A Lagrangian heuristic algorithm for a public healthcare facility location problem

We consider a healthcare facility location problem in which there are two types of patients, low-income patients and middle- and high-income patients. The former can use only public facilities, while the latter can use both public facilities and private facilities. We focus on the problem of determining locations of public healthcare facilities to be established within a given budget and allocating the patients to the facilities for the objective of maximizing the number of served patients while considering preference of the patients for the public and private facilities. We present an integer programming formulation for the problem and develop a heuristic algorithm based on Lagrangian relaxation and subgradient optimization methods. Results of computational experiments on a number of problem instances show that the algorithm gives good solutions in a reasonable computation time and may be effectively used by the healthcare authorities of the government.     

A scatter search methodology for the nurse rostering problem

The benefits of automating the nurse scheduling process in hospitals include reducing the planning workload and associated costs and being able to create higher quality and more flexible schedules. This has become more important recently in order to retain nurses and to attract more people into the profession. Better quality rosters also reduce fatigue and stress due to overwork and poor scheduling and help to maximise the use of leisure time by satisfying more requests. A more contented workforce will lead to higher productivity, increased quality of patient service and a better level of healthcare. This paper presents a scatter search approach for the problem of automatically creating nurse rosters. Scatter search is an evolutionary algorithm, which has been successfully applied across a number of problem domains. To adapt and apply scatter search to nurse rostering, it was necessary to develop novel implementations of some of scatter search's subroutines. The algorithm was then tested on publicly available real-world benchmark instances and compared against previously published approaches. The results show the proposed algorithm is a robust and effective method on a wide variety of real-world instances.     

Minimizing Labor Requirements in a Periodic Vehicle Loading Problem

In this paper, we address a logistics problem that a manufacturer of auto parts in the north of Spain described to the authors. The manufacturer stores products in its warehouse until customers retrieve them. The customers and the manufacturer agree upon an order pickup frequency. The problem is to find the best pickup schedule, which consists of the days and times during the day that each customer is expected to retrieve his/her order. For a given planning horizon, the optimization problem is to minimize the labor requirements to load the vehicles that the customers use to pick up their orders. Heuristically, we approach this situation as a decision problem in two levels. At the first level, customers are assigned to a calendar, consisting of a set of days with the required frequency during the planning horizon. Then, for each day, the decision at the second level is to assign each customer to a time slot. The busiest time slot determines the labor requirement for a given day. Therefore, once customers have been assigned to particular days in the planning horizon, the second-level decision is a multiprocessor scheduling problem, where each time slot is the equivalent of a processor, and where the objective is to minimize the makespan. A metaheuristic procedure is developed for the problem of minimizing labor requirements in this periodic vehicle-loading problem and artificial as well as real data are used to assess its performance.     

A multi-period order selection problem in flexible manufacturing systems

We consider a multi-period order selection problem in flexible manufacturing systems, which is the problem of selecting orders to be produced in each period during the upcoming planning horizon with the objective of minimising earliness and tardiness costs and subcontracting costs. The earliness and tardiness costs are incurred if an order is not finished on time, while subcontracting cost is incurred if an order is not selected within the planning horizon (and must be subcontracted) due to processing time capacity or tool magazine capacity. This problem is formulated as a 0–1 integer program which can be transformed into a generalised assignment problem. To solve the problem, a heuristic algorithm is developed using a Lagrangian relaxation technique. Effectiveness of the algorithm is tested on randomly generated problems and results are reported.     

A variable neighborhood search heuristic for periodic routing problems

The aim of this paper is to propose a new heuristic for the Periodic Vehicle Routing Problem (PVRP) without time windows. The PVRP extends the classical Vehicle Routing Problem (VRP) to a planning horizon of several days. Each customer requires a certain number of visits within this time horizon while there is some flexibility on the exact days of the visits. Hence, one has to choose the visit days for each customer and to solve a VRP for each day. Our method is based on Variable Neighborhood Search (VNS). Computational results are presented, that show that our approach is competitive and even outperforms existing solution procedures proposed in the literature. Also considered is the special case of a single vehicle, i.e. the Periodic Traveling Salesman Problem (PTSP). It is shown that slight changes of the proposed VNS procedure is also competitive for the PTSP.     

Time-dependent discrete road network design with both tactical and strategic decisions

This paper aims to model and investigate the discrete urban road network design problem, using a multi-objective time-dependent decision-making approach. Given a base network made up with two-way links, candidate link expansion projects, and candidate link construction projects, the problem determines the optimal combination of one-way and two-way links, the optimal selection of capacity expansion projects, and the optimal lane allocations on two-way links over a dual time scale. The problem considers both the total travel time and the total CO emissions as the two objective function measures. The problem is modelled using a time-dependent approach that considers a planning horizon of multiple years and both morning and evening peaks. Under this approach, the model allows determining the sequence of link construction, the expansion projects over a predetermined planning horizon, the configuration of street orientations, and the lane allocations for morning and evening peaks in each year of the planning horizon. This model is formulated as a mixed-integer programming problem with mathematical equilibrium constraints. In this regard, two multi-objective metaheuristics, including a modified non-dominated sorting genetic algorithm (NSGA-II) and a multi-objective B-cell algorithm, are proposed to solve the above-mentioned problem. Computational results for various test networks are also presented in this paper.     

An Exact Solution Algorithm for a Class of Production Planning and Scheduling Problems

This paper presents a formulation and an exact solution algorithm for a class of production planning and scheduling problems. The problem is one of optimally specifying production levels for each product in each period of the planning horizon. The objective is to minimize the sum of the set-up, regular time production, overtime and inventory holding costs. The problem has been formulated as a variation of fixed charge transportation problem. The problem discussed here is NP-hard in computational complexity. A numerical example is presented for better understanding of the algorithm.     

A stochastic programming approach for planning horizons of infinite horizon capacity planning problems

Planning horizon is a key issue in production planning. Different from previous approaches based on Markov Decision Processes, we study the planning horizon of capacity planning problems within the framework of stochastic programming. We first consider an infinite horizon stochastic capacity planning model involving a single resource, linear cost structure, and discrete distributions for general stochastic cost and demand data (non-Markovian and non-stationary). We give sufficient conditions for the existence of an optimal solution. Furthermore, we study the monotonicity property of the finite horizon approximation of the original problem. We show that, the optimal objective value and solution of the finite horizon approximation problem will converge to the optimal objective value and solution of the infinite horizon problem, when the time horizon goes to infinity. These convergence results, together with the integrality of decision variables, imply the existence of a planning horizon. We also develop a useful formula to calculate an upper bound on the planning horizon. Then by decomposition, we show the existence of a planning horizon for a class of very general stochastic capacity planning problems, which have complicated decision structure.     

Capacity planning for phased implementation of flexible manufacturing systems under budget restrictions

We consider a problem of gradually replacing conventional dedicated machines with flexible manufacturing modules (FMMs) under budget restrictions over a finite planning horizon assuming that dedicated machines cannot be purchased during the planning horizon and acquired FMMs are kept until the end of the horizon. In the problem, a replacement schedule is to be determined and operations are to be assigned to the FMMs or the dedicated machines with the objective of minimizing the sum of discounted costs of acquisition and operation of FMMs and operation costs of conventional dedicated machines. In this research, the problem is formulated as a mixed integer linear program and solved by a Lagrangean relaxation approach. A subgradient optimization method is employed to obtain lower bounds of solutions and a multiplier adjustment method is devised to improve the lower bounds. We develop a linear programming-based Lagrangean heuristic algorithm to find a good feasible solution of the original problem in a reasonable amount of computation time. The algorithm is tested on randomly generated test problems and the results are reported.     

Optimal Pricing and Production Planning for Multi-product Multi-period Systems with Backorders

In this paper, we develop models for production planning with coordinated dynamic pricing. The application that motivated this research is manufacturing pricing, where the products are non-perishable assets and can be stored to fulfill the future demands. We assume that the firm does not change the price list very frequently. However, the developed model and its solution strategy have the capability to handle the general case of manufacturing systems with frequent time-varying price lists. We consider a multi-product capacitated setting and introduce a demand-based model, where the demand is a function of the price. The key parts of the model are that the planning horizon is discrete-time multi-period, and backorders are allowed. As a result of this, the problem becomes a nonlinear programming problem with the nonlinearities in both the objective function and some constraints. We develop an algorithm which computes the optimal production and pricing policy on a finite time horizon. We illustrate the application of the algorithm through a detailed numerical example.     

The part families problem in flexible manufacturing systems

Parts grouping into families can be performed in flexible manufacturing systems (FMSs) to simplify two classes of problems: long horizon planning and short horizon planning. In this paper the emphasis is on the part families problem applicable to the short horizon planning. Traditionally, parts grouping was based on classification and coding systems, some of which are reviewed in this paper. To overcome the drawbacks of the classical approach to parts grouping, two new methodologies are developed. The methodologies presented are very easy to implement because they take advantage of the information already stored in the CAD system. One of the basic elements of this system is the algorithm for solving the part families problem. Some of the existing clustering algorithms for solving this problem are discussed. A new clustering algorithm has been developed. The computational complexity and some of the computational results of solving the part families problem are also discussed.     

Part type selection problem in flexible manufacturing systems: tabu search algorithms

Production planning in flexible manufacturing systems is concerned with the organization of production in order to satisfy a given master production schedule. The planning problem typically gives rise to several hierarchical subproblems which are then solved sequentially or simultaneously. In this paper, we address one of the subproblems: the part type selection problem. The problem is to determine a subset of part types having production requirements for immediate and simultaneous processing over the upcoming period of the planning horizon, subject to the tool magazine and processing time limitation. Several versions of tabu search (TS) algorithm are proposed for solving the problem. A systematic computational test is conducted to test the performance of the TS algorithms. The best TS algorithm developed is compared to a simulated annealing algorithm.     

A hybrid evolutionary algorithm for the periodic location-routing problem

The well-known vehicle routing problem (VRP) has been studied in depth over the last decades. Nowadays, generalizations of VRP have been developed for tactical or strategic decision levels of companies but not both. The tactical extension or periodic VRP (PVRP) plans a set of trips over a multiperiod horizon, subject to frequency constraints. The strategic extension is motivated by interdependent depot location and routing decisions in most distribution systems. Low-quality solutions are obtained if depots are located first, regardless of the future routes. In the location-routing problem (LRP), location and routing decisions are tackled simultaneously. Here for the first time, except for some conference papers, the goal is to combine the PVRP and LRP into an even more realistic problem covering all decision levels: the periodic LRP or PLRP. A hybrid evolutionary algorithm is proposed to solve large size instances of the PLRP. First, an individual representing an assignment of customers to combinations of visit days is randomly generated. The evolution operates through an Evolutionary Local Search (ELS) on visit day assignments. The algorithm is hybridized with a heuristic based on the Randomized Extended Clarke and Wright Algorithm (RECWA) to create feasible solutions and stops when a given number of iterations is reached. The method is evaluated over three sets of instances, and solutions are compared to the literature on particular cases such as one-day horizon (LRP) or one depot (PVRP). This metaheuristic outperforms the previous methods for the PLRP.     

A Lagrangean heuristic algorithm for disassembly scheduling with capacity constraints

This paper considers the problem of determining the disassembly schedule (quantity and timing) of products in order to satisfy the demand of their parts or components over a finite planning horizon. The objective is to minimize the sum of set-up, disassembly operation, and inventory holding costs. As an extension of the uncapacitated versions of the problem, we consider the resource capacity restrictions over the planning horizon. An integer program is suggested to describe the problem mathematically, and to solve the problem, a heuristic is developed using a Lagrangean relaxation technique together with a method to find a good feasible solution while considering the trade-offs among different costs. The effectiveness of the algorithm is tested on a number of randomly generated problems and the test results show that the heuristic suggested in this paper can give near optimal solutions within a short amount of computation time.     

A Lagrangian relaxation approach to multi-period inventory/distribution planning

We consider a multi-period inventory/distribution planning problem (MPIDP) in a one-warehouse multiretailer distribution system where a fleet of heterogeneous vehicles delivers products from a warehouse to several retailers. The objective of the MPIDP is to minimise transportation costs for product delivery and inventory holding costs at retailers over the planning horizon. In this research, the problem is formulated as a mixed integer linear programme and solved by a Lagrangian relaxation approach. A subgradient optimisation method is employed to obtain lower bounds. We develop a Lagrangian heuristic algorithm to find a good feasible solution of the MPIDP. Computational experiments on randomly generated test problems showed that the suggested algorithm gave relatively good solutions in a reasonable amount of computation time.     

Rotation cycle schedulings for multi-item production systems

Consider the production planning and scheduling on a single machine with finite constant production rate over a planning horizon N. For single-item production problem, we have characterised the structure of the optimal solution when N approaches to infinity. This result suggests a near optimal solution when the planning horizon N is large. For multi-item production problem, we restrict our analysis on the Rotation Cycle policies. Under the assumptions of the policy, we convert the problem into a generalised travelling salesman problem and hence a branch and bound algorithm is proposed to solve the problem. For a given error bound of the solution, the algorithm can be further simplified to determine a near-optimal rotation cycle.     

A heuristic for the multi-period petrol station replenishment problem

In the multi-period petrol station replenishment problem (MPSRP) the aim is to optimize the delivery of several petroleum products to a set of petrol stations over a given planning horizon. One must determine, for each day of the planning horizon, how much of each product should be delivered to each station, how to load these products into vehicle compartments, and how to plan vehicle routes. The objective is to maximize the total profit equal to the revenue, minus the sum of routing costs and of regular and overtime costs. This article describes a heuristic for the MPSRP. It contains a route construction and truck loading procedures, a route packing procedure, and two procedures enabling the anticipation or the postponement of deliveries. The heuristic was extensively tested on randomly generated data and compared to a previously published algorithm. Computational results confirm the efficiency of the proposed methodology.     

Project Scheduling under Time Dependent Costs – A Branch and Bound Algorithm

In a given project network, execution of each activity in normal duration requires utilization of certain resources. If faster execution of an activity is desired then additional resources at extra cost would be required. Given a project network, the cost structure for each activity and a planning horizon, the project compression problem is concerned with the determination of optimal schedule (duration) of performing each activity while satisfying given restrictions and minimizing the total cost of project execution. This paper considers the project compression problem with time dependent cost structure for each activity. The planning horizon is divided into several regular time intervals over which the cost structure of an activity may vary. But the cost structure of the activities remains the same (constant) within a time interval. Key events of the project attract penalty for finishing earlier or later than the corresponding target times. The objective is to find an optimal project schedule minimizing the total project cost. We present a mathematical model for this problem, develop some heuristics and an exact branch and bound algorithm. Using simulated problems we provide an insight into the computational performances of heuristics and the branch and bound algorithm.     

A branch and bound algorithm for disassembly scheduling with assembly product structure

This paper considers a production planning problem in disassembly systems, which is the problem of determining the quantity and timing of disassembling end-of-use/life products in order to satisfy the demand of their parts or components over a planning horizon. The case of single product type without parts commonality is considered for the objective of minimizing the sum of setup and inventory holding costs. To show the complexity of the problem, we prove that the problem is NP-hard. Then, after deriving the properties of optimal solutions, a branch and bound algorithm is suggested that incorporates the Lagrangean relaxation-based upper and lower bounds. Computational experiments are performed on a number of randomly generated problems and the test results indicate that the branch and bound algorithm can give optimal solutions up to moderate-sized problems in a reasonable computation time. A Lagrangean heuristic for a viable alternative for large-sized problems is also suggested and compared with the existing heuristics to show its effectiveness.     

A pre-assignment heuristic algorithm for the Master Surgical Schedule Problem (MSSP)

In this paper a 0–1 linear programming model and a solution heuristic algorithm are developed in order to solve the so-called Master Surgical Schedule Problem (MSSP). Given a hospital department made up of different surgical units (i.e. wards) sharing a given number of Operating Rooms (ORs), the problem herein addressed is determining the assignment among wards and ORs during a given planning horizon, together with the subset of patients to be operated on during each day. Different resource constraints related to operating block time length, maximum OR overtime allowable by collective labour agreement and legislation, patient length of stay (LOS), available OR equipment, number of surgeons, number of stay and ICU beds, are considered. Firstly, a 0–1 linear programming model intended to minimise a cost function based upon a priority score, that takes into proper account both the waiting time and the urgency status of each patient, is developed. Successively, an heuristic algorithm that enables us to embody some pre-assignment rules to solve this NP-hard combinatorial optimisation problem, is presented. In particular, we force the assignment of each patient to a subset of days depending on his/her expected length of stay in order to allow closing some stay areas during the weekend and hence reducing overall hospitalisation cost of the department. The results of an extensive computational experimentation aimed at showing the algorithm efficiency in terms of computational time and solution effectiveness are given and analysed.