An efficient optimal solution method for the joint replenishment problem

During the last two decades, many heuristic procedures for the joint replenishment problem have appeared in the literature. The only available optimal solution procedure was based on an enumerative approach and was computationally prohibitive. In this paper we present an alternative optimal approach based on global optimisation theory. By applying Lipschitz optimisation one can find a solution with an arbitrarily small deviation from an optimal value. An efficient procedure is presented which uses a dynamic Lipschitz constant and generates a solution in little time. The running time of this procedure grows only linearly in the number of items.     

An efficient optimal solution method for the joint replenishment problem with minimum order quantities

We study the joint replenishment problem (JRP) for M items under deterministic demand, with a minimum order quantity constraint for each item in the replenishment order. We derive bounds on the basic cycle time and we propose an efficient global optimisation procedure to solve the JRP with constraints. Moreover, we also consider the case where a correction is made for empty replenishment occasions. The algorithms are tested with data from a real case and some additional numerical experiments are also presented.     

An exact algorithm for the petrol station replenishment problem

In the petrol station replenishment problem (PSRP), the aim is to deliver petroleum products to petrol stations by means of an unlimited heterogeneous fleet of compartmented tank trucks. The problem consists of jointly determining quantities to deliver within a given interval, of allocating products to tank truck compartments and of designing delivery routes to stations. This article describes an exact algorithm which decomposes the PSRP into a truck loading problem and a routing problem. An algorithm which makes use of assignment, optimality tests and possibly standard ILP algorithm is proposed to solve the loading problem. The routing problem is handled using two different strategies, based either on a matching approach or on a column generation scheme. This algorithm was extensively tested on randomly generated data and on a real-life case arising in Eastern Quebec.     

An efficient algorithm for a job shop problem

We study the problem of minimizing makespan in a two-machine job shop with unit processing time operations. An efficient algorithm with respect to a succinct encoding of the problem instances is proposed. The algorithm is an improvement of earlier algorithms proposed for the problem by Brucker [1,2], Hefetz and Adiri [7], and Timkovskiy [15]. The idea behind the algorithm has the potential of extension to job shops with parallel machines.This research is supported in part by NSERC Grants A4619, OGP0105675, OGP0104900, General Motors of Canada, and the Manufacturing Research Corporation of Ontario.     

An efficient algorithm for the three-guard problem

Given a simple polygon P with two vertices u and v, the three-guard problem asks whether three guards can move from u to v such that the first and third guards are separately on two boundary chains of P from u to v and the second guard is always kept to be visible from two other guards inside P. It is a generalization of the well-known two-guard problem, in which two guards move on the boundary chains from u to v and are always kept to be mutually visible. In this paper, we introduce the concept of link-2-ray shots, which can be considered as ray shots under the notion of link-2-visibility. Then, we show a one-to-one correspondence between the structure of the restrictions placed on the motion of two guards and the one placed on the motion of three guards, and generalize the solution for the two-guard problem to that for the three-guard problem. We can decide whether there exists a solution for the three-guard problem in O(nlogn) time, and if so generate a walk in O(nlogn+m) time, where n denotes the number of vertices of P and the size of the optimal walk. This improves upon the previous time bounds O(n²) and O(n²logn), respectively. Moreover, our results can be used to solve other more sophisticated geometric problems.     

Can-order policy for the periodic-review joint replenishment problem

In this paper we study the stochastic joint replenishment problem. We compare the class of periodic replenishment policies and the class of can-order policies for this problem. We present a method, based on Markov decision theory, to calculate near-optimal can-order policies for a periodic-review inventory system. Our numerical study shows that the can-order policy behaves as well as, if not better than, the periodic replenishment policies. In particular, for examples where the demand is irregular, we find cost differences up to 15% in favour of the can-order policy.     

On optimal algorithms for the joint replenishment problem

In this paper, we demonstrate that the algorithm for determining the optimal strict cyclic policy for the Joint Replenishment Problem suggested by Fung and Ma¹ does not guarantee an optimal solution. We propose a modification that will ensure that the algorithm obtains the optimal strict cyclic policy. We then perform a comprehensive computational study to compare the modified Fung and Ma algorithm with other optimal algorithms for the problem. The study reveals that the optimal algorithm of Viswanathan² is computationally more efficient than other methods.     

An approximation algorithm for the generalized assignment problem

The generalized assignment problem can be viewed as the following problem of scheduling parallel machines with costs. Each job is to be processed by exactly one machine; processing jobj on machinei requires timep _ij and incurs a cost ofc _ij ; each machinei is available forT _i time units, and the objective is to minimize the total cost incurred. Our main result is as follows. There is a polynomial-time algorithm that, given a valueC, either proves that no feasible schedule of costC exists, or else finds a schedule of cost at mostC where each machinei is used for at most 2T _i time units.We also extend this result to a variant of the problem where, instead of a fixed processing timep _ij , there is a range of possible processing times for each machine—job pair, and the cost linearly increases as the processing time decreases. We show that these results imply a polynomial-time 2-approximation algorithm to minimize a weighted sum of the cost and the makespan, i.e., the maximum job completion time. We also consider the objective of minimizing the mean job completion time. We show that there is a polynomial-time algorithm that, given valuesM andT, either proves that no schedule of mean job completion timeM and makespanT exists, or else finds a schedule of mean job completion time at mostM and makespan at most 2T. Research partially supported by an NSF PYI award CCR-89-96272 with matching support from UPS, and Sun Microsystems, and by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS-8920550.Research supported in part by a Packard Fellowship, a Sloan Fellowship, an NSF PYI award, and by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS-8920550.     

An algorithm for the symmetric generalized eigenvalue problem

A new method is presented for the solution of the matrix eigenvalue problem Ax=λBx, where A and B are real symmetric square matrices and B is positive semidefinite. It reduces A and B to diagonal form by congruence transformations that preserve the symmetry of the problem. This method is closely related to the QR algorithm for real symmetric matrices.     

An algorithm for the generalized quadratic assignment problem

This paper reports on a new algorithm for the Generalized Quadratic Assignment problem (GQAP). The GQAP describes a broad class of quadratic integer programming problems, wherein M pair-wise related entities are assigned to N destinations constrained by the destinations’ ability to accommodate them. This new algorithm is based on a Reformulation Linearization Technique (RLT) dual ascent procedure. Experimental results show that the runtime of this algorithm is as good or better than other known exact solution methods for problems as large as M=20 and N=15. Current address of P.M. Hahn: 2127 Tryon Street, Philadelphia, PA 19146-1228, USA.     

An efficient algorithm for the bipartite matching problem

The minimum cost bipartite matching problem is considered. An approach based on the solution of a sequence of shortest path sub-problems is proposed. The particular structure of the problem and the use of reduced costs make it possible to devise an efficient “threshold” algorithm to solve these sub-problems. The computational behaviour of the proposed procedure is analyzed.     

An efficient algorithm for the three-dimensional diameter problem

We explore a new approach for computing the diameter of n points in \Bbb R ³ that is based on the restriction of the furthest-point Voronoi diagram to the convex hull. We show that the restricted Voronoi diagram has linear complexity. We present a deterministic algorithm with O(nlog ² n) running time. The algorithm is quite simple and is a good candidate to be implemented in practice. Using our approach the chromatic diameter and all-furthest neighbors in \Bbb R ³ can be found in the same running time. Received February 18, 2000, and in revised form June 27, 2000. Online publication January 17, 2001.     

The joint replenishment problem with resource restriction

There are many resource restrictions in real production/inventory systems (for example, budget, storage, transportation capacity, etc.). But unlike other research areas, there is very little research to handle the joint replenishment problem (JRP) with resource restriction. The purpose of this paper is to develop two efficient algorithms for solving these problems. Firstly, we modify the existing RAND algorithm to be applicable to the JRP with resource restriction. Secondly, we develop a genetic algorithm for the JRP with resource restriction. Extensive computational experiments are performed to test the performances of the algorithms.     

Effective heuristics for the dynamic demand joint replenishment problem

This paper considers the dynamic demand joint replenishment problem where there is a joint setup cost in each time period when any member of the product family is replenished and a separate setup cost for each different item replenished. We present two forward-pass heuristics, a two-phase heuristic, and a simulated annealing metaheuristic (SAM) and investigate their relative effectiveness in solving a comprehensive set of test problems. The experimental results indicate the two-phase heuristic and the SAM perform better than existing approaches for the problem.     

An algorithm for the generalized symmetric tridiagonal eigenvalue problem

In this paper we present an algorithm, parallel in nature, for finding eigenvalues of a symmetric definite tridiagonal matrix pencil. Our algorithm employs the determinant evaluation, split-and-merge strategy and Laguerre's iteration. Numerical results on both single and multiprocessor computers are presented which show that our algorithm is reliable, efficient and accurate. It also enjoys flexibility in evaluating a partial spectrum.This research was supported in part by the Research Grant from University of West Florida.This research was supported in part by NSF under Grant CCR-9024840.     

An efficient evolutionary algorithm for the ring star problem

This paper addresses the ring star problem (RSP). The goal is to locate a cycle through a subset of nodes of a network aiming to minimize the sum of the cost of installing facilities on the nodes on the cycle, the cost of connecting them and the cost of assigning the nodes not on the cycle to their closest node on the cycle. A fast and efficient evolutionary algorithm is developed which is based on a new formulation of the RSP as a bilevel programming problem with one leader and two independent followers. The leader decides which nodes to include in the ring, one follower decides about the connections of the cycle and the other follower decides about the assignment of the nodes not on the cycle. The bilevel approach leads to a new form of chromosome encoding in which genes are associated to values of the upper level variables. The quality of each chromosome is evaluated by its fitness, by means of the objective function of the RSP. Hence, in order to compute the value of the lower level variables, two optimization problems are solved for each chromosome. The computational results show the efficiency of the algorithm in terms of the quality of the solutions yielded and the computing time. A study to select the best configuration of the algorithm is presented. The algorithm is tested on a set of benchmark problems providing very accurate solutions within short computing times. Moreover, for one of the problems a new best solution is found.