Showing papers on "Integer programming published in 1994"

Solving mixed integer nonlinear programs by outer approximation

Abstract: A wide range of optimization problems arising from engineering applications can be formulated as Mixed Integer NonLinear Programming problems (MINLPs). Duran and Grossmann (1986) suggest an outer approximation scheme for solving a class of MINLPs that are linear in the integer variables by a finite sequence of relaxed MILP master programs and NLP subproblems. Their idea is generalized by treating nonlinearities in the integer variables directly, which allows a much wider class of problem to be tackled, including the case of pure INLPs. A new and more simple proof of finite termination is given and a rigorous treatment of infeasible NLP subproblems is presented which includes all the common methods for resolving infeasibility in Phase I. The worst case performance of the outer approximation algorithm is investigated and an example is given for which it visits all integer assignments. This behaviour leads us to include curvature information into the relaxed MILP master problem, giving rise to a new quadratic outer approximation algorithm. An alternative approach is considered to the difficulties caused by infeasibility in outer approximation, in which exact penalty functions are used to solve the NLP subproblems. It is possible to develop the theory in an elegant way for a large class of nonsmooth MINLPs based on the use of convex composite functions and subdifferentials, although an interpretation for thel 1 norm is also given.

Operations research : applications and algorithms

Abstract: 1. INTRODUCTION TO MODEL BUILDING. An Introduction to Modeling. The Seven-Step Model-Building Process. Examples. 2. BASIC LINEAR ALGEBRA. Matrices and Vectors. Matrices and Systems of Linear Equations. The Gauss-Jordan Method for Solving Systems of Linear Equations. Linear Independence and Linear Dependence. The Inverse of a Matrix. Determinants. 3. INTRODUCTION TO LINEAR PROGRAMMING. What is a Linear Programming Problem? The Graphical Solution of Two-Variable Linear Programming Problems. Special Cases. A Diet Problem. A Work-Scheduling Problem. A Capital Budgeting Problem. Short-term Financial Planning. Blending Problems. Production Process Models. Using Linear Programming to Solve Multiperiod Decision Problems: An Inventory Model. Multiperiod Financial Models. Multiperiod Work Scheduling. 4. THE SIMPLEX ALGORITHM AND GOAL PROGRAMMING. How to Convert an LP to Standard Form. Preview of the Simplex Algorithm. The Simplex Algorithm. Using the Simplex Algorithm to Solve Minimization Problems. Alternative Optimal Solutions. Unbounded LPs. The LINDO Computer Package. Matrix Generators, LINGO, and Scaling of LPs. Degeneracy and the Convergence of the Simplex Algorithm. The Big M Method. The Two-Phase Simplex Method. Unrestricted-in-Sign Variables. Karmarkar"s Method for Solving LPs. Multiattribute Decision-Making in the Absence of Uncertainty: Goal Programming. Solving LPs with Spreadsheets. 5. SENSITIVITY ANALYSIS: AN APPLIED APPROACH. A Graphical Introduction to Sensitivity Analysis. The Computer and Sensitivity Analysis. Managerial Use of Shadow Prices. What Happens to the Optimal z-value if the Current Basis is No Longer Optimal? 6. SENSITIVITY ANALYSIS AND DUALITY. A Graphical Introduction to Sensitivity Analysis. Some Important Formulas. Sensitivity Analysis. Sensitivity Analysis When More Than One Parameter is Changed: The 100% Rule. Finding the Dual of an LP. Economic Interpretation of the Dual Problem. The Dual Theorem and Its Consequences. Shadow Prices. Duality and Sensitivity Analysis. 7. TRANSPORTATION, ASSIGNMENT, AND TRANSSHIPMENT PROBLEMS. Formulating Transportation Problems. Finding Basic Feasible Solutions for Transportation Problems. The Transportation Simplex Method. Sensitivity Analysis for Transportation Problems. Assignment Problems. Transshipment Problems. 8. NETWORK MODELS. Basic Definitions. Shortest Path Problems. Maximum Flow Problems. CPM and PERT. Minimum Cost Network Flow Problems. Minimum Spanning Tree Problems. The Network Simplex Method. 9. INTEGER PROGRAMMING. Introduction to Integer Programming. Formulation Integer Programming Problems. The Branch-and-Bound Method for Solving Pure Integer Programming Problems. The Branch-and-Bound Method for Solving Mixed Integer Programming Problems. Solving Knapsack Problems by the Branch-and-Bound Method. Solving Combinatorial Optimization Problems by the Branch-and-Bound Method. Implicit Enumeration. The Cutting Plane Algorithm. 10. ADVANCED TOPICS IN LINEAR PROGRAMMING. The Revised Simplex Algorithm. The Product Form of the Inverse. Using Column Generation to Solve Large-Scale LPs. The Dantzig-Wolfe Decomposition Algorithm. The Simplex Methods for Upper-Bounded Variables. Karmarkar"s Method for Solving LPs. 11. NONLINEAR PROGRAMMING. Review of Differential Calculus. Introductory Concepts. Convex and Concave Functions. Solving NLPs with One Variable. Golden Section Search. Unconstrained Maximization and Minimization with Several Variables. The Method of Steepest Ascent. Lagrange Multiples. The Kuhn-Tucker Conditions. Quadratic Programming. Separable Programming. The Method of Feasible Directions. Pareto Optimality and Tradeoff Curves. 12. REVIEW OF CALCULUS AND PROBABILITY. Review of Integral Calculus. Differentiation of Integrals. Basic Rules of Probability. Bayes" Rule. Random Variables. Mean Variance and Covariance. The Normal Distribution. Z-Transforms. Review Problems. 13. DECISION MAKING UNDER UNCERTAINTY. Decision Criteria. Utility Theory. Flaws in Expected Utility Maximization: Prospect Theory and Framing Effects. Decision Trees. Bayes" Rule and Decision Trees. Decision Making with Multiple Objectives. The Analytic Hierarchy Process. Review Problems. 14. GAME THEORY. Two-Person Zero-Sum and Constant-Sum Games: Saddle Points. Two-Person Zero-Sum Games: Randomized Strategies, Domination, and Graphical Solution. Linear Programming and Zero-Sum Games. Two-Person Nonconstant-Sum Games. Introduction to n-Person Game Theory. The Core of an n-Person Game. The Shapley Value. 15. DETERMINISTIC EOQ INVENTORY MODELS. Introduction to Basic Inventory Models. The Basic Economic Order Quantity Model. Computing the Optimal Order Quantity When Quantity Discounts Are Allowed. The Continuous Rate EOQ Model. The EOQ Model with Back Orders Allowed. Multiple Product Economic Order Quantity Models. Review Problems. 16. PROBABILISTIC INVENTORY MODELS. Single Period Decision Models. The Concept of Marginal Analysis. The News Vendor Problem: Discrete Demand. The News Vendor Problem: Continuous Demand. Other One-Period Models. The EOQ with Uncertain Demand: the (r, q) and (s,S models). The EOQ with Uncertain Demand: the Service Level Approach to Determining Safety Stock Level. Periodic Review Policy. The ABC Inventory Classification System. Exchange Curves. Review Problems. 17. MARKOV CHAINS. What is a Stochastic Process. What is a Markov Chain? N-Step Transition Probabilities. Classification of States in a Markov Chain. Steady-State Probabilities and Mean First Passage Times. Absorbing Chains. Work-Force Planning Models. 18.DETERMINISTIC DYNAMIC PROGRAMMING. Two Puzzles. A Network Problem. An Inventory Problem. Resource Allocation Problems. Equipment Replacement Problems. Formulating Dynamic Programming Recursions. The Wagner-Whitin Algorithm and the Silver-Meal Heuristic. Forward Recursions. Using Spreadsheets to Solve Dynamic Programming Problems. Review Problems. 19. PROBABILISTIC DYNAMIC PROGRAMMING. When Current Stage Costs are Uncertain but the Next Period"s State is Certain. A Probabilistic Inventory Model. How to Maximize the Probability of a Favorable Event Occurring. Further Examples of Probabilistic Dynamic Programming Formulations. Markov Decision Processes. Review Problems. 20. QUEUING THEORY. Some Queuing Terminology. Modeling Arrival and Service Processes. Birth-Death Processes. M/M/1/GD/o/o Queuing System and the Queuing Formula L=o W, The M/M/1/GD/o Queuing System. The M/M/S/ GD/o/o Queuing System. The M/G/ o/GD/oo and GI/G/o/GD/o/oModels. The M/ G/1/GD/o/o Queuing System. Finite Source Models: The Machine Repair Model. Exponential Queues in Series and Opening Queuing Networks. How to Tell whether Inter-arrival Times and Service Times Are Exponential. The M/G/S/GD/S/o System (Blocked Customers Cleared). Closed Queuing Networks. An Approximation for the G/G/M Queuing System. Priority Queuing Models. Transient Behavior of Queuing Systems. Review Problems. 21.SIMULATION. Basic Terminology. An Example of a Discrete Event Simulation. Random Numbers and Monte Carlo Simulation. An Example of Monte Carlo Simulation. Simulations with Continuous Random Variables. An Example of a Stochastic Simulation. Statistical Analysis in Simulations. Simulation Languages. The Simulation Process. 22.SIMULATION WITH PROCESS MODEL. Simulating an M/M/1 Queuing System. Simulating an M/M/2 System. A Series System. Simulating Open Queuing Networks. Simulating Erlang Service Times. What Else Can Process Models Do? 23. SPREADSHEET SIMULATION WITH @RISK. Introduction to @RISK: The Newsperson Problem. Modeling Cash Flows From A New Product. Bidding Models. Reliability and Warranty Modeling. Risk General Function. Risk Cumulative Function. Risktrigen Function. Creating a Distribution Based on a Point Forecast. Forecasting Income of a Major Corporation. Using Data to Obtain Inputs For New Product Simulations. Playing Craps with @RISK. Project Management. Simulating the NBA Finals. 24. FORECASTING. Moving Average Forecasting Methods. Simple Exponential Smoothing. Holt"s Method: Exponential Smoothing with Trend. Winter"s Method: Exponential Smoothing with Seasonality. Ad Hoc Forecasting, Simple Linear Regression. Fitting Non-Linear Relationships. Multiple Regression. Answers to Selected Problems. Index.

Preprocessing and Probing Techniques for Mixed Integer Programming Problems

Abstract: In the first part of the paper, we present a framework for describing basic techniques to improve the representation of a mixed integer programming problem. We elaborate on identification of infeasibility and redundancy, improvement of bounds and coefficients, and fixing of binary variables. In the second part of the paper, we discuss recent extensions to these basic techniques and elaborate on the investigation and possible uses of logical consequences. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

Optimal trellis-based buffered compression and fast approximations

Abstract: The authors formalize the description of the buffer-constrained adaptive quantization problem. For a given set of admissible quantizers used to code a discrete nonstationary signal sequence in a buffer-constrained environment, they formulate the optimal solution. They also develop slightly suboptimal but much faster approximations. These solutions are valid for any globally minimum distortion criterion, which is additive over the individual elements of the sequence. As a first step, they define the problem as one of constrained, discrete optimization and establish its equivalence to some of the problems studied in the field of integer programming. Forward dynamic programming using the Viterbi algorithm is shown to provide a way of computing the optimal solution. Then, they provide a heuristic algorithm based on Lagrangian optimization using an operational rate-distortion framework that, with computing complexity reduced by an order of magnitude, approaches the optimally achievable performance. The algorithms can serve as a benchmark for assessing the performance of buffer control strategies and are useful for applications such as multimedia workstation displays, video encoding for CD-ROMs, and buffered JPEG coding environments, where processing delay is not a concern but decoding buffer size has to be minimized. >

Control of vector discrete-event systems. II. Controller synthesis

Abstract: This paper shows that associated control synthesis problems of Ramadge-Wonham type can be efficiently solved under mild technical restrictions, using elementary methods combined with linear integer programming. Both memoryless and dynamic state feedback are treated. Structural conditions are provided under which the controller can be computed in "closed-form," namely represented as a VDES or "generalized" VDES (GVDES). The results are applied to a model of an AGV system, adapted from work of Holloway and Krogh (1990). >

The Multi-Airport Ground-Holding Problem in Air Traffic Control

Abstract: Motivated by the important problem of congestion costs (they were estimated to be $2 billion in 1991) in air transportation and observing that ground delays are more preferable than airborne delays, we have formulated and studied several integer programming models to assign ground-holding delays optimally in a general network of airports, so that the total (ground plus airborne) delay cost of all flights is minimized. All previous research on this problem has been restricted to the single-airport case, which neglects “down-the-road” effects due to transmission of delays between successive flights performed by the same aircraft. We formulate several models, and then propose a heuristic algorithm which finds a feasible solution to the integer program by rounding the optimal solution of the LP relaxation. Finally, we present extensive computational results with the goal of obtaining qualitative insights on the behavior of the problem under various combinations of the input parameters. We demonstrate that the...

Simple and Fast Algorithms for Linear and Integer Programs with Two Variables Per Inequality

Abstract: The authors present an $O(mn^2 \log m)$ algorithm for solving feasibility in linear programs with up to two variables per inequality which is derived directly from the Fourier--Motzkin elimination method. (The number of variables and inequalities are denoted by $n$ and $m$, respectively.) The running time of the algorithm dominates that of the best known algorithm for the problem, and is far simpler. Integer programming on monotone inequalities, i.e., inequalities where the coefficients are of opposite sign, is then considered. This problem includes as a special case the simultaneous approximation of a rational vector with specified accuracy, which is known to be NP-complete. However, it is shown that both a feasible solution and an optimal solution with respect to an arbitrary objective function can be computed in pseudo-polynomial time.

Minimizing Transportation and Inventory Costs for Several Products on a Single Link

Abstract: This paper deals with the problem of determining the frequencies at which several products have to be shipped on a common link to minimize the sum of transportation and inventory costs. A set of feasible shipping frequencies is given. Transportation costs are supposed to be proportional to the number of journeys performed by vehicles of a given capacity. Vehicles may or may not be supposed to carry out completely all materials available, and products assigned to different frequencies may or may not share the same truck. Integer and mixed integer linear programming models are formulated for each of the resulting four situations, and their properties are investigated. In particular, we show that allowing products to be split among several shipping frequencies makes trucks traveling at high frequencies to be filled up completely. In this situation, trucks may always be loaded with products shipped at the same frequency.

A zero-one implicit enumeration method for optimizing investments in transmission expansion planning

Abstract: This paper presents a zero-one implicit enumeration method applied to an integer programming subproblem which has to be solved as part of a more general process of obtaining an optimal solution for a transmission expansion planning problem by hierarchical Benders decomposition. The proposed algorithm has been successfully implemented and tested in a real-life system. The reasons why the implicit enumeration approach is particularly suited for the static expansion planning problem are fully discussed in the paper. >

Water‐Supply Operations during Drought: Continuous Hedging Rule

Abstract: Demand-management policy rules are sought during drought and impending drought for a water system consisting of a reservoir dedicated only to water supply. The creation of such rules requires solution of a nonlinear, nonseparable mathematical programming problem. A polytope search algorithm using a combination of simulation and optimization is compared to an iterative mixed integer programming method to determine the parameters of continuous demand management rules. The signal used for calling rationing is a trigger volume given in terms of months of demand (as a volume) that are needed in storage. When the sum of actual storage plus anticipated inflow is less than the trigger volume, rationing is initiated. The extent of rationing or demand reduction that is required is determined by the ration of the sum of storage plus inflow to the trigger volume. The two methodologies for parameter determination are compared using as a criteria the maximum shortage that occurs over some planning period.

Dynamic Ground-Holding Policies for a Network of Airports

Abstract: The yearly congestion costs in the U.S. airline industry are estimated to be of the order of $2 billion. In P. B. Vranas, Dimitris J. Bertsimas, and A. R. Odoni, The multi-airport ground-holding problem in air traffic control, Operations Research, Vol. 42, pp. 249–261, 1994, we introduced and studied generic integer programming models for the static multi-airport ground-holding problem (GHP), the problem of assigning optimal ground holding delays in a general network of airports, so that the total (ground plus airborne) delay cost of all flights is minimized. The present paper is the first attempt to address the multi-airport GHP in a dynamic environment. We propose algorithms to update ground-holding decisions as time progresses and more accurate weather (hence capacity) forecasts become available. We propose several pure IP formulations (most of them 0–1), which have the important advantages of being remarkably compact while capturing the essential aspects of the problem and of being sufficiently flexib...

An Evolutionary Algorithm for Integer Programming

Abstract: The mutation distribution of evolutionary algorithms usually is oriented at the type of the search space. Typical examples are binomial distributions for binary strings in genetic algorithms or normal distributions for real valued vectors in evolution strategies and evolutionary programming. This paper is devoted to the construction of a mutation distribution for unbounded integer search spaces. The principle of maximum entropy is used to select a specific distribution from numerous potential candidates. The resulting evolutionary algorithm is tested for five nonlinear integer problems.

EVBDD-based algorithms for integer linear programming, spectral transformation, and function decomposition

Abstract: Edge-Valued Binary-Decision Diagrams (EVBDD's) are directed acyclic graphs that can represent and manipulate integer functions as effectively as Ordered Binary-Decision Diagrams OBDD's) do for Boolean functions. They have been used in logic verification for showing the equivalence between Boolean functions and arithmetic functions. In this paper, we present EVBDD-based algorithms for solving integer linear programs, computing spectral coefficients of Boolean functions, and performing function decomposition. These algorithms have been implemented in C under the SIS environment and experimental results are provided. >

Weighted k-cardinality trees: complexity and polyhedral structure

Abstract: We consider the k-CARD TREE problem, i.e., the problem of finding in a given undirected graph G a subtree with k edges, having minimum weight. Applications of this problem arise in oil-field leasing and facility layout. Although the general problem is shown to be strongly NP hard, it can be solved in polynomial time if G is itself a tree. We give an integer programming formulation of k-CARD TREE and an efficient exact separation routine for a set of generalized subtour elimination constraints. The polyhedral structure of the convex hull of the integer solutions is studied. © 1994 by John Wiley & Sons, Inc.

An Exact Algorithm for Selecting Partial Scan Flip-Flops

Abstract: We develop an exact algorithm for selecting flip-flops in partial scan designs to break all feedback cycles. The main ideas that allowus to solve this hard problemexactly for large, practical instances are - graph transformations, a partitioning scheme used in the branch and bound procedure, and pruning techniques based on an integer linear programming formulation of the minimum feedback vertex set (MFVS) problem.We have obtained optimum solutions for the ISCAS '89 benchmark circuits and several production VLSI circuits within reasonable computation time. For example, the optimal number of scan flip-flops required to eliminate all cycles except self-loops in the circuit s38417 is 374. This optimal solution was obtained in 32 CPU seconds on a SUN Sparc 2 workstation.

Automatic Data Layout Using 0-1 Integer Programming

Abstract: In the optimization world now, even the use of considering the number of processors for parallel programming is carefully considered. The goal of high-level languages is to provide a simple yet efficient machine-independent parallel programming model. The programmer’s data parallel programs should be able to compile and executed with a good performance on many different architectures. However the most sophisticated compiler may not be able to compensate for a poor chosen data layout since many compiler decisions are driven by the data layout specified in the program. Hence the procedure to select a good and efficient data layout is important. The data layout selection problem can be formulated as a 0-1 integer programming problem.

Static scheduling and code generation from dynamic dataflow graphs with integer-valued control streams

Abstract: This paper extends the token flow model of Buck (see Memorandum No. UCB/ERL M93/69 (Ph.D. Thesis), EECS Dept., University of California, Berkeley, September 1993) and Lee (see IEEE Transactions on Parallel and Distributed Systems, vol.2, no.2, 1991), an analytical model for the behavior of dataflow graphs with data-dependent control flow, in two ways: dataflow actor execution may depend on integer rather than Boolean, control tokens, and multiphase implementations of actors are permitted. These extensions permit data-dependent iteration to be modelled more naturally, reduce the memory required for implementations, and result in bounded-memory solutions in more cases than before. A method for generating efficient single-processor programs from the graphs is also described. >

An optimal spare-capacity assignment model for survivable networks with hop limits

Abstract: The paper presents a new algorithm for spare-capacity assignment in survivable networks which use cross-connect systems as transmission hubs. The algorithm minimises total cost of spare capacity for required levels of network restoration following a single link failure (e.g. fibre cut) and limits the restoration routes to any predetermined hop limit. The algorithm is composed of two parts: Part 1 relies on a linear programming (LP) formulation (min-max) from which a lower bound solution is found; Part 2 rounds up the solution of Part 1 and uses a series of related LP schemes (max-flow), aimed at tightening the rounded-up assignment to a practical optimal solution which also supplies optimal restoration routes and capacities. For moderate networks an integer programming formulation of Part 1 can be used to obtain an optimal solution. A network example is analysed to illustrate the algorithm developed and to demonstrate its superiority over other schemes published in this area. In addition, a valuable trade-off between spare capacity and hop limit is presented.

Static and Dynamic Layout Problems with Varying Areas

Abstract: This paper describes a two-step algorithm for solving the layout problem while assuming the departments can have varying areas. The first step solves a quadratic assignment problem formulation of the problem using a heuristic cutting plane routine. The second step solves a mixed-integer linear programming prob- lem to find the desired block diagram layout. The algorithm incorporates two concepts to make the solu- tions more practical. First, rearrangement costs are simultaneously considered along with flow costs in solving a dynamic layout problem involving multiple time periods. It is the only algorithm to solve a general dynamic layout problem with varying department areas. Second, regular department shapes are maintained by requiring all departments to be rectangular. Its formulation for doing this is more efficient than previous algorithms.

Mass exchange networks for waste minimization : a simultaneous approach : Process design

Abstract: The synthesis problem of mass-exchange networks is addressed, where a number of process streams, rich in terms of certain components (typically pollutants) are integrated with lean process or utility streams in order to meet process specifications (e.g. environmental regulations) on their final compositions. A mathematical programming approach is adopted, based on a hyperstructure representation of the mass exchange network. The synthesis problem is then formulated as a mixed integer nonlinear programming optimization problem (MINLP), where both network operating and investment cost are optimized simultaneously. In contrast to all previously published work that simplifies the problem by assuming decomposition based on the concept of pinch, we treat the synthesis of mass exchange networks without decomposition. In the first part, the case of a single component is examined, whereas mass exchange in multiple components can be handled in a straightforward manner. In the second part the approach is applied to reactive mass exchange networks, where chemical sorption takes place. In the third part, the mass exchange hyperstructure is extended to include regeneration of the lean streams. A number of examples illustrate the applicability of the proposed approach to several problems of waste minimization and demonstrate the impact of simultaneously minimizing operating and investment cost

Parallel Branch-and-Bound Algorithms for General Mixed Integer Programming on the CM-5

Abstract: This “proof of concept” paper describes parallel solution of general mixed integer programs by a branch-and-bound algorithm on the CM-5 multiprocessing system. It goes beyond prior parallel branch-and-bound work by implementing a reasonably realistic general-purpose mixed integer programming algorithm, as opposed to a specialized method for a narrow class of problems. It shows how to use the capabilities of the CM-5 to produce an efficient parallel implementation employing centrally controlled search, achieving near-linear speedups using 64–128 processors on a variety of difficult problems derived from real applications. In concrete terms, a problem requiring half an hour to solve on a SPARC-2 workstation might be solved in 15–20 seconds, and a problem originally taking a week might be reduced to about an hour. Central search control does have limitations, and some final computational experiments begin to address the merits of more decentralized options.

Petri net structural analysis for supervisory control

Abstract: The primary motivation for this research is to show how Petri nets may be efficiently used within the framework of supervisory control. In particular, the paper discusses how integer programming techniques for Petri net models may be used to validate supervisors for the control of discrete event systems. We consider a class of place/transition nets, called elementary composed state machines. The reachability problem for this class can be solved by a modification of classical incidence matrix analysis. In fact it is possible to derive a set of linear inequalities that exactly defines the set of reachable markings. Finally, we show how important properties of discrete event systems, such as the absence of blocking states or controllability, may be analyzed by integer programming techniques. >

0-1 Quadratic programming approach for optimum solutions of two scheduling problems

Abstract: Two scheduling problems are considered: (1) scheduling n jobs non-preemptively on a single machine to minimize total weighted earliness and tardiness (WET); (2) scheduling n jobs non-preemptively on two parallel identical processors to minimize weighted mean flow time. In the second problem, a pre-ordering of the jobs is assumed that must be satisfied for any set of jobs scheduled on each specific machine. Both problems are known to be NP-complete. A 0-1 quadratic assignment formulation of the problems is presented. An equivalent 0-1 mixed integer linear programming approach for the problems are considered and a numerical example is given. The formulations presented enable one to use optimal and heuristic available algorithms of 0-1 quadratic assignment for the problems considered here.