Computational study of the multiechelon production planning problem

In this article we try to identify appropriate solution procedures for different types of multiechelon production planning problems. We conduct an extensive computational study on uncapacitated multiechelon production planning problems with serial and assembly types of bill-of-material structures. Problems are formulated as both single-source fixed charge network problems and as multicommodity flow problems with fixed charges. Solution procedures considered are branch and cut, Lagrangean relaxation (for the network formulation), and branch and bound (for the multicommodity formulation). Three hundred problems with various problem structures are tested. Our conclusions suggest the best approach for each type of problem structure. © 1997 John Wiley & Sons, Inc.     

A dual-based optimization procedure for the two-echelon uncapacitated facility location problem

The two-echelon uncapacitated facility location problem (TUFLP) is a generalization of the uncapacitated facility location problem (UFLP) and multiactivity facility location problem (MAFLP). In TUFLP there are two echelons of facilities through which products may flow in route to final customers. The objective is to determine the least-cost number and locations of facilities at each echelon in the system, the flow of product between facilities, and the assignment of customers to supplying facilities. We propose a new dual-based solution procedure for TUFLP that can be used as a heuristic or incorporated into branch-and-bound procedures to obtain optimal solutions to TUFLP. The algorithm is an extension of the dual ascent and adjustment procedures developed by Erlenkotter for UFLP. We report computational experience gained by solving over 420 test problems. The largest problems solved have 25 possible facility locations at each echelon and 35 customer zones, implying 650 integer variables and 21,875 continuous variables.     

The bottleneck transportation problem

The bottleneck transportation problem can be stated as follows: A set of supplies and a set of demands are specified such that the total supply is equal to the total demand. There is a transportation time associated between each supply point and each demand point. It is required to find a feasible distribution (of the supplies) which minimizes the maximum transportaton time associated between a supply point and a demand point such that the distribution between the two points is positive. In addition, one may wish to find from among all optimal solutions to the bottleneck transportation problem, a solution which minimizes the total distribution that requires the maximum time Two algorithms are given for solving the above problems. One of them is a primal approach in the sense that improving fcasible solutions are obtained at each iteration. The other is a “threshold” algorithm which is found to be far superior computationally.     

Use of completely reduced matrices in solving transportation problems with fixed charges

This paper shows how completely reduced matrices can be used in obtaining exact or approximate solutions to transportation problems with fixed charges. It does not treat methods for obtaining reduced matrices, which are available elsewhere, but it does discuss the problem of obtaining a completely reduced matrix, and then a general parametric solution to the primal problem, from any particular solution. Methods for obtaining particular solutions with determinacies of maximum order (solutions for the constant fixed charges problem) are then presented. The paper terminates with a discussion of methods which are useful in obtaining approximations to solutions of fixed charges problems with charges not constant.     

Interval estimation of a global optimum for large combinatorial problems

Consider an “intractable” optimization problem for which no efficient solution technique exists. Given a systematic procedure for generating independent heuristic solutions, we seek to obtain interval estimates for the globally optimal solution using statistical inference. In previous work, accurate point estimates have been derived. Determining interval estimates, however, is a considerably more difficult task. In this paper, we develop straightforward procedures which compute confidence intervals efficiently in order to evaluate heuristic solutions and assess deviations from optimality. The strategy presented is applicable to a host of combinatorial optimization problems. The assumptions of our model, along with computational experience, are discussed.     

Optimal multicommodity network flows with resource allocation

The problem of determining multicommodity flows over a capacitated network subject to resource constraints may be solved by linear programming; however, the number of potential vectors in most applications is such that the standard arc-chain formulation becomes impractical. This paper describes an approach—an extension of the column generation technique used in the multicommodity network flow problem—that simultaneously considers network chain selection and resource allocation, thus making the problem both manageable and optimal. The flow attained is constrained by resource availability and network capacity. A minimum-cost formulation is described and an extension to permit the substitution of resources is developed. Computational experience with the model is discussed.     

Minimizing mean tardiness and earliness in single-machine scheduling problems with unequal due dates

We consider a single-machine scheduling problem with the objective of minimizing the mean (or equivalently, total) tardiness and earliness when due dates may differ among jobs. Some properties of the optimal solution are discussed, and these properties are used to develop both optimal and heuristic algorithms. Results of computational tests indicate that optimal solutions can be found for problems with up to 20 jobs, and that two of the heuristic procedures provide optimal or very near optimal solutions in many instances. © 1994 John Wiley & Sons, Inc.     

The procurement problem: An integer programming problem well suited to a solution using duality

A procurement problem, as formulated by Murty [10], is that of determining how many pieces of equipment units of each of m types are to be purchased and how this equipment is to be distributed among n stations so as to maximize profit, subject to a budget constraint. We have considered a generalization of Murty's procurement problem and developed an approach using duality to exploit the special structure of this problem. By using our dual approach on Murty's original problem, we have been able to solve large problems (1840 integer variables) with very modest computational effort. The main feature of our approach is the idea of using the current evaluation of the dual problem to produce a good feasible solution to the primal problem. In turn, the availability of good feasible solutions to the primal makes it possible to use a very simple subgradient algorithm to solve the dual effectively.     

An optimal procedure for the coordinated replenishment dynamic lot‐sizing problem with quantity discounts

We consider in this paper the coordinated replenishment dynamic lot‐sizing problem when quantity discounts are offered. In addition to the coordination required due to the presence of major and minor setup costs, a separate element of coordination made possible by the offer of quantity discounts needs to be considered as well. The mathematical programming formulation for the incremental discount version of the extended problem and a tighter reformulation of the problem based on variable redefinition are provided. These then serve as the basis for the development of a primal‐dual based approach that yields a strong lower bound for our problem. This lower bound is then used in a branch and bound scheme to find an optimal solution to the problem. Computational results for this optimal solution procedure are reported in the paper. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 686–695, 2000     

A bounded dual (all-integer) integer programming algorithm with an objective cut

In this article, we describe a new algorithm for solving all-integer, integer programming problems. We generate upper bounds on the decision variables, and use these bounds to create an advanced starting point for a dual all-integer cutting plane algorithm. In addition, we use a constraint derived from the objective function to speed progress toward the optimal solution. Our basic vehicle is the dual all-integer algorithm of Gomory, but we incorporate certain row- and column-selection criteria which partially avoid the problem of dual-degenerate iterations. We present the results of computational testing.     

Using grasp to solve the component grouping problem

Component grouping problems, a type of set-partitioning problem, arise in a number of different manufacturing and material logistics application areas. For example, in circuit board assembly, robotic work cells can be used to insert components onto a number of different types of circuit boards. Each type of circuit board requires particular components, with some components appearing on more than one type. The problem is to decide which components should be assigned to each work cell in order to minimize the number of visits by circuit boards to work cells. We describe two new heuristics for this problem, based on so-called greedy random adaptive search procedures (GRASP). With GRASP, a local search technique is replicated many times with different starting points. The starting points are determined by a greedy procedure with a probabilistic aspect. The best result is then kept as the solution. Computational experiments on problems based on data from actual manufacturing processes indicate that these GRASP methods outperform, both in speed and in solution quality, an earlier, network-flow-based heuristic. We also describe techniques for generating lower bounds for the component grouping problem, based on the combinatorial structure of a problem instance. The lower bounds for our real-world test problems averaged within 7%-8% of the heuristic solutions. Similar results are obtained for larger, randomly generated problems. © 1994 John Wiley & Sons. Inc.     

Optimal set partitioning,matchings and lagrangian duality

We formulate the set partitioning problem as a matching problem with simple side constraints. As a result we obtain a Lagrangian relaxation of the set partitioning problem in which the primal problem is a matching problem. To solve the Lagrangian dual we must solve a sequence of matching problems each with different edge-weights. We use the cyclic coordinate method to iterate the multipliers, which implies that successive matching problems differ in only two edge-weights. This enables us to use sensitivity analysis to modify one optimal matching to obtain the next one. We give theoretical and empirical comparisons of these dual bounds with the conventional linear programming ones.     

The simplex method for integral multicommodity networks

The simplex method is interpreted as a labeling procedure for certain classes of multicommodity flow problems in a manner similar to that for single commodity networks. As opposed to general multicommodity algorithms, no explicit matrix inversion is required; all simplex operations are performed graph-theoretically.     

The conjugate gradient technique for certain quadratic network problems

We consider a class of network flow problems with pure quadratic costs and demonstrate that the conjugate gradient technique is highly effective for large-scale versions. It is shown that finding a saddle point for the Lagrangian of an m constraint, n variable network problem requires only the solution of an unconstrained quadratic programming problem with only m variables. It is demonstrated that the number of iterations for the conjugate gradient algorithm is substantially smaller than the number of variables or constraints in the (primal) network problem. Forty quadratic minimum-cost flow problems of various sizes up to 100 nodes are solved. Solution time for the largest problems (4,950 variables and 99 linear constraints) averaged 4 seconds on the CBC Cyber 70 Model 72 computer.     

Optimal service and arrival rates in Jackson queueing networks

In this paper we present an algorithm for solving a class of queueing network design problems. Specifically, we focus on determining both service and arrival rates in an open Jackson network of queueing stations. This class of problems has been widely studied and used in a variety of applications, but not well solved due to the difficulty of the resulting optimization problems. As an example, consider the classic application in computer network design which involves determining the minimum cost line capacities and flow assignments while satisfying a queueing performance measure such as an upper limit on transmission delay. Other application areas requiring the selection of both service and arrival rates in a network of queues include the design of communication, manufacturing, and health care systems. These applications yield optimization problems that are difficult to solve because typically they are nonconvex, which means they may have many locally optimal solutions that are not necessarily globally optimal. Therefore, to obtain a globally optimal solution, we develop an efficient branch and bound algorithm that takes advantage of the problem structure. Computational testing on randomly generated problems and actual problems from a health care organization indicate that the algorithm is able to solve realistic sized problems in reasonable computing time on a laptop computer. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 1–17, 2000     

Solving the generalized knapsack problem with variable coefficients

In this article we present methods based on Lagrangian duality and decomposition techniques for the generalized knapsack problem with variable coefficients. The Lagrangian dual is solved with subgradient optimization or interval bisection. We also describe a heuristic that yields primal feasible solutions. Combining the Lagrangian relaxation with a primal (Benders) subproblem yields the subproblem phase in cross decomposition. By using averages in this procedure, we get the new mean-value cross-decomposition method. Finally, we describe how to insert this into a globally convergent generalized Benders decomposition framework, in the case that there is a duality gap. Encouraging computational results for the optimal generating unit commitment problem are presented. © 1996 John Wiley & Sons, Inc.     

A branch‐and‐price approach for the stochastic generalized assignment problem

In this article, we address a stochastic generalized assignment machine scheduling problem in which the processing times of jobs are assumed to be random variables. We develop a branch‐and‐price (B&P) approach for solving this problem wherein the pricing problem is separable with respect to each machine, and has the structure of a multidimensional knapsack problem. In addition, we explore two other extensions of this method—one that utilizes a dual‐stabilization technique and another that incorporates an advanced‐start procedure to obtain an initial feasible solution. We compare the performance of these methods with that of the branch‐and‐cut (B&C) method within CPLEX. Our results show that all B&P‐based approaches perform better than the B&C method, with the best performance obtained for the B&P procedure that includes both the extensions aforementioned. We also utilize a Monte Carlo method within the B&P scheme, which affords the use of a small subset of scenarios at a time to estimate the “true” optimal objective function value. Our experimental investigation reveals that this approach readily yields solutions lying within 5% of optimality, while providing more than a 10‐fold savings in CPU times in comparison with the best of the other proposed B&P procedures. © 2014 Wiley Periodicals, Inc. Naval Research Logistics 61: 131–143, 2014     

A mutual primal-dual linear programming algorithm

A new primal-dual linear programming algorithm is exhibited. A proof is given that optimal solutions to both primal and dual problems (when such solutions exist) are found in a finite number of steps by this algorithm. A numerical example is included to illustrate the method.     

A streamlined simplex approach to the singly constrained transportation problem

A primal simplex procedure is developed to solve transportation problems with an arbitrary additional linear constraint. The approach is a specialization of the Double Reverse Method of Charnes and Cooper. Efficient procedures for pricing-out the basis, determining representations, and implementing the change of basis are presented. These procedures exploit the pure transportation substructure in such a manner that full advantage may be taken of the computational schemes and list structures used to store and update the basis in codifying the MODI method. Furthermore, the pricing-out and change-of-basis procedures are organized in a manner that permits the calculations for one to be utilized in the other. Computational results are presented which indicate that this method is at least 50 times faster than the state-of-the-art LP code, APEX-III. Methods for obtaining basic primal “feasible” starts and “good” feasible integer solutions are also presented.