Efficient approaches for furnace loading of cylindrical parts

This paper addresses the heat treatment operation in a manufacturing plant that produces different types of cylindrical parts. The immediate prior process to heat treatment is furnace-loading, where parts are loaded into baskets. The furnace-loading process is complex and involves issues relating to geometry, and heterogeneity in the parts and in their processing requirements. Currently, furnace-loading is accomplished by operator ingenuity; consequently, the parts loaded in heat treatment often do not use furnace capacity adequately. Efficiency in furnace operation can be achieved by improving basket utilization, which is determined by the furnace-loading process. This paper describes the development of integer and mixed integer LP models for 3D loading of cylindrical parts into furnace baskets. The models consider the exact location of parts to be loaded on the basket and incorporate three models with different objectives; the first addresses the nesting of parts within one another, the second addresses the number of basket layers used, and the third addresses the number of baskets used.     

The value function of a transportation problem

We investigate the value of an optimal transportation problem with the maximization objective as a function of costs and vectors of production and consumption. The value is concave in production. For generic costs, the numbers of linearity domains and peak points are independent of costs and consumption. The peak points are determined by an auxiliary assignment problem. The volumes of the linearity domains are independent of costs while their dependence on consumption can be expressed via the multinomial distribution.     

A transportation problem formulation for the MAC Airlift Planning problem

The Military Airlift Command (MAC) is responsible for planning the allocation of airlift resources for the movement of cargo and passengers. A heuristic algorithm, the Airlift Planning Algorithm (APA), has recently been developed under subcontract to the Oak Ridge National Laboratory to assist MAC in scheduling airlift resources. In this paper, we present a transportation problem formulation which can be used as a preprocessor to the APA or as an estimator for the APA. This paper examines the robustness and sensitivity of the transportation problem formulation. In particular, the performance of the APA improves by approximately 10% when the transportation problem is used as a preprocessor for two hypothetical problems and improves by up to 50% for derived airlift constrained problems.     

A strongly polynomial algorithm for the transportation problem

For the (linear) transportation problem withm supply nodes,n demand nodes andk feasible arcs we describe an algorithm which runs in time proportional tom logm(k + n logn) (assuming w.l.o.g.mn). The algorithm uses excess scaling. The complexity bound is a slight improvement over the bound achieved by an application of a min-cost-flow algorithm of Orlin to the transportation problem.Corresponding author. Research supported in part by grant no. I-84-095.06/88 of the German—Israeli-Foundation for Scientific Research and Development.     

Optimizing production and transportation in a commit-to-delivery business mode

In problems involving the simultaneous optimization of production and transportation, the requirement that an order can only be shipped once its production has been completed is a natural one. One example is a problem of optimizing shipping costs subject to a production capacity constraint studied recently by Stecke and Zhao. Here we present an integer programming formulation for the case in which only completed orders can be shipped that leads to very tight dual bounds and enables one to solve instances of significant size to optimality.     

A generalized knapsack problem with variable coefficients

The ordinary knapsack problem is to find the optimal combination of items to be packed in a knapsack under a single constraint on the total allowable resources, where all coefficients in the objective function and in the constraint are constant.In this paper, a generalized knapsack problem with coefficients depending on variable parameters is proposed and discussed. Developing an effective branch and bound algorithm for this problem, the concept of relaxation and the efficiency function introduced here will play important roles. Furthermore, a relation between the algorithm and the dynamic programming approach is discussed, and subsequently it will be shown that the ordinary 0–1 knapsack problem, the linear programming knapsack problem and the single constrained linear programming problem with upper-bounded variables are special cases of the interested problem. Finally, practical applications of the problem and its computational experiences will be shown.     

The transportation problem with exclusionary side constraints

We consider the so-called Transportation Problem with Exclusionary Side Constraints (TPESC), which is a generalization of the ordinary transportation problem. We confirm that the TPESC is NP-hard, and we analyze the complexity of different special cases. For instance, we show that in case of a bounded number of suppliers, a pseudo-polynomial time algorithm exists, whereas the case of two demand nodes is already hard to approximate within a constant factor (unless P = NP). This research was partially supported by FWO Grant No. G.0114.03.     

The stochastic transportation problem with single sourcing

We propose a branch-and-price algorithm for solving a class of stochastic transportation problems with single-sourcing constraints. Our approach allows for general demand distributions, nonlinear cost structures, and capacity expansion opportunities. The pricing problem is a knapsack problem with variable item sizes and concave costs that is interesting in its own right. We perform an extensive set of computational experiments illustrating the efficacy of our approach. In addition, we study the cost of the single-sourcing constraints.     

A column generation approach for determining optimal fleet mix,schedules, and transshipment facility locations for a vessel transportation problem

This paper presents a column generation approach for a storage replenishment transportation-scheduling problem. The problem is concerned with determining an optimal combination of multiple-vessel schedules to transport a product from multiple sources to different destinations based on demand and storage information at the destinations, along with cost-effective optimal strategic locations for temporary transshipment storage facilities. Such problems are faced by oil/trucking companies that own a fleet of vessels (oil tankers or trucks) and have the option of chartering additional vessels to transport a product (crude oil or gasoline) to customers (storage facilities or gas stations) based on agreed upon contracts. An integer-programing model that determines a minimum-cost operation of vessels based on implicitly representing feasible shipping schedules is developed in this paper. Due to the moderate number of constraints but an overwhelming number of columns in the model, a column generation approach is devised to solve the continuous relaxation of the model, which is then coordinated with a sequential fixing heuristic in order to solve the discrete problem. Computational results are presented for a range of test problems to demonstrate the efficacy of the proposed approach.     

The transportation problem with exclusionary side constraints and two branch-and-bound algorithms

The transportation problem with exclusionary side constraints, a practical distribution and logistics problem, is formulated as a 0–1 mixed integer programming model. Two branch-and-bound (B&B) algorithms are developed and implemented in this study to solve this problem. Both algorithms use the Driebeek penalties to strengthen the lower bounds so as to fathom some of the subproblems, to peg variables, and to guide the selection of separation variables. One algorithm also strongly exploits the problem structure in selecting separation variables in order to find feasible solutions sooner. To take advantage of the underlying network structure of the problem, the algorithms employ the primal network simplex method to solve network relaxations of the problem. A computational experiment was conducted to test the performance of the algorithms and to characterize the problem difficulty. The commercial mixed integer programming software CPLEX and an existing special purpose algorithm specifically designed for this problem were used as benchmarks to measure the performance of the algorithms. Computational results show that the new algorithms completely dominate the existing special purpose algorithm and run from two to three orders of magnitude faster than CPLEX.     

Constrained integer linear fractional programming problem

An integer linear fractional programming problem, whose integer solution is required to satisfy any h out of given n sets of constraints has been discussed in this paper. Method for ranking and scanning all integer points has also been developed and a numerical illustration is included in support of theory.     

A fuzzy approach to the transportation problem

The transportation problem with fuzzy supply values of the deliverers and with fuzzy demand values of the receivers is analysed. For the solution of the problem the technique of parametric programming is used. This makes it possible to obtain not only the maximizing solution (according to the Bellman-Zadeh criterion) but also other alternatives close to the optimal solution.     

Solving fixed charge transportation problem with interval parameters

In the present paper the fixed charge transportation problem under uncertainty, particularly when parameters are given in interval forms, is formulated. In this case it is assumed that both cost and constraint parameters are arrived in interval numbers. Considering two different order relations for interval numbers, two solution procedures are developed in order to obtain an optimal solution for interval fixed charge transportation problem (IFCTP). In addition, the two order relations are compared to give a better comprehension of their differences. Furthermore, numerical examples are provided to illustrate each of solution procedures.     

Penalty for zero–one integer equivalent problem

The equivalence of zero–one integer programming and a concave quadratic penalty function problem has been shown by Raghavachari, for a sufficiently large value of the penalty. A lower bound for this penalty is obtained here, which in specific cases cannot be reduced.This research was supported in part by the Computer Science Section of the National Science Foundation under Research Grant MCS 8101214.     

Capacitated transportation problem with bounds on RIM conditions

Motivated by dead-mileage problem assessed in terms of running empty buses from various depots to starting points, we consider a class of the capacitated transportation problems with bounds on total availabilities at sources and total destination requirements. It is often difficult to solve such problems and the present paper establishes their equivalence with a balanced capacitated transportation problem which can be easily solved by existing methods. Sometimes, total flow in transportation problem is also specified by some external decision maker because of budget/political consideration and optimal solution of such problem is of practical interest to the decision maker and has motivated us to discuss such problem. Various situations arising in unbalanced capacitated transportation problems have been discussed in the present paper as a particular case of original problem. In addition, we have discussed paradoxical situation in a balanced capacitated transportation problem and have obtained the paradoxical solution by solving one of the unbalanced problems. Numerical illustrations are included in support of theory.     

Incremental and encoding formulations for Mixed Integer Programming

The standard way to represent a choice between n$n$ alternatives in Mixed Integer Programming is through n$n$ binary variables that add up to one. Unfortunately, this approach commonly leads to unbalanced branch-and-bound trees and diminished solver performance. In this paper, we present an encoding formulation framework that encompasses and expands existing approaches to mitigate this behavior. Through this framework, we generalize the incremental formulation for piecewise linear functions to any finite union of polyhedra with identical recession cones.     

Mixed integer programming for a special logic constrained optimal control problem

Integrating logical constraints into optimal control problems is not an easy task. In fact, optimal control problems are usually continuous while logical constraints are naturally expressed by integer (binary) variables. In this article we are interested is a particular form of an LQR optimal control problem: the energy (control L² norm) is to be minimized, system dynamic is linear and logical constraints on the control use are to be fulfilled. Even if the starting continuous problem is not a complicated one, difficulties arise when integrating the additional logical constraints. First, we will present two different ways of modeling the problem, both of them leading us to Mixed Integer Problems. Furthermore, algorithms (Generalized Outer Approximation, Benders Decomposition and Branch and Cut) are applied on each model and results analyzed. We also present a Benders Decomposition algorithm variant that is adapted to our problem (taking into account its particular form) and we will conclude by looking at the optimal solutions obtained in an interesting physical example: the harmonic spring.     

Approximation algorithms for the transportation problem with market choice and related models

Given facilities with capacities and clients with penalties and demands, the transportation problem with market choice consists in finding the minimum-cost way to partition the clients into unserved clients, paying the penalties, and into served clients, paying the transportation cost to serve them.     

Extended Cutset Inequalities for the Network Power Consumption Problem

In this paper, we enhance the MIP formulation for the Network Power Consumption problem, proposed by Giroire et al. We derive cutting planes, extending the wellknown cutset inequalities, and report on preliminary computations.     

Economic planning with institutional price constraints for a decentralized economy

This paper describes the relationship between market prices and shadow prices when the economy has general types of institutional price constraints. We consider a decentralized linear economy where market prices quide the decentralized behavior of each activity and the shadow prices measure the social values of resources. To measure the social values, we introduce a social objective criterion. Hence, our approach could be regarded as a central economic price control with institutional price constraints for a decentralized economy. A simple example is employed to graphically illustrate the wedges between market prices and shadow prices. It has been shown that our problem can be solved through mixed integer linear programming techniques.