Capacitated dynamic lot sizing problems in closed-loop supply chain

In this paper, we address the capacitated dynamic lot sizing problem arising in closed-loop supply chain where returned products are collected from customers. These returned products can either be disposed or be remanufactured to be sold as new ones again; hence the market demands can be satisfied by either newly produced products or remanufactured ones. The capacities of production, disposal and remanufacturing are limited, and backlogging is not allowed. A general model of this problem is formulated, and several useful properties of the problem are characterized when cost functions are concave. Moreover, this problem is analyzed and solved to optimality using dynamic programming algorithms under different scenarios. It is shown that the problem with only disposal or remanufacturing can be converted into a traditional capacitated lot sizing problem and be solved by a polynomial algorithm if the capacities are constant. A pseudo-polynomial algorithm is proposed for the problem with both capacitated disposal and remanufacturing. The problem with capacitated production and remanufacturing and the problem with uncapacitated production and capacitated remanufacturing are also analyzed and solved. Through numerical experiments we show that the proposed algorithms perform well when solving problems of practical sizes. From the experimental results also indicates that it is worthwhile to expand the remanufacturing capacity only when returned products exist in a relatively long planning horizon, and production capacities have little effect on the remanufacturing plan when the demand is mainly satisfied by the production.     

On the resolution of the single product capacitated machine siting problem

The single product capacitated machine siting problem (SPCMSP) is an extension of the simple plant location problem, in which plant production depends on installing capacitated machines. In this paper we compare, both theoretically and computationally, three heuristic algorithms for the SPCMSP based upon Lagrangean relaxation and reduction tests of a mixed-integer formulation of the problem, which is NP-hard. We test the performance of the algorithms with examples involving up to 100 potential plants, 1000 customers and six potential machines per plant, which we obtain encouraging results.     

A composite very large-scale neighborhood structure for the capacitated minimum spanning tree problem

The capacitated minimum spanning tree (CMST) problem is to find a minimum cost spanning tree in a network where nodes have specified demands, with an additional capacity constraints on the subtrees incident to a given source node s. The capacitated minimum spanning tree problem arises as an important subproblem in many telecommunication network design problems. In a recent paper, Ahuja et al. (Math. Program. 91 (2001) 71) proposed two very large-scale neighborhood search algorithms for the capacitated minimum spanning tree problem. Their first node-based neighborhood structure is obtained by performing multi-exchanges involving several trees where each tree contributes a single node. Their second tree-based neighborhood structure is obtained by performing multi-exchanges where each tree contributes a subtree. The computational investigations found that node-based multi-exchange neighborhood gives the best performance for the homogenous demand case (when all nodes have the same demand), and the tree-based multi-exchange neighborhood gives the best performance for the heterogeneous demand case (when nodes may have different demands). In this paper, we propose a composite neighborhood structure that subsumes both the node-based and tree-based neighborhoods, and outperforms both the previous neighborhood search algorithms for solving the capacitated minimum spanning tree problem on standard benchmark instances. We also develop improved dynamic programming based exact algorithms for searching the composite neighborhood.     

Efficient Solution of the Single-item,Capacitated Lot-sizing Problem with Start-up and Reservation Costs

A capacitated dynamic lot-sizing model, where the costs incurred are a start-up cost for switching the production facility on and another reservation cost for keeping the facility on, whether or not it is producing, is considered. The resulting scheduling problem is NP-hard. An efficient shortest path model of the uncapacitated version of the problem is developed. This model is then included, via a redefinition of variables, into a tight capacitated model; tight in the sense that sharp lower bounds can be produced from it. The lower bound problems are solved efficiently by recovering the shortest path structure through column generation, and effective upper bounds are generated by solving a small capacitated trans-shipment problem. The results of computational tests to verify the computational efficiency of the resulting solution scheme are presented.     

Symmetry helps: Bounded bi-directional dynamic programming for the elementary shortest path problem with resource constraints

When vehicle routing problems with additional constraints, such as capacity or time windows, are solved via column generation and branch-and-price, it is common that the pricing subproblem requires the computation of a minimum cost constrained path on a graph with costs on the arcs and prizes on the vertices. A common solution technique for this problem is dynamic programming. In this paper we illustrate how the basic dynamic programming algorithm can be improved by bounded bi-directional search and we experimentally evaluate the effectiveness of the enhancement proposed. We consider as benchmark problems the elementary shortest path problems arising as pricing subproblems in branch-and-price algorithms for the capacitated vehicle routing problem, the vehicle routing problem with distribution and collection and the capacitated vehicle routing problem with time windows.     

An FPTAS for a single-item capacitated economic lot-sizing problem with monotone cost structure

We present a fully polynomial time approximation scheme (FPTAS) for a capacitated economic lot-sizing problem with a monotone cost structure. An FPTAS delivers a solution with a given relative error ɛ in time polynomial in the problem size and in 1/ɛ. Such a scheme was developed by van Hoesel and Wagelmans [8] for a capacitated economic lot-sizing problem with monotone concave (convex) production and backlogging cost functions. We omit concavity and convexity restrictions. Furthermore, we take advantage of a straightforward dynamic programming algorithm applied to a rounded problem.     

Dynamic programming approximation algorithms for the capacitated lot-sizing problem

This paper provides a new idea for approximating the inventory cost function to be used in a truncated dynamic program for solving the capacitated lot-sizing problem. The proposed method combines dynamic programming with regression, data fitting, and approximation techniques to estimate the inventory cost function at each stage of the dynamic program. The effectiveness of the proposed method is analyzed on various types of the capacitated lot-sizing problem instances with different cost and capacity characteristics. Computational results show that approximation approaches could significantly decrease the computational time required by the dynamic program and the integer program for solving different types of the capacitated lot-sizing problem instances. Furthermore, in most cases, the proposed approximate dynamic programming approaches can accurately capture the optimal solution of the problem with consistent computational performance over different instances.     

Multi-exchange neighborhood structures for the capacitated minimum spanning tree problem

The capacitated minimum spanning tree (CMST) problem is to find a minimum cost spanning tree with an additional cardinality constraint on the sizes of the subtrees incident to a given root node. The CMST problem is an NP-complete problem, and existing exact algorithms can solve only small size problems. Currently, the best available heuristic procedures for the CMST problem are tabu search algorithms due to Amberg et al. and Sharaiha et al. These algorithms use two-exchange neighborhood structures that are based on exchanging a single node or a set of nodes between two subtrees. In this paper, we generalize their neighborhood structures to allow exchanges of nodes among multiple subtrees simultaneously; we refer to such neighborhood structures as multi-exchange neighborhood structures. Our first multi-exchange neighborhood structure allows exchanges of single nodes among several subtrees. Our second multi-exchange neighborhood structure allows exchanges that involve multiple subtrees. The size of each of these neighborhood structures grows exponentially with the problem size without any substantial increase in the computational times needed to find improved neighbors. Our approach, which is based on the cyclic transfer neighborhood structure due to Thompson and Psaraftis and Thompson and Orlin transforms a profitable exchange into a negative cost subset-disjoint cycle in a graph, called an improvement graph, and identifies these cycles using variants of shortest path label-correcting algorithms. Our computational results with GRASP and tabu search algorithms based on these neighborhood structures reveal that (i) for the unit demand case our algorithms obtained the best available solutions for all benchmark instances and improved some; and (ii) for the heterogeneous demand case our algorithms improved the best available solutions for most of the benchmark instances with improvements by as much as 18%. The running times our multi-exchange neighborhood search algorithms are comparable to those taken by two-exchange neighborhood search algorithms. Received: September 1998 / Accepted: March 2001?Published online May 18, 2001     

A single-item economic lot-sizing problem with a non-uniform resource: Approximation

We study a generalization of the classical single-item capacitated economic lot-sizing problem to the case of a non-uniform resource usage for production. The general problem and several special cases are shown to be non-approximable with any polynomially computable relative error in polynomial time. An optimal dynamic programming algorithm and its approximate modification are presented for the general problem. Fully polynomial time approximation schemes are developed for two NP-hard special cases: (1) cost functions of total production are separable and holding and backlogging cost functions are linear with polynomially related slopes, and (2) all holding costs are equal to zero.     

A Robust Optimization Approach to Dynamic Pricing and Inventory Control with no Backorders

In this paper, we present a robust optimization formulation for dealing with demand uncertainty in a dynamic pricing and inventory control problem for a make-to-stock manufacturing system. We consider a multi-product capacitated, dynamic setting. We introduce a demand-based fluid model where the demand is a linear function of the price, the inventory cost is linear, the production cost is an increasing strictly convex function of the production rate and all coefficients are time-dependent. A key part of the model is that no backorders are allowed. We show that the robust formulation is of the same order of complexity as the nominal problem and demonstrate how to adapt the nominal (deterministic) solution algorithm to the robust problem.     

Flows on hypergraphs

We consider the capacitated minimum cost flow problem on directed hypergraphs. We define spanning hypertrees so generalizing the spanning tree of a standard graph, and show that, like in the standard and in the generalized minimum cost flow problems, a correspondence exists between bases and spanning hypertrees. Then, we show that, like for the network simplex algorithms for the standard and for the generalized minimum cost flow problems, most of the computations performed at each pivot operation have direct hypergraph interpretations.     

Linear Relaxations of the Capacitated Warehouse Location Problem

The capacitated warehouse location problem consists of the well known transportation problem with the additional feature of a fixed charge associated with each warehouse which is put to use. The problem is usually solved as a special type of mixed integer programme, so that relaxation and lower bounding are a vital part of any algorithm. A deeper insight into the relaxation process may eventually lead to more efficient algorithms for the problem. It is shown here that the LP relaxation of the capacitated warehouse location problem can incorporate constraints of a much more general nature than those previously described.     

Hybrid heuristics for the capacitated lot sizing and loading problem with setup times and overtime decisions

The capacitated lot sizing and loading problem (CLSLP) deals with the issue of determining the lot sizes of product families/end items and loading them on parallel facilities to satisfy dynamic demand over a given planning horizon. The capacity restrictions in the CLSLP are imposed by constraints specific to the production environment considered. When a lot size is positive in a specific period, it is loaded on a facility without exceeding the sum of the regular and overtime capacity limits. Each family may have a different process time on each facility and furthermore, it may be technologically feasible to load a family only on a subset of existing facilities. So, in the most general case, the loading problem may involve unrelated parallel facilities of different classes. Once loaded on a facility, a family may consume capacity during setup time. Inventory holding and overtime costs are minimized in the objective function. Setup costs can be included if setups incur costs other than lost production capacity. The CLSLP is relevant in many industrial applications and may be generalized to multi-stage production planning and loading models. The CLSLP is a synthesis of three different planning and loading problems, i.e., the capacitated lot sizing problem (CLSP) with overtime decisions and setup times, minimizing total tardiness on unrelated parallel processors, and, the class scheduling problem, each of which is NP in the feasibility and optimality problems. Consequently, we develop hybrid heuristics involving powerful search techniques such as simulated annealing (SA), tabu search (TS) and genetic algorithms (GA) to deal with the CLSLP. Results are compared with optimal solutions for 108 randomly generated small test problems. The procedures developed here are also compared against each other in 36 larger size problems.     

An integrated production and preventive maintenance planning model

We are given a set of items that must be produced in lots on a capacitated production system throughout a specified finite planning horizon. We assume that the production system is subject to random failures, and that any maintenance action carried out on the system, in a period, reduces the system’s available production capacity during that period. The objective is to find an integrated lot-sizing and preventive maintenance strategy of the system that satisfies the demand for all items over the entire horizon without backlogging, and which minimizes the expected sum of production and maintenance costs. We show how this problem can be formulated and solved as a multi-item capacitated lot-sizing problem on a system that is periodically renewed and minimally repaired at failure. We also provide an illustrative example that shows the steps to obtain an optimal integrated production and maintenance strategy.     

Heuristics for Multistage Production Planning Problems

Several heuristics for the capacitated multistage production planning problem with concave production costs, based on traditional production planning techniques and linear programming, are stated and empirically evaluated using a new capacity relaxation of the problem to furnish lower bounds on cost.     

Solving the capacitated lot-sizing problem with backorder consideration

In this research, we formulate and solve a type of the capacitated lot-sizing problem. We present a general model for the lot-sizing problem with backorder options, that can take into consideration various types of production capacities such as regular time, overtime and subcontracting. The objective is to determine lot sizes that will minimize the sum of setup costs, holding cost, backorder cost, regular time production costs, and overtime production costs, subject to resource constraints. Most existing formulations for the problem consider the special case of the problem where a single source of production capacity is considered. However, allowing for the use of alternate capacities such as overtime is quite common in many manufacturing settings. Hence, we provide a formulation that includes consideration of multiple sources of production capacity. We develop a heuristic based on the special structure of fixed charge transportation problem. The performance of our algorithm is evaluated by comparing the heuristic solution value to lower bound value. Extensive computational results are presented.     

Simultaneous production and maintenance planning for a single capacitated resource facing both a dynamic demand and intensive wear and tear

This paper presents a generic modeling framework to simultaneously decide about production quantities and maintenance operations for a capacitated resource facing a dynamic demand for different types of products. As the resource needs to be setup for each specific type of product, a lot-sizing problem occurs. In addition it is assumed that production causes intensive wear and tear. For this reason frequent maintenance activities need to be coordinated with the production operations in order to efficiently use the capacitated resource. A single generic model is presented to capture alternative forms of maintenance and different modes of interaction between maintenance and setups. As the model is numerically intractable for standard branch and bound algorithms, we solve it heuristically via a decomposition using a Fix-and-Optimize approach. Numerical results show that the proposed solution method produces high-quality results quickly. We further study the impact of simultaneous vs. sequential decisions about production and maintenance in the case of intensive wear and tear.     

A capacity scaling heuristic for the multicommodity capacitated network design problem

In this paper, we propose a capacity scaling heuristic using a column generation and row generation technique to address the multicommodity capacitated network design problem. The capacity scaling heuristic is an approximate iterative solution method for capacitated network problems based on changing arc capacities, which depend on flow volumes on the arcs. By combining a column and row generation technique and a strong formulation including forcing constraints, this heuristic derives high quality results, and computational effort can be reduced considerably. The capacity scaling heuristic offers one of the best current results among approximate solution algorithms designed to address the multicommodity capacitated network design problem.     

A tabu-based large neighbourhood search methodology for the capacitated examination timetabling problem

Neighbourhood search algorithms are often the most effective known approaches for solving partitioning problems. In this paper, we consider the capacitated examination timetabling problem as a partitioning problem and present an examination timetabling methodology that is based upon the large neighbourhood search algorithm that was originally developed by Ahuja and Orlin. It is based on searching a very large neighbourhood of solutions using graph theoretical algorithms implemented on a so-called improvement graph. In this paper, we present a tabu-based large neighbourhood search, in which the improvement moves are kept in a tabu list for a certain number of iterations. We have drawn upon Ahuja–Orlin's methodology incorporated with tabu lists and have developed an effective examination timetabling solution scheme which we evaluated on capacitated problem benchmark data sets from the literature. The capacitated problem includes the consideration of room capacities and, as such, represents an issue that is of particular importance in real-world situations. We compare our approach against other methodologies that have appeared in the literature over recent years. Our computational experiments indicate that the approach we describe produces the best known results on a number of these benchmark problems.     

Supply chain scheduling with receiving deadlines and non-linear penalty

We study the operations scheduling problem with delivery deadlines in a three-stage supply chain process consisting of (1) heterogeneous suppliers, (2) capacitated processing centres (PCs), and (3) a network of business customers. The suppliers make and ship semi-finished products to the PCs where products are finalized and packaged before they are shipped to customers. Each business customer has an order quantity to fulfil and a specified delivery date, and the customer network has a required service level so that if the total quantity delivered to the network falls below a given targeted fill rate, a non-linear penalty will apply. Since the PCs are capacitated and both shipping and production operations are non-instantaneous, not all the customer orders may be fulfilled on time. The optimization problem is therefore to select a subset of customers whose orders can be fulfilled on time and a subset of suppliers to ensure the supplies to minimize the total cost, which includes processing cost, shipping cost, cost of unfilled orders (if any), and a non-linear penalty if the target service level is not met. The general version of this problem is difficult because of its combinatorial nature. In this paper, we solve a special case of this problem when the number of PCs equals one, and develop a dynamic programming-based algorithm that identifies the optimal subset of customer orders to be fulfilled under each given utilization level of the PC capacity. We then construct a cost function of a recursive form, and prove that the resulting search algorithm always converges to the optimal solution within pseudo-polynomial time. Two numerical examples are presented to test the computational performance of the proposed algorithm.