Local search techniques for the generalized resource constrained project scheduling problem

In this article we address the problem of scheduling a single project network with both precedence and resource constraints through the use of a local search technique. We choose a solution definition which guarantees precedence feasibility, allowing the procedure to focus on overcoming resource infeasibility. We use the 110-problem data set of Patterson to test our procedure. Our results indicate a significant improvement over the best heuristic results reported to date for these problems (Bell and Han [1]). Two major advantages of the local search algorithm are its ability to handle arbitrary objective functions and constraints and its effectiveness over a wide range of problem sizes. We present a problem example with an objective function and resource constraints which include nonlinear and non-continuous components, which are easily considered by the procedure. The results of our algorithm are significantly better than random solutions to the problem. © 1993 John Wiley & Sons, Inc.     

Lagrangian relaxation approach to the targeting problem

In this paper, we consider a new weapon–target allocation problem with the objective of minimizing the overall firing cost. The problem is formulated as a nonlinear integer programming model. We applied Lagrangian relaxation and a branch‐and‐bound method to the problem after transforming the nonlinear constraints into linear ones. An efficient primal heuristic is developed to find a feasible solution to the problem to facilitate the procedure. In the branch‐and‐bound method, three different branching rules are considered and the performances are evaluated. Computational results using randomly generated data are presented. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 640–653, 1999     

An algorithm (GIPC2) for solving integer programming problems with separable nonlinear objective functions

This paper presents an algorithm for solving the integer programming problem possessing a separable nonlinear objective function subject to linear constraints. The method is based on a generalization of the Balas implicit enumeration scheme. Computational experience is given for a set of seventeen linear and seventeen nonlinear test problems. The results indicate that the algorithm can solve the nonlinear integer programming problem in roughly the equivalent time required to solve the linear integer programming problem of similar size with existing algorithms. Although the algorithm is specifically designed to solve the nonlinear problem, the results indicate that the algorithm compares favorably with the Branch and Bound algorithm in the solution of linear integer programming problems.     

Approximating nondominated sets in continuous multiobjective optimization problems

Many important problems in Operations Research and Statistics require the computation of nondominated (or Pareto or efficient) sets. This task may be currently undertaken efficiently for discrete sets of alternatives or for continuous sets under special and fairly tight structural conditions. Under more general continuous settings, parametric characterisations of the nondominated set, for example through convex combinations of the objective functions or ε‐constrained problems, or discretizations‐based approaches, pose several problems. In this paper, the lack of a general approach to approximate the nondominated set in continuous multiobjective problems is addressed. Our simulation‐based procedure only requires to sample from the set of alternatives and check whether an alternative dominates another. Stopping rules, efficient sampling schemes, and procedures to check for dominance are proposed. A continuous approximation to the nondominated set is obtained by fitting a surface through the points of a discrete approximation, using a local (robust) regression method. Other actions like clustering and projecting points onto the frontier are required in nonconvex feasible regions and nonconnected Pareto sets. In a sense, our method may be seen as an evolutionary algorithm with a variable population size. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005.     

The use of dynamic programming methodology for the solution of a class of nonlinear programming problems

This paper presents an application of a method for finding the global solution to a problem in integers with a separable objective function of a very general form. This report shows that there is a relationship between an integer problem with a separable nonlinear objective function and many constraints and a series of nonlinear problems with only a single constraint, each of which can be solved sequentially using dynamic programming. The first solution to any of the individual smaller problems that satisfies the original constraints in addition, will be the optimal solution to the multiply-constrained problem.     

Minimax resource allocation problems with ordering constraints

Resource allocation problems consider the allocation of limited resources among numerous competing activities. We address an allocation problem with multiple knapsack resource constraints. The activities are grouped into disjoint sets. Ordering constraints are imposed on the activities within each set, so that the level of one activity cannot exceed the level of another activity in the same set. The objective function is of the minimax type and each performance function is a nonlinear, strictly decreasing and continuous function of a single variable. Applications for such resource allocation problems are found, for example, in high-tech industries confronted with large-scale and complex production planning problems. We present two algorithms to solve the allocation problem with ordering constraints. The first one uses characterization of the optimal decision variables to apply a search method. The second algorithm solves a sequence of problems, each in the format of the original problem without ordering constraints. Whereas the computational effort of the first algorithm depends on the desired degree of accuracy even for linear performance functions, the effort of the latter algorithm is polynomial for certain classes of performance functions. © 1994 John Wiley & Sons, Inc.     

Solving generalized transportation problems via pure transportation problems

This paper investigates certain issues of coefficient sensitivity in generalized network problems when such problems have small gains or losses. In these instances, it might be computationally advantageous to temporarily ignore these gains or losses and solve the resultant “pure” network problem. Subsequently, the optimal solution to the pure problem could be used to derive the optimal solution to the original generalized network problem. In this paper we focus on generalized transportation problems and consider the following question: Given an optimal solution to the pure transportation problem, under what conditions will the optimal solution to the original generalized transportation problem have the same basic variables? We study special cases of the generalized transportation problem in terms of convexity with respect to a basis. For the special case when all gains or losses are identical, we show that convexity holds. We use this result to determine conditions on the magnitude of the gains or losses such that the optimal solutions to both the generalized transportation problem and the associated pure transportation problem have the same basic variables. For more general cases, we establish sufficient conditions for convexity and feasibility. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 666–685, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10034     

Approximation algorithms for general one‐warehouse multi‐retailer systems

Logistical planning problems are complicated in practice because planners have to deal with the challenges of demand planning and supply replenishment, while taking into account the issues of (i) inventory perishability and storage charges, (ii) management of backlog and/or lost sales, and (iii) cost saving opportunities due to economies of scale in order replenishment and transportation. It is therefore not surprising that many logistical planning problems are computationally difficult, and finding a good solution to these problems necessitates the development of many ad hoc algorithmic procedures to address various features of the planning problems. In this article, we identify simple conditions and structural properties associated with these logistical planning problems in which the warehouse is managed as a cross‐docking facility. Despite the nonlinear cost structures in the problems, we show that a solution that is within ε‐optimality can be obtained by solving a related piece‐wise linear concave cost multi‐commodity network flow problem. An immediate consequence of this result is that certain classes of logistical planning problems can be approximated by a factor of (1 + ε) in polynomial time. This significantly improves upon the results found in literature for these classes of problems. We also show that the piece‐wise linear concave cost network flow problem can be approximated to within a logarithmic factor via a large scale linear programming relaxation. We use polymatroidal constraints to capture the piece‐wise concavity feature of the cost functions. This gives rise to a unified and generic LP‐based approach for a large class of complicated logistical planning problems. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2009     

Minimizing logistic costs in multistage supply chains

We consider the problem in which a set of products has to be shipped from a common origin to a common destination through one or several intermediate nodes with the objective of minimizing the sum of inventory and transportation costs when a set of possible shipping frequencies is given on each link. From the theoretical point of view, the main issue is the computation of the inventory cost in the intermediate nodes. From the computational point of view, given that the simpler single link problem is known to be NP-hard, we present four classes of heuristic algorithms. The first two classes are based on the decomposition of the sequence in links, the third class on the adaptation of the EOQ-type solution known for the continuous case, and the fourth on the optimal solution of a simpler problem through dynamic programming techniques. Finally, we compare them on a set of randomly generated problem instances. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 399–417, 1999     

Solving incremental quantity discounted transportation problems by vertex ranking

Logistics managers often encounter incremental quantity discounts when choosing the best transportation mode to use. This could occur when there is a choice of road, rail, or water modes to move freight from a set of supply points to various destinations. The selection of mode depends upon the amount to be moved and the costs, both continuous and fixed, associated with each mode. This can be modeled as a transportation problem with a piecewise-linear objective function. In this paper, we present a vertex ranking algorithm to solve the incremental quantity discounted transportation problem. Computational results for various test problems are presented and discussed.     

Dual-ascent methods for large-scale multicommodity flow problems

The capacitated multicommodity network flow problem presents itself in a number of problem contexts including transportation, communication, and production. To solve the large-scale multicommodity flow problems encountered in these fields, we develop dual-ascent heuristics and a primal solution generator. The dual-ascent solutions, in addition to determining lower bounds on the optimal objective function value, provide advanced starting solutions for use with primal-based solution techniques. The primal solution generator uses the dual-ascent solution to obtain heuristically primal solutions to the multicommodity flow problems. Computational experiments performed on three test problem sets show that the dual-ascent and primal heuristic procedures typically determine nearoptimal solutions quickly. In addition, by using the dual-ascent procedure to obtain advanced starting solutions, run times for optimal multicommodity flow procedures are reduced significantly and greatly improved solutions are obtained by the new primal solution generator. © 1993 John Wiley & Sons, Inc.     

Solving time/cost trade-off problems with discounted cash flows using generalized benders decomposition

In this article we consider a project scheduling problem where there are cash flows throughout the life of the project and where shorter activity durations can be attained by incurring greater direct costs. In particular, the objective of this problem is to determine the activity durations and a schedule of activity start times so that the net present value of cash flows is maximized. We formulate this problem as a mixed-integer nonlinear program which is amenable to solution using the generalized Benders decomposition technique developed by Geoffrion. We test the algorithm on 140 project scheduling problems, the largest of which contains 30 nodes and 64 activities. Our computational results are quite encouraging inasmuch as 123 of the 140 problems require less than 1 CPU second of solution time. © 1993 John Wiley & Sons, Inc.     

Outbound supply chain network design with mode selection and lead time considerations

We present a large‐scale network design model for the outbound supply chain of an automotive company that considers transportation mode selection (road vs. rail) and explicitly models the relationship between lead times and the volume of flow through the nodes of the network. We formulate the problem as a nonlinear zero‐one integer program, reformulate it to obtain a linear integer model, and develop a Lagrangian heuristic for its solution that gives near‐optimal results in reasonable time. We also present scenario analyses that examine the behavior of the supply chain under different parameter settings and the performance of the solution procedures under different experimental conditions. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Production and transport logistics scheduling with two transport mode choices

This paper considers a new class of scheduling problems arising in logistics systems in which two different transportation modes are available at the stage of product delivery. The mode with the shorter transportation time charges a higher cost. Each job ordered by the customer is first processed in the manufacturing facility and then transported to the customer. There is a due date for each job to arrive to the customer. Our approach integrates the machine scheduling problem in the manufacturing stage with the transportation mode selection problem in the delivery stage to achieve the global maximum benefit. In addition to studying the NP‐hard special case in which no tardy job is allowed, we consider in detail the problem when minimizing the sum of the total transportation cost and the total weighted tardiness cost is the objective. We provide a branch and bound algorithm with two different lower bounds. The effectiveness of the two lower bounds is discussed and compared. We also provide a mathematical model that is solvable by CPLEX. Computational results show that our branch and bound algorithm is more efficient than CPLEX. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005     

A selective newsvendor approach to order management

Consider a supplier offering a product to several potential demand sources, each with a unique revenue, size, and probability that it will materialize. Given a long procurement lead time, the supplier must choose the orders to pursue and the total quantity to procure prior to the selling season. We model this as a selective newsvendor problem of maximizing profits where the total (random) demand is given by the set of pursued orders. Given that the dimensionality of a mixed‐integer linear programming formulation of the problem increases exponentially with the number of potential orders, we develop both a tailored exact algorithm based on the L‐shaped method for two‐stage stochastic programming as well as a heuristic method. We also extend our solution approach to account for piecewise‐linear cost and revenue functions as well as a multiperiod setting. Extensive experimentation indicates that our exact approach rapidly finds optimal solutions with three times as many orders as a state‐of‐the‐art commercial solver. In addition, our heuristic approach provides average gaps of less than 1% for the largest problems that can be solved exactly. Observing that the gaps decrease as problem size grows, we expect the heuristic approach to work well for large problem instances. © 2008 Wiley Periodicals, Inc. Naval Research Logistics 2008     

The continuous‐time single‐sourcing problem with capacity expansion opportunities

This paper develops a new model for allocating demand from retailers (or customers) to a set of production/storage facilities. A producer manufactures a product in multiple production facilities, and faces demand from a set of retailers. The objective is to decide which of the production facilities should satisfy each retailer's demand, in order minimize total production, inventory holding, and assignment costs (where the latter may include, for instance, variable production costs and transportation costs). Demand occurs continuously in time at a deterministic rate at each retailer, while each production facility faces fixed‐charge production costs and linear holding costs. We first consider an uncapacitated model, which we generalize to allow for production or storage capacities. We then explore situations with capacity expansion opportunities. Our solution approach employs a column generation procedure, as well as greedy and local improvement heuristic approaches. A broad class of randomly generated test problems demonstrates that these heuristics find high quality solutions for this large‐scale cross‐facility planning problem using a modest amount of computation time. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005.     

A note on duality in homogeneous fractional programming

For a linear fractional programming problem, Sharma and Swarup have constructed a dual problem, also a linear fractional program, in which the objective functions of both primal and dual problems are the same. Craven and Mond have extended this result to a nonlinear fractional programming problem with linear constraints, and a dual problem for which the objective function is the same as that of the primal. This theorem is now further extended from linear to differentiable convex constraints.     

A lagrangean relaxation algorithm for the constrained matrix problem

We present variants of a convergent Lagrangean relaxation algorithm for minimizing a strictly convex separable quadratic function over a transportation polytope. The algorithm alternately solves two “subproblems,” each of which has an objective function that is defined by using Lagrange multipliers derived from the other. Motivated by the natural separation of the subproblems into independent and very easily solved “subsubproblems,” the algorithm can be interpreted as the cyclic coordinate ascent method applied to the dual problem. We exhibit our computational results for different implementations of the algorithm applied to a set of large constrained matrix problems.     

Algorithms for the minimax transportation problem

In this paper, we consider a variant of the classical transportation problem as well as of the bottleneck transportation problem, which we call the minimax transportation problem. The problem considered is to determine a feasible flow x_ij from a set of origins I to a set of destinations J for which max_(i,j)εIxJ{c_ijx_ij} is minimum. In this paper, we develop a parametric algorithm and a primal-dual algorithm to solve this problem. The parametric algorithm solves a transportation problem with parametric upper bounds and the primal-dual algorithm solves a sequence of related maximum flow problems. The primal-dual algorithm is shown to be polynomially bounded. Numerical investigations with both the algorithms are described in detail. The primal-dual algorithm is found to be computationally superior to the parametric algorithm and it can solve problems up to 1000 origins, 1000 destinations and 10,000 arcs in less than 1 minute on a DEC 10 computer system. The optimum solution of the minimax transportation problem may be noninteger. We also suggest a polynomial algorithm to convert this solution into an integer optimum solution.     

A branch and bound algorithm for allocation problems in which constraint coefficients depend upon decision variables

A branch and bound algorithm is developed for a class of allocation problems in which some constraint coefficients depend on the values of certain of the decision variables. Were it not for these dependencies, the problems could be solved by linear programming. The algorithm is developed in terms of a strategic deployment problem in which it is desired to find a least-cost transportation fleet, subject to constraints on men/materiel requirements in the event of certain hypothesized contingencies. Among the transportation vehicles available for selection are aircraft which exhibit the characteristic that the amount of goods deliverable by an aircraft on a particular route in a given time period (called aircraft productivity and measured in kilotons/aircraft/month) depends on the ratio of type 1 to type 2 aircraft used on that particular route. A model is formulated in which these relationships are first approximated by piecewise linear functions. A branch and bound algorithm for solving the resultant nonlinear problem is then presented; the algorithm solves a sequence of linear programming problems. The algorithm is illustrated by a sample problem and comments concerning its practicality are made.