Exact Solution Procedure Research Articles

Solving reverse logistics vehicle routing problems with time windows

A vehicle routing problem with simultaneous pick-up and delivery in closed-loop logistics network optimization is studied in this paper. Since, in practice, material pick-up and delivery are only allowed to take place on certain time periods, we consider the reverse logistics vehicle routing problem with time windows. A mixed integer programming model is proposed to formulate the considered problem. A heuristic solution approach for solving the model is developed due to the NP-hard nature of solving the model. The heuristic solution is then used as an initial solution of a simulated annealing procedure for improved solutions. The proposed heuristic method and the simulated annealing procedure yield very promising solutions in much less computational time when compared with optimal solutions generated by exact solution procedures. Numerical examples are presented to illustrate the developed model and solution methods.

Value-at-Risk model for hazardous material transportation

This paper introduces a Value-at-Risk (VaR) model to generate route choices for a hazmat shipment based on a specified risk confidence level. VaR is a threshold value such that the probability of the loss exceeding the VaR value is less than a given probability level. The objective is to determine a route which minimizes the likelihood that the risk will be greater than a set threshold. Several properties of the VaR model are established. An exact solution procedure is proposed and tested to solve the single-trip problem. To test the applicability of the approach, routes obtained from the VaR model are compared with those obtained from other hazmat objectives, on a numerical example as well as a hazmat routing scenario derived from the Albany district of New York State. Depending on the choice of the confidence level, the VaR model gives different paths from which we conclude that the route choice is a function of the level of risk tolerance of the decision-maker. Further refinements of the VaR model are also discussed.

Solving Integer Programming Problems with a Variable Number of Switching Costs, Balking and Feedback

We consider a retail service facility with cross trained servers who can perform operations in both front room and back room depending on the length of the queue. Two essential models for service facility are discussed in this paper. The models are represented as a Markovian decision problem and solved using mixed integer programming techniques. Exact solution procedures are developed to solve the models and numerical study is conducted to see the impact of various parameters on performance.

Determining crane areas for balancing workload among interfering and noninterfering cranes

AbstractThis article treats the problem of subdividing an area for storing containers such that the workload is evenly shared among the cranes operating the resulting subareas. We consider two crane sets: while noncrossing constraints between cranes of the same set need to be observed, cranes of different sets do not interfere. Such a problem setting is, for instance, relevant for scheduling the (un‐)loading of vessels by parallel quay cranes operating on opposing berths or in container yards with cross‐over cranes. We formalize the resulting optimization problem, prove computational complexity, and present exact and heuristic solution procedures. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012

Optimal pricing and ordering policy for perishable items with limited storage capacity and partial trade credit

Thangam & Uthayakumar (2010, Optimal pricing and lot-sizing policy for a two warehouse supply chain system with perishable items under partial trade credit financing. Oper. Res. Int. J., 10, 133–161) investigated perishable items with limited storage capacity under a partial trade credit policy. For simplification, they used Taylor series approximations to derive their solution procedures. In our paper, we develop an exact solution procedure that avoids the problem of assuming that the deterioration rate is small, thereby improving the accuracy of the solution but more fundamentally increasing the practical relevance of the solution.

Optimally loading tow trains for just-in-time supply of mixed-model assembly lines

In today's mixed-model assembly production, there are two recent trends—namely, increasing vertical integration and the proliferation of product variety—that shift focus to an efficient just-in-time part supply. In this context, many automobile manufacturers set up decentralized logistics areas referred to as supermarkets. Here, small tow trains are loaded with parts and travel across the shop floor on specific routes to make frequent small-lot deliveries that are needed by the stations of the line. This article investigates the loading problem of tow trains, which aims at minimizing inventory near the line while avoiding material shortages given the limited capacity of the tow trains. An exact polynomial-time solution procedure is presented and interdependencies with production planning, that is, the sequencing problem of product models launched down the line, are investigated in a comprehensive computational study.

A graph-pair representation and MIP-model-based heuristic for the unequal-area facility layout problem

Owing to its theoretical as well as practical significance, the facility layout problem with unequal-area departments has been studied for several decades, with a wide range of heuristic and a few exact solution procedures developed by numerous researchers. In one of the exact procedures, the facility layout problem is formulated as a mixed-integer programming (MIP) model in which binary (0/1) variables are used to prevent departments from overlapping with one another. Obtaining an optimal solution to the MIP model is difficult, and currently only problems with a limited number of departments can be solved to optimality. Motivated by this situation, we developed a heuristic procedure which uses a “graph pair” to determine and manipulate the relative location of the departments in the layout. The graph-pair representation technique essentially eliminates the binary variables in the MIP model, which allows the heuristic to solve a large number of linear programming models to construct and improve the layout in a comparatively short period of time. The search procedure to improve the layout is driven by a simulated annealing algorithm. The effectiveness of the proposed graph-pair heuristic is demonstrated by comparing the results with those reported in recent papers. Possible extensions to the graph-pair representation technique are discussed at the end of the paper.

Optimally routing and scheduling tow trains for JIT-supply of mixed-model assembly lines

In recent years, more and more automobile producers adopted the supermarket-concept to enable a flexible and reliable Just-in-Time (JIT) part supply of their mixed-model assembly lines. Within this concept, a supermarket is a decentralized in-house logistics area where parts are intermediately stored and then loaded on small tow trains. These tow trains travel across the shop floor on specific routes to make frequent small-lot deliveries which are needed by the stations of the line. To enable a reliable part supply in line with the JIT-principle, the interdependent problems of routing, that is, partitioning stations to be supplied among tow trains, and scheduling, i.e., deciding on the start times of each tow train’s tours through its assigned stations, need to be solved. This paper introduces an exact solution procedure which solves both problems simultaneously in polynomial runtime. Additionally, management implications regarding the trade-off between number and capacity of tow trains and in-process inventory near the line are investigated within a comprehensive computational study.

Sequencing mixed-model assembly lines to minimise the number of work overload situations

The mixed-model sequencing problem is to sequence different product models launched down an assembly line, so that work overload at the stations induced by direct succession of multiple labour-intensive models is avoided. As a concept of clearing overload situations, especially applied by Western automobile producers, a team of cross-trained utility workers stands by to support the regular workforce. Existing research assumes that regular and utility workers assemble side-by-side in an overload situation, so that the processing speed is doubled and the workpiece can be finished inside a station's boundaries. However, in many real-world assembly lines the application of utility workers is organised completely differently. Whenever it is foreseeable that a work overload will occur in a production cycle, a utility worker takes over to exclusively execute work, whereas the regular worker omits the respective cycle and starts processing the successive workpiece as soon as possible. This study investigates this more realistic sequencing problem and presents a binary linear program along with a complexity proof. Different exact and heuristic solution procedures are then introduced and tested. Additional experiments show that the new model is preferable from an economic point of view whenever utility work causes considerable setup activities, for example walking to the respective station.

Automated knowledge source selection and service composition

We introduce a new combinatorial problem referred to as the component set identification problem, arising in the context of knowledge discovery, information integration, and knowledge source/service composition. The main motivation for studying this problem is the widespread proliferation of digital knowledge sources and services. Considering a granular knowledge domain consisting of a large number of individual bits and pieces of domain knowledge (properties) and a large number of knowledge sources and services that provide mappings between sets of properties, the objective of the component set identification problem is to select a minimum cost combination of knowledge sources that can provide a joint mapping from a given set of initially available properties (initial knowledge) to a set of initially unknown properties (target knowledge). We position the component set identification problem relative to other combinatorial problems and provide a classification scheme for the different variations of the problem. The problem is next modeled on a directed graph and analyzed in terms of its complexity. The directed graph representation is then augmented and transformed into a time-expanded network representation that is subsequently used to develop an exact solution procedure based on integer programming and branch-and-bound. We enhance the solver by developing preprocessing techniques. All findings are supported by computational experiments.

Scheduling just-in-time part supply for mixed-model assembly lines

With increasing cost competition and product variety, providing an efficient just-in-time (JIT) supply has become one of the greatest challenges in the use of mixed-model assembly line production systems. In the present paper, therefore, we propose a new approach for scheduling JIT part supply from a central storage center. Usually, materials are stored in boxes that are allotted to the consumptive stations of the line by a forklift. For such a real-world problem, a new model, a complexity proof as well as different exact and heuristic solution procedures are provided. Furthermore, a direct comparison with a simple two-bin kanban system is provided. Such a system is currently applied in the real-world industrial process that motivates our research. It becomes obvious that this policy is considerably outperformed according to the resulting inventory- and α-service levels. Moreover, at the interface between logistics and assembly operations, strategic management implications are obtained. Specifically, based on the new approach, it is the first time a statistical analysis is being made as to whether widespread Level Scheduling policies, which are well-known from the Toyota Production System, indeed facilitate material supply. Note that in the literature it is frequently claimed that this causality exists.

Vendor-buyer lot sizing problem with stochastic demand: an exact procedure under service level approach

In this study, we consider the joint economic lot sizing problem (JELP) under stochastic demand. Almost all of the literature on JELP assumes that the demand is deterministic and constant with very limited exceptions. We indicate the shortcomings of the work in the literature in this regard, and suggest a service-level based model for JELP under stochastic demand with an exact solution procedure. We also include numerical examples to illustrate the procedure and the value of the stochastic solution. [Submitted 02 September 2009; Revised 20 December 2009, 25 February 2009; Accepted 26 February 2010]

Rationing mechanisms and inventory control-policy parameters for a divergent supply chain operating with lost sales and costs of review

We consider a static divergent two-stage supply chain with one distributor and many retailers. The unsatisfied demands at the retailers’ end are treated as lost sales, whereas the unsatisfied demand is assumed to be backlogged at the distributor. The distributor uses an inventory rationing mechanism to distribute the available on-hand inventory among the retailers, when the sum of demands from the retailers is greater than the on-hand inventory at the distributor. The present study aims at determining the best installation inventory control-policy or order-policy parameters such as the base-stock levels and review periods, and inventory rationing quantities, with the objective of minimizing the total supply chain costs (TSCC) consisting of holding costs, shortage costs and review costs in the supply chain over a finite planning horizon. An exact solution procedure involving a mathematical programming model is developed to determine the optimum TSCC, base-stock levels, review periods and inventory rationing quantities (in the class of periodic review, order-up-to S policy) for the supply chain model under study. On account of the computational complexity involved in optimally solving problems over a large finite time horizon, a genetic algorithm (GA) based heuristic methodology is presented.

Two Algorithms for Relaxed Structural Balance Partitioning: Linking Theory, Models, and Data to Understand Social Network Phenomena

Understanding social phenomena with the help of mathematical models requires a coherent combination of theory, models, and data together with using valid data analytic methods. The study of social networks through the use of mathematical models is no exception. The intuitions of structural balance were formalized and led to a pair of remarkable theorems giving the nature of partition structures for balanced signed networks. Algorithms for partitioning signed networks, informed by these formal results, were developed and applied empirically. More recently, ‘‘structural balance’’ was generalized to ‘‘relaxed structural balance,’’ and a modified partitioning algorithm was proposed. Given the critical interplay of theory, models, and data, it is important that methods for the partitioning of signed networks in terms of relaxed structural balance model are appropriate. The authors consider two algorithms for establishing partitions of signed networks in terms of relaxed structural balance. One is an older heuristic relocation algorithm, and the other is a new exact solution procedure. The former can be used both inductively and deductively. When used deductively, this requires some prespecification incorporating substantive insights. The new branch-and-bound algorithm is used inductively and requires no prespecification of an image matrix in terms of ideal blocks. Both procedures are demonstrated using several examples from the literature, and their contributions are discussed. Together, the two algorithms provide a sound foundation for partitioning signed networks and yield optimal partitions. Issues of network size and density are considered in terms of their consequences for algorithm performance.

Optimal dynamic price selection under attraction choice models

We present a single-resource finite-horizon Markov decision process approach for a firm that seeks to maximize expected revenues by dynamically adjusting the menu of offered products and their prices to be selected from a finite set of alternative values predetermined as a matter of policy. Consumers choose among available products according to an attraction choice model, a special but widely applied class of discrete choice models. The fractional substructure inherent to attraction choice models enables the derivation of structural properties that significantly simplify the computation of an optimal policy. We show that the more capacity (or less time) is remaining, the lower are the optimal prices. We develop an exact solution procedure for preprocessing and recursively solving price selection problems that clearly outperforms current revenue management approaches with respect to computational complexity. Furthermore, we demonstrate that the potential revenue improvements from dynamic compared to fixed pricing can be significant, ranging from 4.5% up to 149% for selected examples from the literature. Extensions to multiple resources and to related discrete choice models are discussed.

To Wait or Not to Wait—And Who Goes First? Delay Management with Priority Decisions

Delay management is an important issue in the daily operations of any railway company. The task is to update the planned timetable to a disposition timetable in such a way that the inconvenience for the passengers is as small as possible. The two main decisions that have to be made in this respect are the wait-depart decisions, to decide which connections should be maintained in case of delays, and the priority decisions, which determine the order in which trains are allowed to pass a specific piece of track. The latter are necessary to take the limited capacity of the track system into account. While the wait-depart decisions have been intensively studied in the literature, the priority decisions in the capacitated case have been neglected so far in delay management optimization models. In the current paper, we add the priority decisions to the integer programming formulation of the delay management problem and are hence able to deal with the capacitated case. The corresponding constraints are disjunctive constraints leading to cycles in the resulting event-activity network. Nevertheless, we are able to derive reduction techniques for the network that enable us to extend the formulation of the never-meet property from the uncapacitated delay management problem to the capacitated case. We then use our results to derive exact and heuristic solution procedures for solving the delay management problem. The results of the algorithms are evaluated both from a theoretical and a numerical point of view. The latter has been done within a case study using the railway network in the region of Harz, Germany.

Determining crane areas in intermodal transshipment yards: The yard partition problem

At rail–road transshipment yards, gantry cranes move containers from freight trains to trucks and vice versa. They constitute important entities in today’s intermodal transportation systems. Real-world yards are often partitioned into several disjunct crane areas, so that crane interferences during container transshipment are avoided. In practice, the lengths of such crane areas are typically determined by simple rules of thumb, i.e., each crane receives an equally sized area, which might result in an unleveled division of labor among cranes and, thus, prolong train processing times. This paper provides an exact solution procedure which determines disjunct yard areas of varying size for multiple gantry cranes in polynomial runtime, so that the workload for a given pulse of trains is equally distributed among cranes. Furthermore, we investigate the potential acceleration of train processing as compared to equally sized areas in a yard simulation.

An exact solution procedure for multi-item two-echelon spare parts inventory control problem with batch ordering in the central warehouse

We consider a multi-item two-echelon inventory system in which the central warehouse operates under a ( Q , R ) policy, and the local warehouses implement basestock policy. An exact solution procedure is proposed to find the inventory control policy parameters that minimize the system-wide inventory holding and fixed ordering cost subject to an aggregate mean response time constraint at each facility.

An efficient Lagrangean heuristic for rental vehicle scheduling

In this article we present a heuristic based on Lagrangean relaxation for scheduling rental vehicles. The scheduling problem can be formulated as a set of large assignment problems with linking constraints. We discuss the theory behind the heuristic, including the ability to obtain lower bounds. The heuristic is based on ideas first introduced by D. Wedelin. Empirical testing shows that our heuristic is superior to the exact solution procedure of the MILP formulation with the commercial ILOG Cplex solver as it generally produces better solutions within a set time limit. Moreover, the heuristic can easily be parallelized to take advantage of multicore CPUs. The work presented here has been motivated by a real-life application and is currently being used by companies that rent out recreational vehicles. Testing on data sets from these companies shows the practicality of our algorithm.

Scheduling Truck Arrivals at an Air Cargo Terminal

We consider the scheduling of truck arrivals at an air cargo terminal. By coordinating arrivals of cargo delivery trucks with outbound flight departure schedules, some of the shipments can be transferred directly to the departing flights, while others will be stored at the terminal's storage facility and incur extra handling and storage costs. The objective is to obtain a feasible schedule so as to minimize the total cost of operations. We formulate the problem as a time‐indexed integer program and show that, even with limited number of unloading docks at the terminal, the problem is non‐trivial (NP‐hard in the strong sense). Our solution method includes an exact solution procedure to determine an optimal unloading sequence for the shipments carried by each truck, together with a Lagrangian relaxation‐based heuristic for assigning trucks to truck docks and determining truck arrival times. We conducted computational experiments to test the performance of our solution method. Computational results show that our method can generate near‐optimal solutions efficiently. Our simulation results indicate that the scheduling approach proposed in this paper has the potential to generate significant cost savings over a first‐come, first‐served approach currently used at the air cargo terminal that we observed.