An algorithm for minimizing flow time and maximum earliness on a single machine

This study presents an algorithm for efficient scheduling in terms of total flow time and maximum earliness. All the algorithms in the literature for solving this problem are based on heuristic procedures, and cannot necessarily generate all efficient schedules. This study shows that this problem can actually be solved in pseudo-polynomial time, and develops an algorithm for so doing. The complexity of the algorithm is O (n²p? log n). Its computational performance in solving problems of various sizes is determined.     

Single machine earliness and tardiness scheduling

We examine the problem of scheduling a given set of jobs on a single machine to minimize total early and tardy costs without considering machine idle time. We decompose the problem into two subproblems with a simpler structure. Then the lower bound of the problem is the sum of the lower bounds of two subproblems. A lower bound of each subproblem is obtained by Lagrangian relaxation. Rather than using the well-known subgradient optimization approach, we develop two efficient multiplier adjustment procedures with complexity O(nlog n) to solve two Lagrangian dual subproblems. A branch-and-bound algorithm based on the two efficient procedures is presented, and is used to solve problems with up to 50 jobs, hence doubling the size of problems that can be solved by existing branch-and-bound algorithms. We also propose a heuristic procedure based on the neighborhood search approach. The computational results for problems with up to 3 000 jobs show that the heuristic procedure performs much better than known heuristics for this problem in terms of both solution efficiency and quality. In addition, the results establish the effectiveness of the heuristic procedure in solving realistic problems to optimality or near optimality.     

The stochastic economic lot sizing problem for non-stop multi-grade production with sequence-restricted setup changeovers

We study a variant of the stochastic economic lot scheduling problem (SELSP) encountered in process industries, in which a single production facility must produce several different grades of a family of products to meet random stationary demand for each grade from a common finished-goods (FG) inventory buffer that has limited storage capacity. When the facility is set up to produce a particular grade, the only allowable changeovers are from that grade to the next lower or higher grade. Raw material is always available, and the production facility produces continuously at a constant rate even during changeover transitions. All changeover times are constant and equal to each other, and demand that cannot be satisfied directly from inventory is lost. There is a changeover cost per changeover occasion, a spill-over cost per unit of product in excess whenever there is not enough space in the FG buffer to store the produced grade, and a lost-sales cost per unit short whenever there is not enough FG inventory to satisfy the demand. We model the SELSP as a discrete-time Markov decision process (MDP), where in each time period the decision is whether to initiate a changeover to a neighboring grade or keep the set up of the production facility unchanged, based on the current state of the system, which is defined by the current set up of the facility and the FG inventory levels of all the grades. The goal is to minimize the (long-run) expected average cost per period. For problems with more than three grades, we develop a heuristic solution procedure which is based on decomposing the original multi-grade problem into several 3-grade MDP sub-problems, numerically solving each sub-problem using value iteration, and constructing the final policy for the original problem by combining parts of the optimal policies of the sub-problems. We present numerical results for problem examples with 2–5 grades. For the 2- and 3-grade examples, we numerically solve the exact MDP problem using value iteration to obtain insights into the structure of the optimal changeover policy. For the 4- and 5-grade examples, we compare the performance of the decomposition-based heuristic (DBH) solution procedure against that obtained by numerically solving the exact problem. We also compare the performance of the DBH method against the performance of three simpler parameterized heuristics. Finally, we compare the performance of the DBH and the exact solution procedures for the case where the FG inventory storage consists of a number of separate general-purpose silos capable of storing any grade as long as it is not mixed with any other grade.     

A new exact method for the two-dimensional orthogonal packing problem

The two-dimensional orthogonal packing problem (2OPP) consists in determining if a set of rectangles (items) can be packed into one rectangle of fixed size (bin). In this paper we propose two exact algorithms for solving this problem. The first algorithm is an improvement on a classical branch&bound method, whereas the second algorithm is based on a new relaxation of the problem. We also describe reduction procedures and lower bounds which can be used within enumerative methods. We report computational experiments for randomly generated benchmarks which demonstrate the efficiency of both methods: the second method is competitive compared to the best previous methods. It can be seen that our new relaxation allows an efficient detection of non-feasible instances.     

A method of adjoints for solving some Ill-posed equations of the first kind

We consider solving the operator (1)

where f may be badly behaved. It is shown that if one needs only linear functionals of the solution u, this may be accomplished by solving a related adjoint problem. This problem may be better posed than (1). Applications are made to several integral equations of the first kind, and it is shown that several apparently distinct procedures are particular cases of our result.     

Balancing assembly lines effectively – A computational comparison

We report on results of a numerical experiment including the most effective branch and bound procedures for solving type 1 of the simple assembly line balancing problem (SALBP-1), namely Johnson's (1988) FABLE, Nourie and Venta's (1991) OptPack, Hoffmann's (1992) Eureka and Scholl and Klein's (1997) SALOME. It appears that SALOME clearly outperforms the other procedures with respect to hard instances of a new challenging data set. Beyond that, it is shown that the performance of SALOME can greatly be improved by adding some further features including dynamic renumbering and dominance rules.     

Solving large single allocation p-hub problems with two or three hubs

In this paper we present an efficient approach for solving single allocation p-hub problems with two or three hubs. Two different variants of the problem are considered: the uncapacitated single allocation p-hub median problem and the p-hub allocation problem. We solve these problems using new mixed integer linear programming formulations that require fewer variables than those formerly used in the literature. The problems that we solve here are the largest single allocation problems ever solved. The numerical results presented here will demonstrate the superior performance of our mixed integer linear programs over traditional approaches for large problems. Finally we present the first mixed integer linear program for solving single allocation hub location problems that requires only O(n²) variables and O(n²) constraints that is valid for any number of hubs.     

Lagrangean duals and exact solution to the capacitated p-center problem

In this work, we address the capacitated p-center problem (CpCP). We study two auxiliary problems, discuss their relation to CpCP, and analyze the lower bounds obtained with two different Lagrangean duals based on each of these auxiliary problems. We also compare two different strategies for solving exactly CpCP, based on binary search and sequential search, respectively. Various data sets from the literature have been used for evaluating the performance of the proposed algorithms.     

Assessing solution quality in stochastic programs

Determining whether a solution is of high quality (optimal or near optimal) is fundamental in optimization theory and algorithms. In this paper, we develop Monte Carlo sampling-based procedures for assessing solution quality in stochastic programs. Quality is defined via the optimality gap and our procedures' output is a confidence interval on this gap. We review a multiple-replications procedure that requires solution of, say, 30 optimization problems and then, we present a result that justifies a computationally simplified single-replication procedure that only requires solving one optimization problem. Even though the single replication procedure is computationally significantly less demanding, the resulting confidence interval might have low coverage probability for small sample sizes for some problems. We provide variants of this procedure that require two replications instead of one and that perform better empirically. We present computational results for a newsvendor problem and for two-stage stochastic linear programs from the literature. We also discuss when the procedures perform well and when they fail, and we propose using ɛ-optimal solutions to strengthen the performance of our procedures.     

Sequential quadratic programming methods for parametric nonlinear optimization

Sequential quadratic programming (SQP) methods are known to be efficient for solving a series of related nonlinear optimization problems because of desirable hot and warm start properties—a solution for one problem is a good estimate of the solution of the next. However, standard SQP solvers contain elements to enforce global convergence that can interfere with the potential to take advantage of these theoretical local properties in full. We present two new predictor–corrector procedures for solving a nonlinear program given a sufficiently accurate estimate of the solution of a similar problem. The procedures attempt to trace a homotopy path between solutions of the two problems, staying within the local domain of convergence for the series of problems generated. We provide theoretical convergence and tracking results, as well as some numerical results demonstrating the robustness and performance of the methods.     

A comparison of two dual-based procedures for solving the p-median problem

We compare two dual-based procedures for solving the p-median problem. They rely upon alternative Lagrangean relaxations of the problem. We describe the algorithms and report on our computational experiments.     

Minimizing maximum completion time in a proportionate flow shop with one machine of different speed

A flow shop with identical machines is called a proportionate flow shop. In this paper, we consider the variant of the n-job, m-machine proportionate flow shop scheduling problem in which only one machine is different and job processing times are inversely proportional to machine speeds. The objective is to minimize maximum completion time. We describe some optimality conditions and show that the problem is NP-complete. We provide two heuristic procedures whose worst-case performance ratio is less than two. Extensive experiments with various sizes are conducted to show the performance of the proposed heuristics.     

A hybrid three-stage flowshop problem: Efficient heuristics to minimize makespan

We treat a problem of scheduling n jobs on a three stages hybrid flowshop of particular structure (one machine in the first and third stages and two dedicated machines in stage two). The objective is to minimize the makespan. This problem is NP-complete. We propose two heuristic procedures to cope with realistic problems. Extensive experimentation with various problem sizes are conducted and the computational results show excellent performance of the proposed heuristics.     

Estimation methods for choice-based conjoint analysis of consumer preferences

Conjoint analysis, a preference measurement method typical in marketing research, has gradually expanded to other disciplines. Choice-based conjoint analysis (CBC) is currently the most popular type. Very few alternative estimation approaches have been suggested since the introduction of the Hierarchical Bayes (HB) method for estimating CBC utility functions. Studies that compare the performance of more than one of the proposed approaches and the HB are almost non- existing. We compare the performance of four published optimization-based procedures and additionally we introduce a new one called CP. The CP is an estimation approach based on convex penalty minimization. In comparison with HB as the benchmark we use eight field data sets. We base the performance comparisons on holdout validation, i.e. predictive performance. Among the optimization based procedures CP performs best. We run simulations to compare the extent to which CP and HB can recover the true utilities. With the field data on the average, the CP and HB results are equally good. However, depending on the problem characteristics, one may perform better than the other. In terms of average performance, the other four methods were inferior to CP and HB.     

Achieving sharp deliveries in supply chains through variance pool allocation

Variability reduction and business process synchronization are acknowledged as key to achieving sharp and timely deliveries in supply chain networks. In this paper, we develop an approach that facilitates variability reduction and business process synchronization for supply chains in a cost effective way. The approach developed is founded on an analogy between mechanical design tolerancing and supply chain lead time compression. We first present a motivating example to describe this analogy. Next, we define, using process capability indices, a new index of delivery performance called delivery sharpness which, when used with the classical performance index delivery probability, measures the accuracy as well as the precision with which products are delivered to the customers. Following this, we solve the following specific problem: how do we compute the allowable variability in lead time for individual stages of the supply chain so that specified levels of delivery sharpness and delivery probability are achieved in a cost-effective way? We call this the variance pool allocation (VPA) problem. We suggest an efficient heuristic approach for solving the VPA problem and also show that a variety of important supply chain design problems can be posed as instances of the VPA problem. One such problem, which is addressed in this paper, is the supply chain partner selection problem. We formulate and solve the VPA problem for a plastics industry supply chain and demonstrate how the solution can be used to choose the best mix of supply chain partners.     

The p-median problem: A survey of metaheuristic approaches

The p-median problem is one of the basic models in discrete location theory. As with most location problems, it is classified as NP-hard, and so, heuristic methods are usually used to solve it. Metaheuristics are frameworks for building heuristics. In this survey, we examine the p-median, with the aim of providing an overview on advances in solving it using recent procedures based on metaheuristic rules.     

A New Approach to Analysis of Polling Systems

In this paper, we consider polling systems with J stations with Poisson arrivals and general service distributions attended by a cyclic server. The service discipline at each station is either exhaustive or gated. We propose a new approach to analysis of the mean waiting times in the polling systems. The outline of our method is as follows. We first define the stochastic process Q that represents an evolution of the system state, and define three types of the performance measures W _i ,H _i and F _i , which are the expected waiting times conditioned on the system state. Then from the analysis of customers at polling instants, we find their linear functional expressions. The steady state average waiting times can be derived from the performance measures by simple limiting procedures. Their actual values can be obtained by solving J(J+1) linear equations.     

A column generation based decomposition algorithm for a parallel machine just-in-time scheduling problem

We propose a column generation based exact decomposition algorithm for the problem of scheduling n jobs with an unrestrictively large common due date on m identical parallel machines to minimize total weighted earliness and tardiness. We first formulate the problem as an integer program, then reformulate it, using Dantzig–Wolfe decomposition, as a set partitioning problem with side constraints. Based on this set partitioning formulation, a branch and bound exact solution algorithm is developed for the problem. In the branch and bound tree, each node is the linear relaxation problem of a set partitioning problem with side constraints. This linear relaxation problem is solved by column generation approach where columns represent partial schedules on single machines and are generated by solving two single machine subproblems. Our computational results show that this decomposition algorithm is capable of solving problems with up to 60 jobs in reasonable cpu time.     

Finding a minimum independent dominating set in a permutation graph

We give a linear time reduction of the problem of finding a minimum independent dominating set in a permutation graph, into that of finding a shortest maximal increasing subsequence. We then give an O(n log²n)-time algorithm for solving the second (and hence the first) problem. This improves on the O(n³)-time algorithm given in [4] for solving the problem of finding a minimum independent dominating set in a permutation graph.     

A periodic review inventory system with two supply modes

We describe a periodic review inventory system in which there are two modes of resupply, namely a regular mode and an emergency mode. Orders placed through the emergency channel have a shorter supply lead time but are subject to higher ordering costs compared to orders placed through the regular channel. We analyze this problem within the framework of an order-up-to-R inventory control policy. At each epoch, the inventory manager must decide which of the two supply modes to use and then order enough units to raise the inventory position to a level R. We show that given any non-negative order-up-to level, either only the regular supply mode is used, or there exists an indifference inventory level such that if the inventory position at the review epoch is below the indifference inventory level, the emergency supply mode is used. We also develop procedures for solving for the two policy parameters, i.e., the order-up-to level and the indifference inventory level.