Chandra A. Poojari

Computational solution of capacity planning models under uncertainty

Abstract The traditional supply chain network planning problem is stated as a multi-period resource allocation model involving 0–1 discrete strategic decision variables. The MIP structure of this problem makes it fairly intractable for practical applications, which involve multiple products, factories, warehouses and distribution centres (DCs). The same problem formulated and studied under uncertainty makes it even more intractable. In this paper we consider two related modelling approaches and solution techniques addressing this issue. The first involves scenario analysis of solutions to “wait and see” models and the second involves a two-stage integer stochastic programming (ISP) representation and solution of the same problem. We show how the results from the former can be used in the solution of the latter model. We also give some computational results based on serial and parallel implementations of the algorithms.

Genetic Algorithm based technique for solving Chance Constrained Problems

Management and measurement of risk is an important issue in almost all areas that require decisions to be made under uncertain information. Chance Constrained Programming (CCP) have been used for modelling and analysis of risks in a number of application domains. However, the resulting mathematical problems are non-trivial to represent using algebraic modelling languages and pose significant computational challenges due to their non-linear, non-convex, and the stochastic nature. We develop and implement C++ classes to represent such CCP problems. We propose a framework consisting of Genetic Algorithm and Monte Carlo Simulation in order to process the problems. The non-linear and non-convex nature of the CCP problems are processed using Genetic Algorithm, whereas the stochastic nature is addressed through Simulation. The computational investigations have shown that the framework can efficiently represent and obtain good solutions for seven test problems.

Improving benders decomposition using a genetic algorithm

We develop and investigate the performance of a hybrid solution framework for solving mixed-integer linear programming problems. Benders decomposition and a genetic algorithm are combined to develop a framework to compute feasible solutions. We decompose the problem into a master problem and a subproblem. A genetic algorithm along with a heuristic are used to obtain feasible solutions to the master problem, whereas the subproblem is solved to optimality using a linear programming solver. Over successive iterations the master problem is refined by adding cutting planes that are implied by the subproblem. We compare the performance of the approach against a standard Benders decomposition approach as well as against a stand-alone solver (Cplex) on MIPLIB test problems.

Robust solutions and risk measures for a supply chain planning problem under uncertainty

We consider a strategic supply chain planning problem formulated as a two-stage stochastic integer programming (SIP) model. The strategic decisions include site locations, choices of production, packing and distribution lines, and the capacity increment or decrement policies. The SIP model provides a practical representation of real-world discrete resource allocation problems in the presence of future uncertainties which arise due to changes in the business and economic environment. Such models that consider the future scenarios (along with their respective probabilities) not only identify optimal plans for each scenario, but also determine a hedged strategy for all the scenarios. We 1)exploit the natural decomposable structure of the SIP problem through Benders’ decomposition,2)approximate the probability distribution of the random variables using the generalized lambda distribution, and3)through simulations, calculate the performance statistics and the risk measures for the two models, namely the expected-value and the here-and-now.

An application of Lagrangian relaxation to a capacity planning problem under uncertainty

A supply chain network-planning problem is presented as a two-stage resource allocation model with 0-1 discrete variables. In contrast to the deterministic mathematical programming approach, we use scenarios, to represent the uncertainties in demand. This formulation leads to a very large scale mixed integer-programming problem which is intractable. We apply Lagrangian relaxation and its corresponding decomposition of the initial problem in a novel way, whereby the Lagrangian relaxation is reinterpreted as a column generator and the integer feasible solutions are used to approximate the given problem. This approach addresses two closely related problems of scenario analysis and two-stage stochastic programs. Computational solutions for large data instances of these problems are carried out successfully and their solutions analysed and reported. The model and the solution system have been applied to study supply chain capacity investment and planning.

A tabu search algorithm for the single vehicle routing allocation problem

In the single vehicle routing allocation problem (SVRAP) we have a single vehicle, together with a set of customers, and the problem is one of deciding a route for the vehicle (starting and ending at given locations) such that it visits some of the customers. Customers not visited by the vehicle can either be allocated to a customer on the vehicle route, or they can be isolated. The objective is to minimize a weighted sum of routing, allocation and isolation costs. One special case of the general SVRAP is the median cycle problem, also known as the ring star problem, where no isolated vertices are allowed. Other special cases include the covering tour problem, the covering salesman problem and the shortest covering path problem. In this paper, we present a tabu search algorithm for the SVRAP. Our tabu search algorithm includes aspiration, path relinking and frequency-based diversification. Computational results are presented for test problems used previously in the literature and our algorithm is compared with the results obtained by other researchers. We also report results for much larger problems than have been considered by others.

Strategic and Tactical Planning Models for Supply Chain: An Application of Stochastic Mixed Integer Programming

Stochastic mixed integer programming (SMIP) models arise in a variety of applications. In particular they are being increasingly applied in the modelling and analysis of financial planning and supply chain management problems. SMIP models explicitly consider discrete decisions and model uncertainty and thus provide hedged decisions that perform well under several scenarios. Such SMIP models are relevant for industrial practitioners and academic researchers alike. From an industrial perspective, the need for well-hedged solutions cannot be overemphasized. On the other hand the NP-Hard nature of the underlying model (due to the discrete variables) together with the curse of dimensionality (due to the underlying random variables) make these models important research topics for academics. In this chapter we discus the contemporary developments of SMIP models and algorithms. We introduce a generic classification scheme that can be used to classify such models. Next we discuss the use of such models in supply chain planning and management. We present a case study of a strategic supply chain model modelled as a two-stage SMIP model. We propose a heuristic based on Lagrangean relaxation for processing the underlying model. Our heuristic can be generalized to an optimum-seeking branch-and-price algorithm, but we confine ourselves to a simpler approach of ranking the first-stage decisions which use the columns generated by the Lagrangean relaxation. In addition to providing integer feasible solutions, this approach has the advantage of mimicking the use of scenario analysis, while seeking good “here-and-now” solutions. The approach thus provides industrial practitioners an approach that they are able to incorporate within their decision-making methodology. Our computational investigations with this model are presented.

An overview of the issues in the airline industry and the role of optimization models and algorithms

Optimization application has revolutionized the airline industry in all phases of the planning process. One of the current issues facing the airline industry is planning under uncertainty, especially in the context of schedule disruptions. We discuss the robust models and solution algorithms that have been proposed and developed to handle the uncertain parameters. We show that stochastic programming (SP) provides an ideal paradigm for capturing the uncertainties and making robust decisions. We develop and investigate a prototype fleet assignment model formulated as a two-stage SP with recourse.

Risk based methods for supply chain planning and management

The strategic, tactical and operational decision-making within the supply chain are recognized to be important OR/Management Science problems. In the development of analytic models, these decisions have been often considered separately. Recently, practitioners and researchers have realized the close interactions between these three levels of decision-making and new research activities are aimed at modelling these strategic, tactical and operational decisions. Planning decisions are under-pinned by three salient concepts—namely, irreversibility, uncertainty and the choice of timing. First, an investment planning decision is partially if not completely irreversible. Second, the future returns from a given investment (decision) are uncertain. Finally, it is always possible to make a choice in respect of the timing of the (investment) decisions. Also the focus of decision-making has moved from planning for the expected return to the issues which concern the risks of the underlying decisions. Whereas excellent tools for measuring and managing risk have been developed for finance professionals; investment bankers, fund managers, credit analysts; only recently such analyses in the manufacturing and retail industries have attracted attention and consideration of managers and planners. We believe that there will be an increasing relevance of business analytics which encapsulates the risk and uncertainty in the supply chain. The collection of papers that make up this special issue clearly justifies this assertion. We now briefly consider the focus of the papers in this collection: