ex-ante constraint newsvendor problem and extensions

This paper provides an approximating programming technique to solve the multi-product newsvendor model in which product demands are independent and stocking quantities are subject to two or more ex-ante linear contraints, such as budget or volume constraints. Previous research has attempted to solve this problem with Lagrange relaxation techniques or by limiting the distribution of demand. However, by taking advantage of the separable nature of the problem, a close approximation of the optimal solution can be found using convex separable programming for any demand distribution in the traditional newsvendor model and extensions. Sensitivity analysis of the linear program provides managerial insight into the effects of parameters of the problem on the optimal solution and future decisions.     

Probabilistic programming models for traffic incident management operations planning

This paper proposes mathematical programming models with probabilistic constraints in order to address incident response and resource allocation problems for the planning of traffic incident management operations. For the incident response planning, we use the concept of quality of service during a potential incident to give the decision-maker the flexibility to determine the optimal policy in response to various possible situations. An integer programming model with probabilistic constraints is also proposed to address the incident response problem with stochastic resource requirements at the sites of incidents. For the resource allocation planning, we introduce a mathematical model to determine the number of service vehicles allocated to each depot to meet the resource requirements of the incidents by taking into account the stochastic nature of the resource requirement and incident occurrence probabilities. A detailed case study for the incident resource allocation problem is included to demonstrate the use of proposed model in a real-world context. The paper concludes with a summary of results and recommendations for future research.     

含直觉模糊弹性约束的广义模糊变量线性规划

本文基于模糊结构元方法建立并讨论了一类含有直觉模糊弹性约束的广义模糊变量线性 规划问题。首先,简单介绍了结构元方法并对结构元加权排序中权函数表征决策者风险态度进行了深入分析。然后,通过选取风险中立型决策态度来定义序关系并拓展Verdegay模糊线性规划方法,将新型模糊变量线性规划问题转化为两个含一般模糊弹性约束的模糊变量线性规划模型,给出了此类规划最优直觉模糊解的求法。最后,通过数值算例进一步说明该方法的有效性。