Faiz A. Al-Khayyal

Inventory constrained maritime routing and scheduling for multi-commodity liquid bulk, Part I: Applications and model

Abstract This paper formulates a model for finding a minimum cost routing in a network for a heterogeneous fleet of ships engaged in pickup and delivery of several liquid bulk products. The problem is frequently encountered by maritime chemical transport companies, including oil companies serving an archipelago of islands. The products are assumed to require dedicated compartments in the ship. The problem is to decide how much of each product should be carried by each ship from supply ports to demand ports, subject to the inventory level of each product in each port being maintained between certain levels that are set by the production rates, the consumption rates, and the storage capacities of the various products in each port. This important and challenging inventory constrained multi-ship pickup–delivery problem is formulated as a mixed-integer nonlinear program. We show that the model can be reformulated as an equivalent mixed-integer linear program with special structure. Over 100 test problems are randomly generated and solved using CPLEX 7.5. The results of our numerical experiments illuminate where problem structure can be exploited in order to solve larger instances of the model. Part II of the sequel will deal with new algorithms that take advantage of model properties.

Machine parameter selection for turning with constraints : an analytical approach based on geometric programming

This paper describes the design and development of an analytical tool for the selection of machine parameters in turning. The problem is to determine values for machine speed and feed that minimize the total cost of machining; specifically the costs associated with machining time and cutting tool wear. Geometric programming is used as the basic methodology, and the solution approach for the selection of machine parameters is based on an analysis of the complementary slackness conditions and realistic machining conditions. The quality of the solution is illustrated on several examples and compared to solutions obtained by some optimization methods proposed in the literature. In general, our technique is simple and straightforward and indicates how sensitive the solution is to the machine power consumed and surface finish attained.

A relaxation method for nonconvex quadratically constrained quadratic programs

We present an algorithm for finding approximate global solutions to quadratically constrained quadratic programming problems. The method is based on outer approximation (linearization) and branch and bound with linear programming subproblems. When the feasible set is non-convex, the infinite process can be terminated with an approximate (possibly infeasible) optimal solution. We provide error bounds that can be used to ensure stopping within a prespecified feasibility tolerance. A numerical example illustrates the procedure. Computational experiments with an implementation of the procedure are reported on bilinearly constrained test problems with up to sixteen decision variables and eight constraints.

Global optimization of concave functions subject to quadratic constraints: an application in nonlinear bilevel programming

When the followers optimality conditions are both necessary and sufficient, the nonlinear bilevel program can be solved as a global optimization problem. The complementary slackness condition is usually the complicating constraint in such problems. We show how this constraint can be replaced by an equivalent system of convex and separable quadratic constraints. In this paper, we propose different methods for finding the global minimum of a concave function subject to quadratic separable constraints. The first method is of the branch and bound type, and is based on rectangular partitions to obtain upper and lower bounds. Convergence of the proposed algorithm is also proved. For computational purposes, different procedures that accelerate the convergence of the proposed algorithm are analysed. The second method is based on piecewise linear approximations of the constraint functions. When the constraints are convex, the problem is reduced to global concave minimization subject to linear constraints. In the case of non-convex constraints, we use zero-one integer variables to linearize the constraints. The number of integer variables depends only on the concave parts of the constraint functions.

Generalized bilinear programming: Part I. Models, applications and linear programming relaxation

Abstract The evolution of the bilinear programming problem is reviewed and a new, more general model is discussed. The model involves two decision vectors and reduces to a linear program when one of the decision vectors is fixed. The class of problems under consideration contains traditional bilinear programs, general quadratic programs, and bilinearly constrained and quadratically constrained extensions of these problems. We describe how several important applications from the literature, including the multiple modular design problem, can be modeled as generalized bilinear programs. Finally, we derive a linear programming relaxation that can be used as a subproblem in algorithmic solution schemes based on outer approximation and branch and bound.

Monotonic Optimization: Branch and Cut Methods

Monotonic optimization is concerned with optimization problems dealing with multivariate monotonic functions and differences of monotonic functions. For the study of this class of problems a general framework (Tuy, 2000a) has been earlier developed where a key role was given to a separation property of solution sets of monotonic inequalities similar to the separation property of convex sets. In the present paper the separation cut is combined with other kinds of cuts, called reduction cuts, to further exploit the monotonic structure. Branch and cuts algorithms based on an exhaustive rectangular partition and a systematic use of cuts have proved to be much more efficient than the original polyblock and copolyblock outer approximation algorithms.

A piecewise linearization framework for retail shelf space management models

Managing shelf space is critical for retailers to attract customers and optimize profits. This article develops a shelf-space allocation optimization model that explicitly incorporates essential in-store costs and considers space- and cross-elasticities. A piecewise linearization technique is used to approximate the complicated nonlinear space-allocation model. The approximation reformulates the non-convex optimization problem into a linear mixed integer programming (MIP) problem. The MIP solution not only generates near-optimal solutions for large scale optimization problems, but also provides an error bound to evaluate the solution quality. Consequently, the proposed approach can solve single category-shelf space management problems with as many products as are typically encountered in practice and with more complicated cost and profit structures than currently possible by existing methods. Numerical experiments show the competitive accuracy of the proposed method compared with the mixed integer nonlinear programming shelf-space model. Several extensions of the main model are discussed to illustrate the flexibility of the proposed methodology.

A D.C. optimization method for single facility location problems

The single facility location problem with general attraction and repulsion functions is considered. An algorithm based on a representation of the objective function as the difference of two convex (d.c.) functions is proposed. Convergence to a global solution of the problem is proven and extensive computational experience with an implementation of the procedure is reported for up to 100,000 points. The procedure is also extended to solve conditional and limited distance location problems. We report on limited computational experiments on these extensions.

A branch-reduce-cut algorithm for the global optimization of probabilistically constrained linear programs

We consider probabilistically constrained linear programs with general distributions for the uncertain parameters. These problems involve non-convex feasible sets. We develop a branch-and-bound algorithm that searches for a global optimal solution to this problem by successively partitioning the non-convex feasible region and by using bounds on the objective function to fathom inferior partition elements. This basic algorithm is enhanced by domain reduction and cutting plane strategies to reduce the size of the partition elements and hence tighten bounds. The proposed branch-reduce-cut algorithm exploits the monotonicity properties inherent in the problem, and requires solving linear programming subproblems. We provide convergence proofs for the algorithm. Some illustrative numerical results involving problems with discrete distributions are presented.

Linear, quadratic, and bilinear programming approaches to the linear complementarity problem

Abstract Traditional pivoting procedures for solving the linear complementarity problem can only guarantee convergence for problems having well defined structures. Recently, optimization procedures based on linear, quadratic, and bilinear programming have been developed to extend the class of problems that can be solved efficiently. These latter approaches are the focus of this paper. The strengths and weaknesses of each of the approaches are discussed. The linear programming approach, advanced by Mangasarian, is the most efficient once an appropriate objective function is found. This requires the solution of a system of linear and bilinear equations that is easily solvable only in some cases. Extensions to this approach, due to Shiau, show some promise but are still limited to special cases of the general problem. The quadratic programming approaches discussed here are restricted to specialized procedures for the complementarity problem. One, proposed by Cheng, is based on the levitin-Poljak gradient projection method, and the other, due to Cirina, is based on Karush-Kuhn-Tucker theory. Both are only successful on some problems. The two bilinear programming algorithms discussed are the most general. For any problem, they are guaranteed to find at least one solution or conclude that none exist. One is a specialization of the recent biconvex programming algorithm of Al-Khayyal and Falk and the other is an entirely new implicit enumeration procedure.