Solving the uncapacitated facility location problem using tabu search

A tabu search heuristic procedure is developed to solve the uncapacitated facility location problem. Tabu search is used to guide the solution process when evolving from one solution to another. A move is defined to be the opening or closing of a facility. The net cost change resulting from a candidate move is used to measure the attractiveness of the move. After a move is made, the net cost change of a candidate move is updated from its old value. Updating, rather than re-computing, the net cost changes substantially reduces computation time needed to solve a problem when the problem is not excessively large. Searching only a small subset of the feasible solutions that contains the optimal solution, the procedure is computationally very efficient. A computational experiment is conducted to test the performance of the procedure and computational results are reported. The procedure can easily find optimal or near optimal solutions for benchmark test problems from the literature. For randomly generated test problems, this tabu search procedure not only obtained solutions completely dominating those obtained with other heuristic methods recently published in the literature but also used substantially less computation time. Therefore, this tabu search procedure has advantage over other heuristic methods in both solution quality and computation speed.     

Minimizing total weighted flowtime subject to minimum makespan on two identical parallel machines

We study the problem of scheduling n jobs on two identical parallel processors or machines where an optimal schedule is defined as one with the shortest total weighted flowtime (i.e., the sum of the weighted completion time of all jobs), among the set of schedules with minimum makespan (i.e., the completion time of the last job finished). We present a two phase non-linear Integer Programming formulation for its solution, admittedly not to be practical or useful in most cases, but theoretically interesting since it models the problem. Thus, as an alternative, we propose an optimization algorithm, for small problems, and a heuristic, for large problems, to find optimal or near optimal solutions. Furthermore, we perform a computational study to evaluate and compare the effectiveness of the two proposed methods.     

The class Steiner minimal tree problem: a lower bound and test problem generation

The class Steiner minimal tree problem is an extension of the standard Steiner minimal tree problem in graphs, motivated by the problem of wire routing in the area of physical design of very large scale integration (VLSI). This problem is NP-hard, even if there are no Steiner nodes and the graph is a tree; moreover, there exists no polynomial time approximation algorithm with a constant bound on the relative error under the hypothesis that P NP [16]. Hence, fast and good heuristic algorithms are needed in practice. In this paper, we present an integer programming formulation of the problem. Using Lagrangean relaxation and subgradient optimization, we derive a lower bound. In order to test the lower bound, we present a procedure for generating test problems for the class Steiner minimal tree problem that have known optimal solutions. The computational experiments for the test problems demonstrate that the lower bound is very tight and differs from the optimal solutions by only a few percent on average for sparse graphs. Received: 5 July 1999 / Revised version: 14 July 2000     

Multi-level lot sizing in material requirements planning: an empirical investigation

The research report in this paper was a result of a recent article by Yelle [8]. He reports computational results using all combinations of four single level heuristic lot sizing rules, which are applied sequentially to a two level problem in a Material Requirements Planning (MRP) environment. To test the impact of the different combinations of these lot sizing rules he employed six different demand patterns for the end-items which range from level demand to demand which is intermittent throughout the planning horizon. In our research we utilized the same data as Yelle and conducted a more extensive analysis by adding (a) two more recent and powerful single level heuristic rules, namely, the least unit cost heuristic and Silver and Meal's least-cost per period heuristic, (b) the very well known single level optimum method of Wagner-Whitin[7], and (c) a multi-level heuristic proposed recently by McLaren and Whybark [4]. The results show a significant improvement in many cases over the best solutions reported by Yelle.Further in this paper, we also provide a much simpler mathematical formulation of the multi-level lot sizing problem than found in the literature. This formulation was advantageously used to determine optimum solutions using IBM's Mathematical Programming System Extended (MPSX) to the aforementioned problems.     

GRASP and hybrid GRASP-Tabu heuristics to solve a maximal covering location problem with customer preference ordering

In this study, a maximal covering location problem is investigated. In this problem, we want to maximize the demand of a set of customers covered by a set of p facilities located among a set of potential sites. It is assumed that a set of facilities that belong to other firms exists and that customers freely choose allocation to the facilities within a coverage radius. The problem can be formulated as a bilevel mathematical programming problem, in which the leader locates facilities in order to maximize the demand covered and the follower allocates customers to the most preferred facility among those selected by the leader and facilities from other firms. We propose a greedy randomized adaptive search procedure (GRASP) heuristic and a hybrid GRASP-Tabu heuristic to find near optimal solutions. Results of the heuristic approaches are compared to solutions obtained with a single-level reformulation of the problem. Computational experiments demonstrate that the proposed algorithms can find very good quality solutions with a small computational burden. The most important feature of the proposed heuristics is that, despite their simplicity, optimal or near-optimal solutions can be determined very efficiently.     

Scheduling algorithms for heterogeneous batch processors with incompatible job-families

We consider the problem of scheduling heterogeneous batch processors (i.e., batch processors with different capacity) with incompatible job-families and non-identical job sizes to maximize the utilization of the batch processors. We analyzed the computational complexity of this problem and showed that it is NP-hard and proposed eight variants of a fast greedy heuristic. A series of computational experiments were carried out to compare the performance of the heuristics and showed that the heuristics are capable of consistently obtaining near (estimated) optimal solutions with very low-computational burden for large-scale problems. We also carried out a study to find the effect of family processing time changes on the performance of the heuristics. This sensitivity analysis indicated that the processing time set of job-families influences the performance of the heuristic algorithms.     

A linear programming-based method for job shop scheduling

We present a decomposition heuristic for a large class of job shop scheduling problems. This heuristic utilizes information from the linear programming formulation of the associated optimal timing problem to solve subproblems, can be used for any objective function whose associated optimal timing problem can be expressed as a linear program (LP), and is particularly effective for objectives that include a component that is a function of individual operation completion times. Using the proposed heuristic framework, we address job shop scheduling problems with a variety of objectives where intermediate holding costs need to be explicitly considered. In computational testing, we demonstrate the performance of our proposed solution approach.     

Uplink scheduling for joint wireless orthogonal frequency and time division multiple access networks

In this paper, we present a deterministic resource allocation model for a hybrid uplink wireless orthogonal frequency and time division multiple access network. Since the input data of the model may be affected by uncertainty, we further consider a stochastic formulation of the problem which we transform into an equivalent deterministic binary second-order conic program (SOCP). Subsequently, we use this binary SOCP to derive an equivalent integer linear programming formulation. The proposed models are aimed at maximizing the total bandwidth channel capacity subject to user power and sub-carrier assignment constraints while simultaneously scheduling users in time. As such, the models are best suited for non-real-time applications where sub-channel multiuser diversity can be further exploited simultaneously in frequency and time domains. Finally, in view of the large execution times required by CPLEX to solve the proposed models, we propose a variable neighborhood search metaheuristic procedure. Our numerical results show tight bounds and near optimal solutions for most of the instances when compared to the optimal solution of the problem. Moreover, we obtain better feasible solutions than CPLEX in the stochastic case. Finally, these bounds are obtained at a very low computational cost.     

LINEAR PROGRAMMING AND RECURRENT ASSOCIATIVE MEMORIES

Many optimization procedures presume the availability of an initial approximation in the neighborhood of a local or global optimum. Unfortunately, finding a set of good starting conditions is itself a nontrivial proposition. We describe a procedure for identifying approximate solutions to constrained optimization problems. Recurrent neural network structures are interpreted in the context of linear associative memory matrices. A recurrent associative memory (RAM) is trained to map the inputs of closely related transportation linear programs to optimal solution vectors. The procedure performs well when training cases are selected according to a simple rule, identifying good heuristic solutions for representative test cases. Modest infeasibilities exist in some of these estimated solutions, but the basic variables associated with true optimums are usually apparent. In the great majority of cases, rounding identifies the true optimum.     

Robust supply chain design under uncertain demand in agile manufacturing

This paper considers a supply chain design problem for a new market opportunity with uncertain demand in an agile manufacturing setting. We consider the integrated optimization of logistics and production costs associated with the supply chain members. These problems routinely occur in a wide variety of industries including semiconductor manufacturing, multi-tier automotive supply chains, and consumer appliances to name a few. There are two types of decision variables: binary variables for selection of companies to form the supply chain and continuous variables associated with production planning. A scenario approach is used to handle the uncertainty of demand. The formulation is a robust optimization model with three components in the objective function: expected total costs, cost variability due to demand uncertainty, and expected penalty for demand unmet at the end of the planning horizon. The increase of computational time with the numbers of echelons and members per echelon necessitates a heuristic. A heuristic based on a k-shortest path algorithm is developed by using a surrogate distance to denote the effectiveness of each member in the supply chain. The heuristic can find an optimal solution very quickly in some small- and medium-size cases. For large problems, a “good” solution with a small gap relative to our lower bound is obtained in a short computational time.     

Effective replenishment policies for the multi-item dynamic lot-sizing problem with storage capacities

We address the dynamic lot-sizing problem considering multiple items and storage capacity. Despite we can easily characterize a subset of optimal solutions just extending the properties of the single-item case, these results are not helpful to design an efficient algorithm. Accordingly, heuristics are appropriate approaches to obtain near-optimal solutions for this NP-hard problem. Thus, we propose a heuristic procedure based on the smoothing technique, which is tested on a large set of randomly generated instances. The computational results show that the method is able to build policies that are both easily implemented and very effective, since they are on average 5% above the best solution reported by CPLEX. Moreover, an additional computational experiment is carried out to show that the performance of this new heuristic is on average better and more robust than other methods previously proposed for this problem.     

Designing Partially Survivable Cellular Telecommunications Networks

In a cellular telecommunications network, a mobile telephone switching office (MTSO) co-ordinates the activities of the mobile units either directly or via intermediate base transceivers or hubs. In order to increase reliability, cells often split their traffic via multiple hubs—this ensures partial survivability in the face of equipment failures. One version of this problem was presented in Dutta and Kubat [4] where it was solved heuristically. In this paper, we show how this problem can be reduced to a significantly smaller one—one which can be solved to optimality almost instantaneously even for instances much larger than those presented in Dutta and Kubat [4]. Results from extensive computational tests are presented on various channel capacities, ranging from DS-1 to OC-192. The optimal approach is observed to provide better solutions in less time than the heuristic approach, thereby rendering the use of heuristic approaches unnecessary.     

Handling biological sequence alignments on networked computing systems: A divide-and-conquer approach

In this paper, we address the biological sequence alignment problem, which is one of the most commonly used steps in several bioinformatics applications. We employ the Divisible Load Theory (DLT) paradigm that is suitable for handling large-scale processing on network-based systems to achieve a high degree of parallelism. Using the DLT paradigm, we propose a strategy in which we carefully partition the computation work load among the processors in the system so as to minimize the overall computation time of determining the maximum similarity between the DNA/protein sequences. We consider handling such a computational problem on networked computing platforms connected as a linear daisy chain. We derive the individual load quantum to be assigned to the processors according to computation and communication link speeds along the chain. We consider two cases of sequence alignment where post-processes, i.e., trace-back processes that are required to determine an optimal alignment, may or may not be done at individual processors in the system. We derive some critical conditions to determine if our strategies are able to yield an optimal processing time. We apply three different heuristic strategies proposed in the literature to generate sub-optimal solutions for processing time when the above conditions cannot be satisfied. To testify the proposed schemes, we use real-life DNA samples of house mouse mitochondrion and the DNA of human mitochondrion obtained from the public database GenBank [GenBank, http://www.ncbi.nlm.nih.gov] in our simulation experiments. By this study, we conclusively demonstrate the applicability and potential of the DLT paradigm to such biological sequence related computational problems.     

Full-shipload tramp ship routing and scheduling with variable speeds

This paper investigates the simultaneous optimization problem of routing and sailing speed in the context of full-shipload tramp shipping. In this problem, a set of cargoes can be transported from their load to discharge ports by a fleet of heterogeneous ships of different speed ranges and load-dependent fuel consumption. The objective is to determine which orders to serve and to find the optimal route for each ship and the optimal sailing speed on each leg of the route so that the total profit is maximized. The problem originated from a real-life challenge faced by a Danish tramp shipping company in the tanker business. To solve the problem, a three-index mixed integer linear programming formulation as well as a set packing formulation is presented. A novel Branch-and-Price algorithm with efficient data preprocessing and heuristic column generation is proposed. The computational results on the test instances generated from real-life data show that the heuristic provides optimal solutions for small test instances and near-optimal solutions for larger test instances in a short running time. The effects of speed optimization and the sensitivity of the solutions to the fuel price change are analyzed. It is shown that speed optimization can improve the total profit by 16% on average and the fuel price has a significant effect on the average sailing speed and total profit.     

Heuristic algorithms for the general nonlinear separable knapsack problem

We consider the nonlinear knapsack problem with separable nonconvex functions. Depending on the assumption on the integrality of the variables, this problem can be modeled as a nonlinear programming or as a (mixed) integer nonlinear programming problem. In both cases, this class of problems is very difficult to solve, both from a theoretical and a practical viewpoint. We propose a fast heuristic algorithm, and a local search post-optimization procedure. A series of computational comparisons with a heuristic method for general nonconvex mixed integer nonlinear programming and with global optimization methods shows that the proposed algorithms provide high-quality solutions within very short computing times.     

A hybrid method for the Probabilistic Maximal Covering Location–Allocation Problem

This paper presents a hybrid algorithm that combines a metaheuristic and an exact method to solve the Probabilistic Maximal Covering Location–Allocation Problem. A linear programming formulation for the problem presents variables that can be partitioned into location and allocation decisions. This model is solved to optimality for small- and medium-size instances. To tackle larger instances, a flexible adaptive large neighborhood search heuristic was developed to obtain location solutions, whereas the allocation subproblems are solved to optimality. An improvement procedure based on an integer programming method is also applied. Extensive computational experiments on benchmark instances from the literature confirm the efficiency of the proposed method. The exact approach found new best solutions for 19 instances, proving the optimality for 18 of them. The hybrid method performed consistently, finding the best known solutions for 94.5% of the instances and 17 new best solutions (15 of them optimal) for a larger dataset in one-third of the time of a state-of-the-art solver.     

A two-level location–allocation problem in designing local access fiber optic networks

This paper deals with a two-level facility location–allocation problem on tree topology arising from the design of access networks. The access network design problem seeks to find an optimal location of switch and allocation of demands such that the total cost of switch and fiber cable is minimized, while satisfying switch port constraint, switch capacity constraint, and no-split routing constraint. The problem is formulated as a mixed-integer programming model and alternative formulations are developed by the reformulation-linearization technique (RLT) for improving computational effectiveness. By exploiting the tree structure of the problem, we develop some valid inequalities that have complementary strength and devise separation problems for finding effective valid inequalities that cut off fractional LP solutions. Also, we develop an effective MIP-based tree partitioning heuristic for finding good quality solutions for large size problems. Computational results demonstrate the efficacy of the valid inequalities and the proposed heuristic.     

Single facility scheduling with nonlinear processing times

This paper considers the static single facility scheduling problem where the processing times of jobs are a monotonically increasing function of their starting (waiting) times and the objective is to minimize the total elapsed time (called the makespan) in which all jobs complete their processing. Based on the combinatorial analysis of the problem, an exact optimization algorithm is developed for the general processing time function which is then specialized for the linear case. In view of the excessive computational burden of the exact optimization algorithm for the nonlinear processing time functions, heuristic algorithms are proposed. The effectiveness of these proposed alogrithms is empirically evaluated and found to indicate that these heuristic algorithms yield optimal or near optimal schedules in many cases.     

A new mixed-integer linear programming model for rescue path planning in uncertain adversarial environment

Efficient vehicle path planning in hostile environment to carry out rescue or tactical logistic missions remains very challenging. Most approaches reported so far rely on key assumptions and heuristic procedures to reduce problem complexity. In this paper, a new model is proposed to solve the discrete rescue path planning problem for a single agent navigating in uncertain adversarial environment. It relies on a novel and simplified mathematical mixed-integer linear programming formulation aimed at minimizing traveled distance and threat exposure. Exploiting a user-defined survivability function approximation and survivability threshold, the approximate model allows constructing a solution providing an adjustable optimality gap interval on the optimal solution. Experimental results show the value of the proposed approach in computing near optimal solutions reasonably fast for various problem instances.