Jonathan F. Bard

An explicit solution to the multi-level programming problem☆

Abstract The multi-level programming problem is defined as an n-person nonzero-sum game with perfect information in which the players move sequentially. The bi-level linear case is addressed in detail. Solutions are obtained by recasting this problem as a standard mathematical probram and appealing to its implicitly separable structure. The reformulated optimization problem is linear save for a complementarity constraint of the form 〈u, g〉 = 0. This constraint is decomposed in a manner that permits us to achieve separability with very little cost in dimensionality. A general branch and bound algorithm is then applied to obtain solutions. Unlike the conventional mathematical program though, the multi-level program may fail to have a solution even when the decision variables are defined over a compact set. An auxiliary optimization problem is employed to detect such failure. Finally, the general max-min problem is discussed within the bi-level programming framework. Examples are given for a variety of related problems.

Nondifferentiable and two-level mathematical programming

Preface. 1. Introduction. 2. Mathematical Preliminaries. 3. Differentiable Nonlinear Programming. 4. Nondifferentiable Nonlinear Programming. 5. Linear Programming. 6. Optimal-Value Functions. 7. Two-Level Mathematical Programming Problem. 8. Large-Scale Nonlinear Programming: Decomposition Methods. 9. Min-Max Problem. 10. Satisfaction Optimization Problem. 11. Two-Level Design Problem (Mathematical Programming with Optimal-Value Functions). 12. General Resource Allocation Problem for Decentralized Systems. 13. Min-Max Type Multi-Objective Programming Problem. 14. Best Approximation Problem by Chebyshev Norm. 15. The Stackelberg Problem: General Case. 16. The Stackelberg Problem: Linear and Convex Case. References. Index.

A branch and bound algorithm for the bilevel programming problem

The bilevel programming problem is a static Stackelberg game in which two players try to maximize their individual objective functions. Play is sequential and uncooperative in nature. This paper presents an algorithm for solving the linear/quadratic case. In order to make the problem more manageable, it is reformulated as a standard mathematical program by exploiting the followers Kuhn–Tucker conditions. A branch and bound scheme suggested by Fortuny-Amat and McCarl is used to enforce the underlying complementary slackness conditions. An example is presented to illustrate the computations, and results are reported for a wide range of problems containing up to 60 leader variables, 40 follower variables, and 40 constraints. The main contributions of the paper are in the step-by-step details of the implementation, and in the scope of the testing.

Preference scheduling for nurses using column generation

The purpose of this paper is to present a new methodology for scheduling nurses in which several conflicting factors guide the decision process. Unlike manufacturing facilities where standard shifts and days off are the rule, hospitals operate 24 hours a day, 7 days a week and face widely fluctuating demand. A more flexible arrangement for working hours and days off is needed, especially in light of the growing nursing shortage. To improve retention, management must now take into account individual preferences and requests for days off in a way that is perceived as fair, while ensuring sufficient coverage at all times. This multi-objective problem is solved with a column generation approach that combines integer programming and heuristics. The integer program is formulated as a set covering-type problem whose columns correspond to alternative schedules that a nurse can work over the planning horizon. A double swapping heuristic is used to generate the columns. The objective coefficients are determined by the degree to which the individual preferences of a nurse are violated. As part of the computational scheme, feasible solutions are refined to minimize the use of outside nurses, but when gaps in coverage exist, the outside nurses are distributed as evenly as possible over the shifts. The methodology was tested on a series of problems with up to 100 nurses using data provided by a large hospital in the US. The results indicate that high-quality solutions can be obtained within a few minutes in the majority of cases.

The mixed integer linear bilevel programming problem

A two-person, noncooperative game in which the players move in sequence can be modeled as a bilevel optimization problem. In this paper, we examine the case where each player tries to maximize the individual objective function over a jointly constrained polyhedron. The decision variables are variously partitioned into continuous and discrete sets. The leader goes first, and through his choice may influence but not control the responses available to the follower. For two reasons the resultant problem is extremely difficult to solve, even by complete enumeration. First, it is not possible to obtain tight upper bounds from the natural relaxation; and second, two of the three standard fathoming rules common to branch and bound cannot be applied fully. In light of these limitations, we develop a basic implicit enumeration scheme that finds good feasible solutions within relatively few iterations. A series of heuristics are then proposed in an effort to strike a balance between accuracy and speed. The computational results suggest that some compromise is needed when the problem contains more than a modest number of integer variables.

Short-Term Scheduling of Thermal-Electric Generators Using Lagrangian Relaxation

This paper presents an expanded formulation of the unit commitment problem in which hundreds of thermal-electric generators must be scheduled on an hourly basis, for up to 7 days at a time. The underlying model incorporates the full set of costs and constraints including setup, production, ramping, and operational status, and takes the form of a mixed integer nonlinear control problem. Lagrangian relaxation is used to disaggregate the model by generator into separate subproblems which are then solved with a nested dynamic program. The strength of the methodology lies partially in its ability to construct good feasible solutions from information provided by the dual. Thus, the need for branch-and-bound is eliminated. In addition, the inclusion of the ramping constraint provides new insight into the limitations of current techniques. Computational experience with the proposed algorithm indicates that problems containing up to 100 units and 48 time periods can be readily solved in reasonable times. Duality gaps of less than 1% were achieved in all cases.

An Efficient Point Algorithm for a Linear Two-Stage Optimization Problem

This paper presents an algorithm using sensitivity analysis to solve a linear two-stage optimization problem. The underlying theory rests on a set of first order optimality conditions that parallel the Kuhn-Tucker conditions associated with a one-dimensional parametric linear program. The solution to the original problem is uncovered by systematically varying the parameter over the unit interval and solving the corresponding linear program. Finite convergence is established under nondegenerate assumptions. The paper also discusses other solution techniques including branch and bound and vertex enumeration and gives an example highlighting their computational and storage requirements. By these measures, the algorithm presented here has an overall advantage. Finally, a comparison is drawn between bicriteria and bilevel programming, and underscored by way of an example.

Some properties of the bilevel programming problem

The purpose of this paper is to elaborate on the difficulties accompanying the development of efficient algorithms for solving the bilevel programming problem (BLPP). We begin with a pair of examples showing that, even under the best of circumstances, solutions may not exist. This is followed by a proof that the BLPP is NP-hard.

A Decomposition Approach to the Inventory Routing Problem with Satellite Facilities

This paper presents a comprehensive decomposition scheme for solving the inventory routing problem in which a central supplier must restock a subset of customers on an intermittent basis. In this setting, the customer demand is not known with certainty and routing decisions taken over the short run might conflict with the long-run goal of minimizing annual operating costs. A unique aspect of the short-run subproblem is the presence of satellite facilities where vehicles can be reloaded and customer deliveries continued until the closing time is reached. Three heuristics have been developed to solve the vehicle routing problem with satellite facilities (randomized Clarke-Wright, GRASP, modified sweep). After the daily tours are derived, a parametric analysis is conducted to investigate the tradeoff between distance and annual costs. This leads to the development of the efficient frontier from which the decision maker is free to choose the most attractive alternative. The proposed procedures are tested on data sets generated from field experience with a national liquid propane distributor.

An analytic framework for sequencing mixed model assembly lines

The problem of sequencing mixed model assembly lines is characterized by a set of parameters whose values are dictated by the actual manufacturing environment. In some cases, it may be desirable to minimize the size of the facility, while in others, the throughput time is paramount. Important design considerations include operator schedules, the product mix, station boundaries, and the launching discipline. The intent of this paper is to present a common mathematical framework in which each possible variant can be addressed. By implication, a solution technique developed for one can be readily adopted for the others. Virtually all of the previous work on mixed model sequencing has focused on the development of heuristics. While these may work well in specific instances, it is difficult to assess their performance without a frame of reference. Moreover, they cannot be universally applied. In this paper, we show that it is possible to obtain optimal solutions at very little cost. The presentation is concern...