Dynamic Slope Scaling Procedure Research Articles

Service network design in short and local fresh food supply chain

This paper aims at developing efficient solving methods for an original service network design problem imbued with sustainable issues. Indeed the network has to be designed for short and local supply chain and for fresh food products. The original features of the problem are the seasonality of supply, the limitation of transshipments for a product and no possibility of storage between consecutive periods. Decisions at strategic and tactical level are (1) decisions on a subset of hubs to open among a given set of potential locations, (2) transportation services to open between the actors and (3) flow quantities for the fresh food products. We propose for this problem a Mixed Integer Programming formulation and two solving techniques: Benders Decomposition and Dynamic Slope Scaling Procedure. These techniques are adapted to the problem and some experimental tests are conducted in order to compare the approaches on large-scale instances.

Dynamic Slope Scaling Procedure for Stochastic Integer Programming Problem

Mathematical programming has been applied to various problems. For many actual problems, the assumption that the parameters involved are deterministic known data is often unjustified. In such cases, these data contain uncertainty and are thus represented as random variables, since they represent information about the future. Decision-making under uncertainty involves potential risk. Stochastic programming is a commonly used method for optimization under uncertainty. A stochastic programming problem with recourse is referred to as a two-stage stochastic problem. In this study, we consider a stochastic programming problem with simple integer recourse in which the value of the recourse variable is restricted to a multiple of a nonnegative integer. The algorithm of a dynamic slope scaling procedure for solving this problem is developed by using a property of the expected recourse function. Numerical experiments demonstrate that the proposed algorithm is quite efficient. The stochastic programming model defined in this paper is quite useful for a variety of design and operational problems. Keywords—stochastic programming problem with recourse, simple integer recourse, dynamic slope scaling procedure

Inventory-Transportation Integrated Optimization Problem in Many-to-Many Distribution Network

In order to solve the Inventory-transportation Integrated Optimization problem (ITIO problem) in a distribution network consisting of many warehouses and many retailers, the Lagrange multiplier method, the scenario-based Dynamic Slope Scaling Procedure (DSSP) heuristics and the Lagrangian relaxation-based DSSP heuristics were applied to integrate the system of inventory control and transportation scheduling respectively and in proper sequence. The more efficient Lagrangian relaxation-based DSSP heuristics is found in solving the ITIO problem. And furthermore, the comparative experiments prove the heuristics can get the optimized results in less calculating time and the efficiency of this method is also proved in solving the ITIO problem in a many-to-many distribution network.

A heuristic method for the minimum toll booth problem

This paper addresses the toll pricing problem in which the objective is to minimize the number of required toll facilities in a traffic network. The problem is shown to be NP-hard. To obtain a solution in a reasonable time, an effective metaheuristic algorithm is developed. The algorithm uses a local search technique in which the neighborhood function employs the dynamic slope scaling procedure to deal with the fixed charge nature of the objective function. Numerical results from 50 randomly generated and three real networks are reported.

Adaptive dynamic cost updating procedure for solving fixed charge network flow problems

We approximate the objective function of the fixed charge network flow problem (FCNF) by a piecewise linear one, and construct a concave piecewise linear network flow problem (CPLNF). A proper choice of parameters in the CPLNF problem guarantees the equivalence between those two problems. We propose a heuristic algorithm for solving the FCNF problem, which requires solving a sequence of CPLNF problems. The algorithm employs the dynamic cost updating procedure (DCUP) to find a solution to the CPLNF problems. Preliminary numerical experiments show the effectiveness of the proposed algorithm. In particular, it provides a better solution than the dynamic slope scaling procedure in less CPU time.

A bilinear relaxation based algorithm for concave piecewise linear network flow problems

We present a continuous relaxation technique for the Concave Piecewise Linear Network Flow Problem (CPLNFP), which has a bilinear objective function and network constraints. We show that a global optimum of the resulting problem is a solution of CPLNFP. The theoretical results are generalized for a concave minimization problem with a separable objective function. An efficient and effective Dynamic Cost Updating Procedure (DCUP) is considered to find a local minimum of the relaxation problem, which converges in a finite number of iterations. We show that the CPLNFP is equivalent to a Network Flow Problem with Flow Dependent Cost Functions (NFPwFDCF), and we prove that the solution of the Dynamic Slope Scaling Procedure (DSSP) is an equilibrium solution of the NFPwFDCF. The numerical experiments show that the proposed algorithm can provide a better solution than DSSP using less amount of CPU time and iterations.

An Enhanced Dynamic Slope Scaling Procedure with Tabu Scheme for Fixed Charge Network Flow Problems

A heuristic algorithm for solving large scale fixed charge network flow problems (FCNFP) based on the dynamic slope scaling procedure (DSSP) and tabu search strategies is presented. The proposed heuristic integrates the DSSP with short-term memory intensification and long-term memory diversification mechanisms in the tabu scheme to improve the performance of the pure DSSP. With the feature of dynamically evolving memory, the enhanced DSSP evaluates the solutions in the search history and iteratively adjusts the linear factors in the linear approximation of the FCNFP to produce promising search neighborhoods for good quality solutions. The comprehensive numerical experiments on various test problems ranging from sparse to dense network structures are reported. The overall comparison of the pure DSSP, the enhanced DSSP, and branch and bound (B&B by cutting-edge MIP optimizer in CPLEX) is shown in terms of solution quality and CPU time. The results show that the enhanced DSSP approach has a higher solution quality than the pure DSSP for larger scale problems.

Dynamic Slope Scaling Procedure and Lagrangian Relaxation with Subproblem Approximation

The dynamic slope scaling procedure (DSSP) is an efficient heuristic algorithm that provides good solutions to the fixed-charge transportation or network flow problem. However, the procedure is graphically motivated and appears unrelated to other optimization techniques. In this paper, we formulate the fixed-charge problem as a mathematical program with complementarity constraints (MPCC) and show that DSSP is equivalent to solving MPCC using Lagrangian relaxation with subproblem approximation.

Dynamic slope scaling and trust interval techniques for solving concave piecewise linear network flow problems

In this paper, we propose an efficient heuristic approach for solving concave Piecewise Linear Network Flow Problems (PLNFP) in which the cost is separable and each arc cost is concave piecewise linear function of the total flow along the arcs. The problem is well known to be WNP-hard and exact global optimization algorithms are of limited use. In this paper, PLNFP is transformed into an equivalent Fixed-Charge Network Flow Problem (FCNFP) using an arc separation procedure, and we develop an algorithm combining the Dynamic Slope Scaling Procedure (DSSP) and Trust Interval techniques. Due to the fast growing size of the transformed FCNFP, exact branch-and-bound methods are not able to handle the problems of practical size. Based on the dynamic slope scaling algorithm developed for solving FCNFP in previous work, we add a trust interval technique based on solution tendencies to speed up the algorithm. The numerical experiments show that the proposed algorithm is quite efficient and its performance is very stable. © 2000 John Wiley & Sons, Inc.

A solution approach to the fixed charge network flow problem using a dynamic slope scaling procedure

In this paper, we consider the Fixed Charge Network Flow Problem (FCNFP) which is known to be NP -hard. It has many practical applications including transportation, network design, communication, and production scheduling. Although many exact methods have been developed to solve the FCNFP, their computational requirements increase exponentially with the size of the problem. Hence, heuristic approaches are needed to solve suboptimally large-scale problems. Here, we propose a new approach for solving the general capacitated (or uncapacitated) FCNFP by adapting an economic viewpoint of the fixed cost. A new concept of the dynamic slope scaling procedure is presented and some computational results on a wide range of test problems are reported. The largest problem has 202 nodes and 10 200 arcs. The results show that the proposed procedure generates solutions within 0% to 0.65% of optimality in all cases.