Mark H. Karwan

On two-level optimization

Decentralized planning has long been recognized as an important decision making problem. Many approaches based on the concepts of large scale system decomposition have generally lacked the ability to model the type of truly independent subsystems which have often existed in practice. Similarly, tractable game-theoretic approaches have not allowed for a continuous solution space in which one players decision restricts the feasible choices of another. This correspondence offers a formulation to help model this type of decision making hierarchy. Characterizations and algorithmic approaches for a class of two-level problems are discussed.

Developing conflict-free routes for automated guided vehicles

Automated guided vehicles AGVs are a highly sophisticated and increasingly popular type of material handling device in flexible manufacturing systems. This paper details solution methodologies for the static routing problem in which demand assignment of the AGVs are known; the focus is to obtain an implementable solution within a reasonable amount of computer time. The objective is to minimize the makespan, while routing AGVs on a bidirectional network in a conflict-free manner. This problem is solved via column generation. The master problem in this column generation procedure has the makespan and vehicle interference constraints. Columns in the master problem are routes iteratively generated for each AGV. The subproblem is a constrained shortest path problem with time-dependent costs on the edges. An improvement procedure is developed to better the solution obtained at the end of the master-subproblem interactions. Several methods of iterating between the master and subproblem are experimented with in-depth computational experiments. Our empirical results indicate that the procedure as a whole usually generates solutions that are within a few percent of a proposed bound, within reasonable computer time.

Modeling equity of risk in the transportation of hazardous materials

In this paper, we develop and analyze a model to generate an equitable set of routes for hazardous material shipments. The objective is to determine a set of routes that will minimize the total risk of travel and spread the risk equitably among the zones of the geographical region in which the transportation network is embedded, when several trips are necessary from origin to destination. An integer programming formulation for the problem is proposed. We develop and test a heuristic that repeatedly solves single-trip problems: a Lagrangian dual approach with a gap-closing procedure is used to optimally solve single-trip problems. We report a sampling of our computational experience, based on a real-life routing scenario in the Albany district of New York State. Our findings indicate that one can achieve a high degree of equity by modestly increasing the total risk and by embarking on different routes to evenly spread the risk among the zones. Furthermore, it appears that our heuristic procedure is excellent in terms of computational requirements as well as solution quality. We also suggest some directions for future research.

Multicriteria integer programming: A (hybrid) dynamic programming recursive approach

Dynamic programming recursive equations are used to develop a procedure to obtain the set of efficient solutions to the multicriteria integer linear programming problem. An alternate method is produced by combining this procedure with branch and bound rules. Computational results are reported.

Predicting search performance for multiple targets

Performance in a visual search task is usually measured by the cumulative probability of locating a target, F(t), in a given time (t). Two extreme F(t) against (t) relationships have been postulated, one assuming that search is random, and the other assuming that search is systematic. However, these relationships have only been available for the situation in which each search field contains a single occurrence of a single type of target. This paper extends both search models (random and systematic) first to the case of multiple occurrences of a single fault type within a search field and second to the case of multiple fault types. For systematic search, these two cases can be combined to predict the effects of multiple occurrences of multiple fault types. The general F(t) relationships are given in each case and illustrated with a worked example.

Some relationships between lagrangian and surrogate duality in integer programming

Lagrangian dual approaches have been employed successfully in a number of integer programming situations to provide bounds for branch-and-bound procedures. This paper investigates some relationship between bounds obtained from lagrangian duals and those derived from the lesser known, but theoretically more powerful surrogate duals. A generalization of Geoffrions integrality property, some complementary slackness relationships between optimal solutions, and some empirical results are presented and used to argue for the relative value of surrogate duals in integer programming. These and other results are then shown to lead naturally to a two-phase algorithm which optimizes first the computationally easier lagrangian dual and then the surrogate dual.

A variable-memory model of visual search

Previous models of visual search have hypothesized either a random search or a repeated systematic search strategy. Although both models reproduce well the cumulative search time distribution, F(t), neither fully accords with eye movement data. A new model is proposed in which search is intended to be systematic but suffers from imperfect memory. Systematic search is then a special case in which the memory is perfect, and random search a special case in which the memory is totally lacking. The model was derived for single and multiple occurrences of a single fault (or target) type. Where the model could be proved to be insoluble, a simulation model was used. Simulation results were compared with the previous calculations of Morawski, Drury, and Karwan (1980) and were shown to give identical results for pure random and pure systematic search. As the parameters of the memory model were varied, a family of curves between these extremes was produced.

Capacitated lot-sizing and scheduling by Lagrangean relaxation

Abstract Development of new models and solution procedures for production planning has been of research interest for several decades. Implementation of these models has resulted in lower production costs by reducing inventories, number of setups and labor costs. In this paper, we develop several optimal/near-optimal procedures for the Capacitated Lot-Sizing and Scheduling Problem (CLSP) with setup times, limited regular time and limited overtime. We formulate a mixed-integer linear programming model of the problem and solve it by Lagrangean relaxation. We experiment with alternative Lagrangean relaxations and develop new procedures to solve these relaxations. Overall, the capacity constraints relaxation seems to be superior to the demand constraints relaxation. Our results show that large problems can be solved in reasonable computer times and within one-percent accuracy of the optimal solutions. We solved 99 × 8 (i.e., 99 items and 8 periods), 50 × 12 and 50 × 8 problems in 30.61, 36.25 and 12.65 seconds of CDC Cyber 730 computer time, respectively. Our procedures are general enough to be applied directly or with slight modifications in real-life production settings.

Operations planning with real time pricing of a primary input

Abstract We examine the problem of planning the operation of an air separation plant where the price of its primary production input, electricity, changes hour to hour, a situation commonly referred to as real time pricing (RTP). We present a solution approach where operating decisions are obtained from optimizing a mixed integer program embedded in a rolling horizon procedure. A simulation study is conducted to assess the effect of unreliable and finite information on the efficiency of the operations plans generated by the procedure. Results of the study suggest that the rolling horizon procedure generates robust plans. An additional simulation study is conducted to identify the conditions under which RTP is attractive vis-a-vis other selected electricity pricing schemes. Results of the study indicate that RTP is most appealing when there is substantial flexibility in the operating environment in terms of the load placed on the plant (customer demand) and with regard to ramp-up (akin to set-up) times. Although this appeal diminishes with increased loads and longer ramp-ups, it is nevertheless the case that the operational inflexibility must be significant before RTP loses its allure. Scope and purpose: This paper considers the plant operation problem faced by firms in the industrial gas industry with production facilities where the price of the primary production input, that is electricity, changes hour to hour, which is often referred to as real time pricing. The purpose of this work is to present an optimization based planning approach that rigorously takes into account the realities of this environment. In addition this work seeks to identify, through the use of simulation, the conditions under which real time pricing is most appealing vis-a-vis other electricity pricing schemes, and also the degree to which planning horizon length and uncertainty in electricity prices impact the efficiency of the operations plans generated by our planning approach.

Surrogate Dual Multiplier Search Procedures in Integer Programming

Search procedures for optimal Lagrange multipliers are highly developed and provide good bounds in branch and bound procedures that have led to the successful application of Lagrangean duality in integer programming. Although the surrogate dual generally provides a better objective bound, there has been little development of surrogate multiplier search procedures. This paper develops and empirically analyzes several surrogate multiplier search procedures. Results indicate that the procedures can produce possibly superior bounds in an amount of time comparable to other techniques. Our discussion also highlights the similarity of the procedures to some well known Lagrangean search techniques.