Exact Solution Procedure Research Articles

Solving symmetric mixed-model multi-level just-in-time scheduling problems

The generation of leveled production schedules is of high importance for mixed-model assembly lines whose parts and materials are supplied just-in-time by multi-level production processes. The Output Rate Variation problem is the standard mathematical representation of this complex level scheduling problem and has been extensively studied by research thus far. This work identifies novel symmetries in solution sequences of this problem class and shows how these insights can be used to improve exact solution procedures presented in the literature. The effectiveness of the modifications is evaluated by a computational study.

An optimal deterministic control policy of servers in front and back rooms with a variable number of switching points and switching costs

In this paper we consider a retail service facility with cross-trained workers who can perform operations in both the front room and back room. Workers are brought from the back room to the front room and vice versa depending on the number of customers in the system. A loss of productivity occurs when a worker returns to the back room. Two problems are studied. In the first problem, given the number of workers available, we determine an optimal deterministic switching policy so that the expected number of customers in queue is minimized subject to a constraint ensuring that there is a sufficient workforce to fulfill the functions in the back room. In the second problem, the number of workers needed is minimized subject to an additional constraint requiring that the expected number of customers waiting in queue is bounded above by a given threshold value. Exact solution procedures are developed and illustrative numerical examples are presented.

Optimisation of the exploitation period of individual vehicles in freight transportation companies

The individual vehicle replacement problem typical for freight transportation companies is discussed in the paper. Two characteristic features of such problem are that transportation companies utilise vehicles with intensity decreasing with an age of vehicles and that managers of such companies first of all take into account economical criteria when planning vehicle replacements. The paper presents a single criterion, nonlinear, deterministic and discrete mathematical model of such a problem that minimises a total exploitation and ownership costs calculated per kilometre. The exact solution procedure is proposed here. The problem is solved as a real life case study. As a result, an average, economically optimal 5-year exploitation period of vehicles has been determined.

Level Scheduling for batched JIT supply

A mixed-model assembly line requires the solution of a short-term sequencing problem which decides on the succession of different models launched down the line. A famous solution approach stemming from the Toyota Production System is the so-called Level Scheduling, which aims at distributing the part consumption induced by the model sequence evenly over time. Traditional Level Scheduling seeks to closely approximate target demand rates at every production cycle, however, such a strict leveling is only required if parts are directly pulled from a connected feeder line. In real-world assembly lines, parts are predominately delivered in (small) batches at certain points in time. In such a situation, a Just-in-Time supply is already facilitated whenever the cumulative consumption is leveled in accordance with each part’s delivery schedule, while the exact consumption pattern between two delivery points seems irrelevant. The paper on hand provides new Level Scheduling models, proves complexity, presents exact and heuristic solution procedures and shows inferiority of traditional Level Scheduling for such a batched JIT-supply of parts.

Truck scheduling at zero-inventory cross docking terminals

Handling freight at cross docking terminals constitutes a complex planning task which comprises several executive steps as shipments delivered by inbound trucks are to be unloaded, sorted according to their designated destinations, moved across the dock and finally loaded onto outbound trucks for an immediate delivery elsewhere in the distribution system. To enable an efficient synchronization of inbound and outbound flows, a careful planning of operations, e.g. by computerized scheduling procedures, becomes indispensable. This work treats a special truck scheduling problem arising in the (zero-inventory) cross docks of the food industry, where strict cooling requirements forbid an intermediate storage inside the terminal, so that all products are to be instantaneously loaded onto refrigerated outbound trucks. The problem is formalized such that different operational objectives, i.e. the flow time, processing time and tardiness of outbound trucks, are minimized. To solve the resulting truck scheduling problem suited exact and heuristic solution procedures are presented.

A Binary Integer Program to Maximize the Agreement Between Partitions

This research note focuses on a problem where the cluster sizes for two partitions of the same object set are assumed known; however, the actual assignments of objects to clusters are unknown for one or both partitions. The objective is to find a contingency table that produces maximum possible agreement between the two partitions, subject to constraints that the row and column marginal frequencies for the table correspond exactly to the cluster sizes for the partitions. This problem was described by H. Messatfa (Journal of Classification, 1992, pp. 5---15), who provided a heuristic procedure based on the linear transportation problem. We present an exact solution procedure using binary integer programming. We demonstrate that our proposed method efficiently obtains optimal solutions for problems of practical size.

Exact solution procedures for the balanced unidirectional cyclic layout problem

In this paper, we consider the balanced unidirectional cyclic layout problem (BUCLP) arising in the determination of workstation locations around a closed loop conveyor system, in the allocation of cutting tools on the sites around a turret, in the positioning of stations around a unidirectional single loop AGV path. BUCLP is known to be NP-Complete. One important property of this problem is the balanced material flow assumption where the material flow is conserved at every workstation. We first develop a branch-and-bound procedure by using the special material flow property of the problem. Then, we propose a dynamic programming algorithm, which provides optimum solutions for instances with up to 20 workstations due to memory limitations. The branch and bound procedure can solve problems with up to 50 workstations.

Exact and inexact solution procedures for the order picking in an automated carousal conveyor

In many modern logistic systems, automated carousal conveyors are frequently used for storing/retrieving small to medium items to fulfil the customer's demand. The system is used to process a sequence of orders from customers in a dynamic fashion. Each order includes one or several items to be picked from the carousal storage or bin. The sequence between orders and the sequences among items within each order are to be determined, to minimise the total order processing or associated picking cost. This multi-order picking problem is formulated as 0–1 integer programming model, which resembles the well-known Multi-travelling salesman problem. Some important properties of this unique order processing system are described. New theoretical properties are established to characterise the optimal retrieval sequence in an order with multiple items to be picked. Based on these properties, efficient and effective heuristics are developed to solve this multi-order picking problem. The computational detail of these approaches is illustrated by an example. Numerical experiments with problem sizes up to 500 items for a batch of 50 orders are performed, where different order densities and order rates are considered.

A column generation method for the multiple-choice multi-dimensional knapsack problem

In this paper, we propose to solve large-scale multiple-choice multi-dimensional knapsack problems. We investigate the use of the column generation and effective solution procedures. The method is in the spirit of well-known local search metaheuristics, in which the search process is composed of two complementary stages: (i) a rounding solution stage and (ii) a restricted exact solution procedure. The method is analyzed computationally on a set of problem instances of the literature and compared to the results reached by both Cplex solver and a recent reactive local search. For these instances, most of which cannot be solved to proven optimality in a reasonable runtime, the proposed method improves 21 out of 27.

Exact and Approximate Solutions of Source Localization Problems

We consider least squares (LS) approaches for locating a radiating source from range measurements (which we call R-LS) or from range-difference measurements (RD-LS) collected using an array of passive sensors. We also consider LS approaches based on squared range observations (SR-LS) and based on squared range-difference measurements (SRD-LS). Despite the fact that the resulting optimization problems are nonconvex, we provide exact solution procedures for efficiently computing the SR-LS and SRD-LS estimates. Numerical simulations suggest that the exact SR-LS and SRD-LS estimates outperform existing approximations of the SR-LS and SRD-LS solutions as well as approximations of the R-LS and RD-LS solutions which are based on a semidefinite relaxation.

O uso da programação inteira 0-1 para o balanceamento de linhas de montagem: modelagem, estudos de caso e avaliação

O presente trabalho trata do problema do balanceamento de linha de montagem. Dentro deste contexto o objetivo é mostrar que nos dias atuais, em muitos casos práticos, é possível, mesmo em problemas relativamente grandes, a aplicação de procedimentos de solução exatos (no caso, programação inteira). Isto é feito por meio de dois estudos de caso em grandes empresas do Estado de São Paulo, as quais apresentam problemas de balanceamento de linha relativamente grandes (linhas com 80 e 104 tarefas). Em um destes problemas foi incluída no modelo uma restrição, inédita na literatura, para melhor adequá-lo à situação real. Esta restrição se refere a situações nas quais a linha de montagem possui tanto operações manuais quanto automáticas. Os resultados conseguidos foram muito bons, conseguindo-se redução de mais de 10% no número de estações de trabalho necessárias em cada caso.

Demand Allocation in Systems with Multiple Inventory Locations and Multiple Demand Sources

We consider the problem of allocating demand that originates from multiple sources among multiple inventory locations. Demand from each source arrives dynamically according to an independent Poisson process. The cost of fulfilling each order depends on both the source of the order and its fulfillment location. Inventory at all locations is replenished from a shared production facility with a finite production capacity and stochastic production times. Consequently, supply lead times are load dependent and affected by congestion at the production facility. Our objective is to determine an optimal demand allocation and optimal inventory levels at each location so that the sum of transportation, inventory, and backorder costs is minimized. We formulate the problem as a nonlinear optimization problem and characterize the structure of the optimal allocation policy. We show that the optimal demand allocations are always discrete, with demand from each source always fulfilled entirely from a single inventory location. We use this discreteness property to reformulate the problems as a mixed-integer linear program and provide an exact solution procedure. We show that this discreteness property extends to systems with other forms of supply processes. However, we also show that supply systems exist for which the property does not hold. Using numerical results, we examine the impact of different parameters and provide some managerial insights.

The β-Reliable Median on a Network with Discrete Probabilistic Demand Weights

This study extends the classical network median problem by considering the stochastic nature of demand. Assuming that the demand weights associated with nodes are independent discrete random variables, we introduce a chance-constrained programming model to define a β-reliable median of the network. It is shown that the β-reliable median problem is NP-hard. Exact solution procedures and a normal approximation algorithm are developed to search for the β-reliable median. Their performance is evaluated by computational experiments.

A Survey of the Generalized Assignment Problem and Its Applications

Given n items and m knapsacks, the Generalized Assignment Problem (GAP) is to find the optimum assignment of each item to exactly one knapsack, without exceeding the capacity of any knapsack. This problem can also be described as the optimal assignment of n jobs to m capacitated agents. During the last three decades, many papers have been published on the GAP. In this survey we mainly concentrate on its real-life applications in scheduling, timetabling, telecommunication, facility location, transportation, production planning, etc. We also mention some of the most recent solution approaches: from state-of-the-art metaheuristics to variable neighborhood search algorithms and from exact solution procedures to simple heuristic algorithms.

A Survey of the Generalized Assignment Problem and Its Applications

Given n items and m knapsacks, the Generalized Assignment Problem (GAP) is to find the optimum assignment of each item to exactly one knapsack, without exceeding the capacity of any knapsack. This problem can also be described as the optimal assignment of n jobs to m capacitated agents. During the last three decades, many papers have been published on the GAP. In this survey we mainly concentrate on its real-life applications in scheduling, timetabling, telecommunication, facility location, transportation, production planning, etc. We also mention some of the most recent solution approaches: from state-of-the-art metaheuristics to variable neighborhood search algorithms and from exact solution procedures to simple heuristic algorithms.

Optimal Control of Serial Inventory Systems with Fixed Replenishment Intervals

We consider a single-item, periodic-review, serial inventory/production system, with linear inventory-holding and penalty costs. To facilitate shipment consolidation and capacity planning, we assume that the system has implemented fixed replenishment intervals; each stage is allowed to order only at given equidistant times. Further, for each stage except the most downstream one, the replenishment interval is assumed to be an integer multiple of the replenishment interval of the next downstream stage. This reflects the fact that the further upstream in a supply chain, the higher setup times and costs tend to be, and thus larger batches are desired. Our model with fixed replenishment intervals is a direct generalization of the serial model of Clark and Scarf (1960). For this generalized model, we prove the optimality of base-stock policies, we derive newsboy equations for the optimal base-stock levels, and we describe an efficient exact solution procedure for the case with mixed Erlang demands. Finally, we present extensions to assembly systems and to systems with a modified fill-rate constraint instead of backorder costs.

An exact solution procedure for a cluster hub location problem

This paper proposes a new (MIP) model formulation and a new solution procedure for the hub network design problem under a non-restrictive policy introduced by Sung and Jin [Sung, C.S., Jin, H.W., 2001. Dual-based approach for a hub network design problem under non-restrictive policy. European Journal of Operational Research 132 (1), 88–105]. The model formulation contains significantly fewer variables so that optimal solutions for the LP-relaxation of the model can be determined for large instances using standard procedures for LP-models. Furthermore, the LP-relaxation provides very tight lower bounds. Computational results are given, which demonstrate that the new model formulation allows for solving much larger instances. It turned out that the new (exact) solution procedure, which utilises the new model formulation, is faster than the heuristic proposed by Sung and Jin (2001). It is also shown that the problem is np-hard.

The joint replenishment problem with variable production costs

This paper discusses an approach to solve the joint replenishment problem in a production environment with variable production cost. These variable production costs occur due to economies of scale in production. Under this environment, the model leads to a global optimization problem, which is investigated by using some standard results from convex analysis. Consequently, an effective and exact solution procedure is proposed. The proposed procedure is guaranteed to return a solution with a predetermined quality in terms of the objective function value. A computational study is provided to illustrate the performance of the proposed solution procedure with respect to the running time.

State-of-the-art exact and heuristic solution procedures for simple assembly line balancing

The assembly line balancing problem arises and has to be solved when an assembly line has to be configured or redesigned. It consists of distributing the total workload for manufacturing any unit of the product to be assembled among the work stations along the line. The so-called simple assembly line balancing problem (SALBP), a basic version of the general problem, has attracted attention of researchers and practitioners of operations research for almost half a century. In this paper, we give an up-to-date and comprehensive survey of SALBP research with a special emphasis on recent outstanding and guiding contributions to the field.

An exact solution procedure to determine the optimal dispatching times for complex rail networks

Trains operating in densely populated metropolitan areas typically encounter complex track configurations. To make optimal use of the available rail capacity, some portions of the rail network may consist of single-track lines whereas other locations may consist of double- or triple-track lines. This paper develops a branch-and-bound procedure that is able to determine the optimal dispatching times for trains traveling in complex rail networks. We demonstrate the efficiency of our branch-and-bound algorithm by comparing it to CPLEX, a commercially available integer program solver, on an actual rail network in Los Angeles County.