Mathematical models and solution methods for optimal container terminal yard layouts

In this paper, we introduce an integer linear program for planning the layout of container yards. We concentrate on a special layout class of container yards which we call yard layout with transfer lanes. For those layouts typically rubber tired gantry cranes are used for stacking operations and trucks for horizontal transports. We show that the optimization model can be formulated as a special type of a resource constrained shortest path problem for which the LP relaxation always has at least one integer optimal solution. This model is restricted to a rectangular storage yard which allows a linear formulation. For an arbitrary shaped container yard we adopt the model and develop a variable neighborhood descent (VND) heuristic for solving non-rectangular instances. Concerning the rectangular case, we show that the VND heuristic achieves optimal solutions for 38% of the realistic test instances.     

Methods for optimum and near optimum disassembly sequencing

This paper considers disassembly sequencing problems subjected to sequence dependent disassembly costs. In practice, the methods for dealing with such problems rely mainly on metaheuristic and heuristic methods, which intrinsically generate suboptimum solutions. Exact methods are NP-hard and therefore unsuitable to most of the practical problems. Nevertheless, it is useful to have exact methods available that can be applied in order to check, at least medium sized problems, to what extent the heuristically obtained solutions deviate from the optimum solution. The existing exact approaches, which are based on integer linear programming (ILP), become unmanageable, even for the cases of modest product complexity. To alleviate this problem to some extent, the iterative method that has been proposed by Lambert (2006) is applied here. This method is based on repeatedly solving a binary integer linear programming (BILP) problem instead of an ILP problem. The method appears to converge sufficiently quickly to be valuable for dealing with medium sized problems. We then use the iterative method for the validation of a new heuristic method that is also proposed in this paper. Finally, both the heuristic and the iterative BILP methods are implemented on a cellphone from practice consisting of 25 components that are represented, according to a set of precedence relationships, via a disassembly precedence graph.     

A tabu search algorithm for simultaneous machine/AGV scheduling problem

Machines and automated guided vehicles (AGVs) scheduling problems are two essential issues that need to be addressed for the efficiency of the overall production system. The purpose of this paper is to study the simultaneous scheduling problem of machines and AGVs in a flexible manufacturing system (FMS) since the global optimum cannot be reached by considering each of them individually. In this paper, a mixed integer linear programming (MILP) model is developed with the objective of makespan minimisation. The MILP model consists of the following two constraint sets: machines and AGVs scheduling sub-problems. As both sub-problems are known to be NP-hard, a heuristic algorithm based on tabu search (TS) is proposed to get optimal or near to optimal solution for large-size problems within reasonable computation time. The proposed algorithm includes a novel two-dimensional solution representation and the generation of two neighbour solutions, which are alternately and iteratively applied to improve solutions. Moreover, an improved lower bound calculation method is introduced for the large-size problems. Computational results show the superior performance of the TS algorithm for the simultaneous scheduling problem.     

Linear Programming Based Genetic Algorithm for the Unequal Area Facility Layout Problem

The facility layout problem (FLP) is generally defined as locating a set of departments in a facility with a given dimension. In this paper, a hybrid genetic algorithm (GA)/linear programming (LP) approach is proposed to solve the FLP on the continuous plane with unequal area departments. This version of the FLP is very difficult to solve optimally due to the large number of binary decision variables in mixed integer programming (MIP) models as well as the lack of tight lower bounds. In this paper, a new encoding scheme, called the location/shape representation, is developed to represent layouts in a GA. This encoding scheme represents relative department positions in the facility based on the centroids and orientations of departments. Once relative department positions are set by the GA, actual department locations and shapes are determined by solving an LP problem. Finally, the output of the LP solution is incorporated into the encoding scheme of the GA. Numerical results are provided for test problems with varying sizes and department shape constraints. The proposed approach is able to either improve on or find the previously best known solutions of several test problems.     

Operator Scheduling

The development of mechanized scheduling of operators at Illinois Bell Telephone Company is discussed in this paper. The solution of the scheduling problem using integer linear programming was uneconomical so that a two step procedure was developed to solve the problem. First, the optimal solution with fractional operators is found by linear programming. Then a heuristic algorithm is used to find an integer solution close to this optimal solution. This method is in use at all Illinois Bell Telephone operator offices.     

A Disjunctive Model for Scheduling in a Manufacturing and Packing Facility with Intermediate Storage

The problem considered in this paper deals with the short term scheduling of a two stage continuous process with intermediate storage tanks. The major scheduling decisions in this problem are: a) the assignment of orders to various storage tanks; b) the sequence of orders in each unit; c) the timing of various operations in different stages. The problem is highly combinatorial in nature. The major challenge is to develop strong integer programming formulations and to devise efficient solution techniques. An initial model is presented in the form of a disjunctive program which is later transformed to a Mixed Integer Linear Programming (MILP) problem. A number of example problems are solved which highlight the limitations of this model as the number of orders increases. A heuristic based on partial preordering is considered which solves industrial sized problems very quickly. The objective function values for the heuristic solutions are within 7% of the optimal values.     

The simple and multiple job assignment problems

This paper addresses two real-life assignment problems. In both cases, the number of employees to whom tasks should be assigned is significantly greater than the number of tasks. In the simple job assignment problem, at most one task (job) should be assigned to each employee; this constraint is relaxed in the multiple job assignment problem. In both cases, the goal is to minimize the time the last task is completed: these problems are known as Bottleneck Assignment Problems (BAPs for short). We show that the simple job assignment problem can be solved optimally using an iterative approach based on dichotomy. At each iteration, a linear programming problem is solved: in this case the solution is integer. We propose a fast heuristic to solve the multiple job assignment problem, as well as a branchand-bound approach which leads to an optimal solution. Numerical examples are presented. They show that the heuristic is satisfactory for the application at hand.     

The Segregated Storage Problem: Some Properties and an Effective Heuristic

The segregated storage problem is formulated as an integer programming problem. Some special cases are shown to reduce to ordinary assignment problems. A special property of the integer formulation is used to develop a new heuristic procedure which produces near optimal solutions and which can be used as a tight bound in branch and bound procedures. Computational comparisons with heuristics proposed by other authors indicate this method to be superior in solving large problems.     

A Branch-and-price algorithm for placement routing for a multi-head beam-type component placement tool

We develop a branch-and-price procedure for a placement routing problem for a multi-head beam-type component placement tool. The problem is modelled as an integer programming model with a huge number of variables, each of which corresponds to a placement route. Its linear programming relaxation is solved by a column generation method. For the column generation subproblem to determine the columns to be added, we develop a dynamic programming procedure. We also propose an effective branching rule to partition the current solution space to eliminate the current fractional solution. Through experiments using real tool data, we observe that the LP relaxation solution value is noticeably close to an integer optimal solution value and hence the integer program formulation and the column generation approach are effective.     

A multi-stage facility location problem with staircase costs and splitting of commodities: model,heuristic approach and application

In this paper a multi-stage facility location problem with staircase costs and splitting of commodities is introduced and formulated as a mixed integer program. The problem is motivated by an application in the context of a reverse supply chain for end-of-life vehicles. We propose a two-phase heuristic solution approach: The greedy construction heuristic utilizes the solution obtained by the LP-relaxation of the problem. In the improvement heuristic we combine ADD, DROP and SWAP neighborhoods with a diversification strategy to a Variable Neighborhood Descent (VND) and to a Variable Neighborhood Search (VNS) approach. Computational results show that this approach is able reduce the duality gap provided by state-of-the-art MIP solver CPLEX for small and medium-sized instances and is also able to provide high-quality solutions for large-scale instances with up to 2,900 candidate facilities. The building blocks of the solution approach can easily be rearranged in order to solve other facility location problems.     

A minimum violations ranking method

We present a rating method that, given information on the pairwise comparisons of n items, minimizes the number of inconsistencies in the ranking of those items. Our Minimum Violations Ranking (MVR) Method uses a binary linear integer program (BILP) to do this. We prove conditions when the relaxed LP will give an optimal solution to the original BILP. In addition, the LP solution gives information about ties and sensitivities in the ranking. Lastly, our MVR method makes use of bounding and constraint relaxation techniques to produce a fast algorithm for the linear ordering problem, solving an instance with about one thousand items in less than 10 minutes.     

A Scheduling Algorithm for the Unbalanced Production Line

This paper presents procedures for obtaining feasible fractional and integer assignments which can be used to schedule manpower for the unbalanced production line. The objective is to minimize the cost of the total in-process inventory. A mathematical model is formulated and then used to calculate fractional assignments using linear programming and integer assignments using integer programming. An algorithm is then developed which provides the equivalent fractional solution as obtained from linear programming followed by a heuristic algorithm which converts these fractional assignments to feasible integer assignments. Sample problems and their solutions are provided to illustrate the assignment procedures.     

Solving large-scale fixed cost integer linear programming models for grid-based location problems with heuristic techniques

Grid-based location problems (GBLPs) can be used to solve location problems in business, engineering, resource exploitation, and even in the field of medical sciences. To solve these decision problems, an integer linear programming (ILP) model is designed and developed to provide the optimal solution for GBLPs considering fixed cost criteria. Preliminary results show that the ILP model is efficient in solving small to moderate-sized problems. However, this ILP model becomes intractable in solving large-scale instances. Therefore, a decomposition heuristic is proposed to solve these large-scale GBLPs, which demonstrates significant reduction of solution runtimes. To benchmark the proposed heuristic, results are compared with the exact solution via ILP. The experimental results show that the proposed method significantly outperforms the exact method in runtime with minimal (and in most cases, no) loss of optimality.     

Single machine scheduling problem with batch setups involving positional deterioration effects and multiple rate-modifying activities

This article considers a single machine scheduling problem with batch setups, positional deterioration effects, and multiple optional rate-modifying activities to minimize the total completion time. This problem is formulated as an integer quadratic programming problem. In view of the complexity of optimally solving this problem, a two-phase heuristic algorithm is proposed where an optimal but non-integer solution is obtained in the first phase by solving a continuous relaxed version of the problem. This solution serves as a lower bound for the optimal value of the total completion time. The second phase of the algorithm generates an integer solution using a simple rounding scheme that is optimum or very close to optimum for this problem. Empirical evaluation and comparison with an existing heuristic algorithm show that the proposed heuristic algorithm is substantially more effective in solving large-size problem instances.     

The capacitated lot sizing problem with setup carry-over

Although there is a significant amount of literature on the capacitated lot sizing problem, there has been insufficient consideration of planning problems in which it is possible for a lot size, or production run, to continue over consecutive time periods without incurring multiple setups. While there are papers that consider this feature, they typically restrict production to at most one product in each period. We present a set of mixed integer linear programs for the capacitated lot sizing problem that incorporate setup carry-over without restricting the number of products produced in each time period. Efficient reformulations are developed for finding optimal solutions, and a Lagrangian decomposition heuristic is provided that quickly generates near-optimal solutions. The computational results demonstrate that incorporating setup carry-over has a significant effect on both cost and lot sizes.     

Heuristic Procedures for Multi-Item Inventory Planning with Limited Storage

The determination of replenishment quantities for multiple products with dynamic demand, subject to storage constraints, is addressed. A lower bound is obtained by solving the dual problem. Both subgradient optimization of the Lagrangean relaxation and LP relaxation of the convexified solution space are considered. Dantzig-Wolfe decomposition is used to solve the LP relaxation. A heuristic is proposed for the generation of feasible solutions obtained by modifying solutions created at each step of either subgradient optimization or Dantzig-Wolfe decomposition. An experimental investigation of 428 test problems indicates that the heuristic coupled with sub-gradient optimization gives consistently good solutions.     

A heuristic for dynamic yard crane deployment in a container terminal

Rubber Tired Gantry Cranes (RTGCs) are the most widely used pieces of equipment in the Hong Kong sea-freight container yards. Workload distribution in the yard changes continuously over time. The dynamic deployment of RTGCs is an important issue in yard operation management. This paper investigates the dynamic crane deployment problem with the objective of determining the crane deployment frequency and routes over a planning horizon to minimize the total workload overflow. The problem is formulated as a mixed integer programming model. A heuristic algorithm is then developed to solve problems of practical sizes. The heuristic quickly finds a near optimal solution for crane deployment operation.     

Genetic algorithms for job shop scheduling problems with alternative routings

The purpose of this research is to solve a general job shop problem with alternative machine routings. We consider four performance measures: mean flow time, makespan, maximum lateness, and total absolute deviation from the due dates. We first develop mixed-integer linear programming (MILP) formulations for the problems. The MILP formulations can be used either to compute optimal solutions for small-sized problems or to test the performance of existing heuristic algorithms. In addition, we have developed a genetic algorithm that can be used to generate relatively good solutions quickly. Further, computational experiments have been performed to compare the solution of the MILP formulations with that of existing algorithms.     

Quay crane scheduling in container terminals

The problem of scheduling identical quay cranes moving along a common linear rail to handle containers for a ship is studied. The ship has a number of container-stacking compartments called bays, and only one quay crane can work on a bay at the same time. The objective of the scheduling problem is to find the work schedule for each quay crane which minimizes the ship’s stay time in port. Finding the optimal solution of the scheduling problem is computationally intractable and a heuristic is proposed to solve it. The heuristic first decomposes the difficult multi-crane scheduling problem into easier subproblems by partitioning the ship into a set of non-overlapping zones. The resulting subproblems for each possible partition are solved optimally by a simple rule. An effective algorithm for finding tight lower bounds is developed by modifying and enhancing an effective lower-bounding procedure proposed in the literature. Computational experiments were carried out to evaluate the performance of the heuristic on a set of test problems randomly generated based on typical terminal operations data. The computational results show that the heuristic can indeed find effective solutions for the scheduling problem, with the heuristic solutions on average 4.8% above their lower bounds.     

An optimal multi-machine replacement policy in a serially dependent production system

Here we investigate the serially dependent, multi-machine replacement problem. Given a planning horizon with T periods, M machines of T+ 1 possible vintages, costs for machine operations, machine replacements, and shutdown times, we investigate a linear programming reformulation which involves half the variables and a factor of T fewer constraints than earlier forms. Only T binary variables are necessary, a factor of M fewer than currently employed by heuristic procedures on alternate formulations. Specialized monotone cost structures are no longer necessary, thus extending the class of problems which can be solved efficiently. Computations done using the reformulation's linear programming relaxation on randomly generated problems typically produced integer solutions without branching, even in problems with 75 machines and 15 time periods. In situations where exact optimal solutions are required, the reformulation partially remedies the slow convergence witnessed in earlier studies using branch and bound techniques.