Joint Randomized Decisions in Chance-Constrained Programming

Mukherjee has considered the problem (P) of minimizing E[X₁ + X₂] subject to

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where b₁ is uniform over (0,2), and b₂ is uniform over (0,4), assuming b₁ and b₂ to be independent. It is shown in this paper that his solution is incorrect, and a correct solution is presented under the less restrictive assumption that X₁ and X₂ have any joint distribution.     

Mixed Strategies in Chance-Constrained Programming

In usual stochastic programming problems involving randomly distributed "resources" and chance constraints, decision variables are taken as deterministic. With the help of simple illustrations involving a single decision variable, Vajda and Greenberg showed that minimum expected values could be made smaller by treating the decision variable as random. This idea has been extended here to the case of two decision variables. The difficulty of finding an optimal mixed strategy in the case of an infinite-range distribution of the decision variable(s) has also been shown.     

A Note on Chance-Constrained Programming

A stochastic linear programme with chance constraints is considered. By allowing for the possibility of sampling, a so-called dynamic statistical decision model is developed, and the existence of an optimal decision rule is shown under appropriate continuity and compactness assumptions.     

A Linear Approximation for Chance-Constrained Programming

Decision environments involve the need to solve problems with varying degrees of uncertainty as well as multiple, potentially conflicting objectives. Chance constraints consider the uncertainty encountered. Codes incorporating chance constraints into a mathematical programming model are not available on a widespread basis owing to the non-linear form of the chance constraints. Therefore, accurate linear approximations would be useful to analyse this class of problems with efficient linear codes. This paper presents an approximation formula for chance constraints which can be used in either the single- or multiple-objective case. The approximation presented will place a bound on the chance constraint at least as tight as the true non-linear form, thus overachieving the chance constraint at the expense of other constraints or objectives.     

On Accurate Linear Approximations for Chance-Constrained Programming

This paper introduces the CHAPS (chance-constrained programming system) algorithm, which uses linearization techniques but gives more accurate solutions than earlier, similar methods.     

A Chance-Constrained Goal Programming Model for Capital Budgeting

A chance-constrained formulation is presented for a zero-one goal programming problem whose coefficients in the technological matrix are stochastic. The model is presented with a numerical example. A capital budgeting problem is taken for illustration.     

模糊批量生产计划问题的机会约束规划

描述了模糊单位利润、模糊生产能力以及模糊需求下的批量生产计划,并应用模糊机会约束规划规划建立了模型.当模糊变量是梯形模糊数时,我们将模糊模型转化为确定意义下的模型.为了求解优化模型,我们设计了基于模糊模拟的遗传算法.最后,通过一个数值例子说明算法的有效性.     

Programming for Optimal Decisions.

Journal of the Operational Research Society -     

A Chance-Constrained Integer Goal Programming Model for Capital Budgeting in the Production Area

The selection of capital projects in a production environment is complicated by the existence of multiple and conflicting goals. Typical production objectives for cost minimization often conflict with goals for quality, environmental standards, labor relations, etc. This problem of project selection is further complicated by the uncertainty inherent in product demand, the key factor in production management. This paper approaches these complications by employing an integer goal programming (to compensate for multiple conflicting objectives) with chance-constrained capabilities (to reflect uncertainty in product demand). The approach is demonstrated via an in-depth case example of a production problem.     

Some Randomized Algorithms for Convex Quadratic Programming

We adapt some randomized algorithms of Clarkson [3] for linear programming to the framework of so-called LP-type problems, which was introduced by Sharir and Welzl [10]. This framework is quite general and allows a unified and elegant presentation and analysis. We also show that LP-type problems include minimization of a convex quadratic function subject to convex quadratic constraints as a special case, for which the algorithms can be implemented efficiently, if only linear constraints are present. We show that the expected running times depend only linearly on the number of constraints, and illustrate this by some numerical results. Even though the framework of LP-type problems may appear rather abstract at first, application of the methods considered in this paper to a given problem of that type is easy and efficient. Moreover, our proofs are in fact rather simple, since many technical details of more explicit problem representations are handled in a uniform manner by our approach. In particular, we do not assume boundedness of the feasible set as required in related methods. Accepted 7 May 1997     

A Heuristic for Purchasing of Sheet Metal Decisions Based on Mathematical Programming

An operational research project at the Oskarshamn plant of Saab Scania AB Truckdivision Sweden is reported. The decision problem concerns the product mix of sheet metal purchased. A mathematical model that approximates the real-world problem is derived, the solution of which includes a p-median problem. This optimization problem is solved using Lagrangean relaxation and a subgradient optimization procedure. Numerical results for problems of real-world size and structure are reported and indicate satisfactory efficiency.     

A New Chance-Constrained Maximum Capture Location Problem

The paper presents a new model based on the basic Maximum Capture model, MAXCAP. The new Chance-Constrained Maximum Capture model introduces a stochastic threshold constraint, which recognises the fact that a facility can be open only if a minimum level of demand is captured. A metaheuristic based on Max-Min Ant System and Tabu Search procedure is presented to solve the model. This is the first time that the Max-Min Ant system is adapted to solve a location problem. Computational experience and an application to 55-node network are also presented.     

On Distributionally Robust Chance-Constrained Linear Programs

In this paper, we discuss linear programs in which the data that specify the constraints are subject to random uncertainty. A usual approach in this setting is to enforce the constraints up to a given level of probability. We show that, for a wide class of probability distributions (namely, radial distributions) on the data, the probability constraints can be converted explicitly into convex second-order cone constraints; hence, the probability-constrained linear program can be solved exactly with great efficiency. Next, we analyze the situation where the probability distribution of the data is not completely specified, but is only known to belong to a given class of distributions. In this case, we provide explicit convex conditions that guarantee the satisfaction of the probability constraints for any possible distribution belonging to the given class.Communicated by B. T. PolyakThis work was supported by FIRB funds from the Italian Ministry of University and Research.     

Scenario Approximation of Robust and Chance-Constrained Programs

We consider scenario approximation of problems given by the optimization of a function over a constraint that is too difficult to be handled but can be efficiently approximated by a finite collection of constraints corresponding to alternative scenarios. The covered programs include min-max games, and semi-infinite, robust and chance-constrained programming problems. We prove convergence of the solutions of the approximated programs to the given ones, using mainly epigraphical convergence, a kind of variational convergence that has demonstrated to be a valuable tool in optimization problems.     

A Smoothing Function Approach to Joint Chance-Constrained Programs

In this article, we consider a DC (difference of two convex functions) function approach for solving joint chance-constrained programs (JCCP), which was first established by Hong et al. (Oper Res 59:617–630, 2011). They used a DC function to approximate the probability function and constructed a sequential convex approximation method to solve the approximation problem. However, the DC function they used was nondifferentiable. To alleviate this difficulty, we propose a class of smoothing functions to approximate the joint chance-constraint function, based on which smooth optimization problems are constructed to approximate JCCP. We show that the solutions of a sequence of smoothing approximations converge to a Karush–Kuhn–Tucker point of JCCP under a certain asymptotic regime. To implement the proposed method, four examples in the class of smoothing functions are explored. Moreover, the numerical experiments show that our method is comparable and effective.     

“Decisions,Decisions.....”

The decision analyst can provide valuable support to the decisionmaking process and, increasingly, the impact of this support is becoming more forceful through the use of decision support software packages available on microcomputers. What are the opportunities for using decision support software and what software is currently available? This paper discusses these questions and outlines the part decision support software can play, its value to the decisionmaker, and some of the pitfalls in practical application.     

A Sampling-and-Discarding Approach to Chance-Constrained Optimization: Feasibility and Optimality

In this paper, we study the link between a Chance-Constrained optimization Problem (CCP) and its sample counterpart (SP). SP has a finite number, say N, of sampled constraints. Further, some of these sampled constraints, say k, are discarded, and the final solution is indicated by x^*_N,kx^{\ast}_{N,k}. Extending previous results on the feasibility of sample convex optimization programs, we establish the feasibility of x^*_N,kx^{\ast}_{N,k} for the initial CCP problem.     

Integer Programming Duality in Multiple Objective Programming

The weighted sums approach for linear and convex multiple criteria optimization is well studied. The weights determine a linear function of the criteria approximating a decision makers overall utility. Any efficient solution may be found in this way. This is not the case for multiple criteria integer programming. However, in this case one may apply the more general e-constraint approach, resulting in particular single-criteria integer programming problems to generate efficient solutions. We show how this approach implies a more general, composite utility function of the criteria yielding a unified treatment of multiple criteria optimization with and without integrality constraints. Moreover, any efficient solution can be found using appropriate composite functions. The functions may be generated by the classical solution methods such as cutting plane and branch and bound algorithms.     

Optimal Investment Decisions

A discrete investment model of a production project is considered. Stoppage and recovery of production are allowed. The project is scrapped if the maintenance costs exceed some limit. The optimal maintenance cost threshold is calculated.