Integrated supply chain planning under uncertainty using an improved stochastic approach

This paper proposes an integrated model and a modified solution method for solving supply chain network design problems under uncertainty. The stochastic supply chain network design model is provided as a two-stage stochastic program where the two stages in the decision-making process correspond to the strategic and tactical decisions. The uncertainties are mostly found in the tactical stage because most tactical parameters are not fully known when the strategic decisions have to be made. The main uncertain parameters are the operational costs, the customer demand and capacity of the facilities. In the improved solution method, the sample average approximation technique is integrated with the accelerated Benders’ decomposition approach to improvement of the mixed integer linear programming solution phase. The surrogate constraints method will be utilized to acceleration of the decomposition algorithm. A computational study on randomly generated data sets is presented to highlight the efficiency of the proposed solution method. The computational results show that the modified sample average approximation method effectively expedites the computational procedure in comparison with the original approach.     

Supply chain optimization of petroleum organization under uncertainty in market demands and prices

A multi-period stochastic planning model has been developed and implemented for a supply chain network of a petroleum organization operating in an oil producing country under uncertain market conditions. The proposed supply chain network consists of all activities related to crude oil production, processing and distribution. Uncertainties were introduced in market demands and prices. A deterministic optimization model was first developed and tested. The impact of uncertainty on the supply chain was studied by performing a sensitivity analysis in which ±20% deviations were introduced in market demands and prices of different commodities. A stochastic formulation was then proposed, which is based on the two-stage problem with finite number of realizations. The proposed stochastic programming approach proved to be quite effective in developing resilient production plans in light of high degree of uncertainty in market conditions. The anticipated production plans have a considerably lower expected value of perfect information (EVPI). The main conclusion of this study is that for an oil producing country with oil processing capabilities, the impact of economic uncertainties may be tolerated by an appropriate balance between crude exports and processing capacities.     

The sample average approximation method for empty container repositioning with uncertainties

One of the challenges faced by liner operators today is to effectively operate empty containers in order to meet demand and to reduce inefficiency in an uncertain environment. To incorporate uncertainties in the operations model, we formulate a two-stage stochastic programming model with random demand, supply, ship weight capacity, and ship space capacity. The objective of this model is to minimize the expected operational cost for Empty Container Repositioning (ECR). To solve the stochastic programs with a prohibitively large number of scenarios, the Sample Average Approximation (SAA) method is applied to approximate the expected cost function. To solve the SAA problem, we consider applying the scenario aggregation by combining the approximate solution of the individual scenario problem. Two heuristic algorithms based on the progressive hedging strategy are applied to solve the SAA problem. Numerical experiments are provided to show the good performance of the scenario-based method for the ECR problem with uncertainties.     

Smooth sample average approximation of stationary points in nonsmooth stochastic optimization and applications

Inspired by a recent work by Alexander et al. (J Bank Finance 30:583–605, 2006) which proposes a smoothing method to deal with nonsmoothness in a conditional value-at-risk problem, we consider a smoothing scheme for a general class of nonsmooth stochastic problems. Assuming that a smoothed problem is solved by a sample average approximation method, we investigate the convergence of stationary points of the smoothed sample average approximation problem as sample size increases and show that w.p.1 accumulation points of the stationary points of the approximation problem are weak stationary points of their counterparts of the true problem. Moreover, under some metric regularity conditions, we obtain an error bound on approximate stationary points. The convergence result is applied to a conditional value-at-risk problem and an inventory control problem.      

Logarithmic sample bounds for Sample Average Approximation with capacity- or budget-constraints

Sample Average Approximation (SAA) is used to approximately solve stochastic optimization problems. In practice, SAA requires much fewer samples than predicted by existing theoretical bounds that ensure the SAA solution is close to optimal. Here, we derive new sample-size bounds for SAA that, for certain problems, are logarithmic (existing bounds are polynomial) in problem dimension. Notably, our new bounds provide a theoretical explanation for the success of SAA for many capacity- or budget-constrained optimization problems.     

Collaborative production planning of supply chain under price and demand uncertainty

This research is motivated by an automobile manufacturing supply chain network. It involves a multi-echelon production system with material supply, component fabrication, manufacturing, and final product distribution activities. We address the production planning issue by considering bill of materials and the trade-offs between inventories, production costs and customer service level. Due to its complexity, an integrated solution framework which combines scatter evolutionary algorithm, fuzzy programming and stochastic chance-constrained programming are combined to jointly take up the issue. We conduct a computational study to evaluate the model. Numerical results using the proposed algorithm confirm the advantage of the integrated planning approach. Compared with other solution methodologies, the supply chain profits from the proposed approach consistently outperform, in some cases up to 13% better. The impacts of uncertainty in demand, material price, and other parameters on the performance of the supply chain are studied through sensitivity analysis. We found the proposed model is effective in developing robust production plans under various market conditions.     

Strategic robust supply chain design based on the Pareto-optimal tradeoff between efficiency and risk

The strategic design of a robust supply chain has to determine the configuration of the supply chain so that its performance remains of a consistently high quality for all possible future conditions. The current modeling techniques often only consider either the efficiency or the risk of the supply chain. Instead, we define the strategic robust supply chain design as the set of all Pareto-optimal configurations considering simultaneously the efficiency and the risk, where the risk is measured by the standard deviation of the efficiency. We model the problem as the Mean–Standard Deviation Robust Design Problem (MSD-RDP). Since the standard deviation has a square root expression, which makes standard maximization algorithms based on mixed-integer linear programming non-applicable, we show the equivalency to the Mean–Variance Robust Design Problem (MV-RDP). The MV-RDP yields an infinite number of mixed-integer programming problems with quadratic objective (MIQO) when considering all possible tradeoff weights. In order to identify all Pareto-optimal configurations efficiently, we extend the branch-and-reduce algorithm by applying optimality cuts and upper bounds to eliminate parts of the infeasible region and the non-Pareto-optimal region. We show that all Pareto-optimal configurations can be found within a prescribed optimality tolerance with a finite number of iterations of solving the MIQO. Numerical experience for a metallurgical case is reported.     

Selection of contract suppliers under price and demand uncertainty in a dynamic market

In this paper, we consider a supply contracting problem in which the buyer firm faces non-stationary stochastic price and demand. First, we derive analytical results to compare two pure strategies: (i) periodically purchasing from the spot market; and (ii) signing a long-term contract with a single supplier. The results from the pure strategies show that the selection of suppliers can be complicated by many parameters, and is particularly affected by price uncertainty. We then develop a stochastic dynamic programming model to incorporate mixed strategies, purchasing commitments and contract cancellations. Computational results show that increases in price (demand) uncertainty favor long-term (short-term) suppliers. By examining the two-way interactions of contract factors (price, demand, purchasing bounds, learning and technology effect, salvage values and contract cancellation), both intuitive and non-intuitive managerial insights in outsourcing strategies are derived.     

Stochastic mathematical programs with equilibrium constraints,modelling and sample average approximation

In this article, we discuss the sample average approximation (SAA) method applied to a class of stochastic mathematical programs with variational (equilibrium) constraints. To this end, we briefly investigate the structure of both–the lower level equilibrium solution and objective integrand. We show almost sure convergence of optimal values, optimal solutions (both local and global) and generalized Karush–Kuhn–Tucker points of the SAA program to their true counterparts. We also study uniform exponential convergence of the sample average approximations, and as a consequence derive estimates of the sample size required to solve the true problem with a given accuracy. Finally, we present some preliminary numerical test results.     

Environmental supply chain network design using multi-objective fuzzy mathematical programming

The concern about environmental impact of business activities has spurred an interest in designing environmentally conscious supply chains. This paper proposes a multi-objective fuzzy mathematical programming model for designing an environmental supply chain under inherent uncertainty of input data in such problem. The proposed model is able to consider the minimization of multiple environmental impacts beside the traditional cost minimization objective to make a fair balance between them. A life cycle assessment-based (LCA-based) method is applied to assess and quantify the environmental impact of different options for supply chain network configuration. Also, to solve the proposed multi-objective fuzzy optimization model, an interactive fuzzy solution approach is developed. A real industrial case is used to demonstrate the significance and applicability of the developed fuzzy optimization model as well as the usefulness of the proposed solution approach.     

Sample average approximation of expected value constrained stochastic programs

We propose a sample average approximation (SAA) method for stochastic programming problems with expected value constraints. Such problems arise, for example, in portfolio selection with constraints on conditional value-at-risk (CVaR). We provide a convergence analysis and a statistical validation scheme for the proposed method.     

Supply chain design considering economies of scale and transport frequencies

In this paper we consider a 3-echelon, multi-product supply chain design model with economies of scale in transport and warehousing that explicitly takes transport frequencies into consideration. Our model simultaneously optimizes locations and sizes of tank farms, material flows, and transport frequencies within the network. We consider all relevant costs: product cost, transport cost, tank rental cost, tank throughput cost, and inventory cost. The problem is based on a real-life example from a chemical company. We show that considering economies of scale and transport frequencies in the design stage is crucial and failing to do so can lead to substantially higher costs than optimal. We solve a wide variety of problems with branch-and-bound and with the efficient solution heuristics based on iterative linearization techniques we develop. We show that the heuristics are superior to the standard branch-and-bound technique for large problems like the one of the chemical company that motivated our research.     

Stochastic programming approach to optimization under uncertainty

In this paper we discuss computational complexity and risk averse approaches to two and multistage stochastic programming problems. We argue that two stage (say linear) stochastic programming problems can be solved with a reasonable accuracy by Monte Carlo sampling techniques while there are indications that complexity of multistage programs grows fast with increase of the number of stages. We discuss an extension of coherent risk measures to a multistage setting and, in particular, dynamic programming equations for such problems.      

Solving multistage asset investment problems by the sample average approximation method

The vast size of real world stochastic programming instances requires sampling to make them practically solvable. In this paper we extend the understanding of how sampling affects the solution quality of multistage stochastic programming problems. We present a new heuristic for determining good feasible solutions for a multistage decision problem. For power and log-utility functions we address the question of how tree structures, number of stages, number of outcomes and number of assets affect the solution quality. We also present a new method for evaluating the quality of first stage decisions.     

Integrated design and operation of remnant inventory supply chains under uncertainty

We consider the simultaneous design and operation of remnant inventory supply chains. Remnant inventory is generated when demand for various lengths of a product may be satisfied by existing inventory, or by cutting a large piece into smaller pieces. We formulate our problem as a two-stage stochastic mixed-integer program. In solving our stochastic program, we enhance the standard L-shaped method in two ways. Our computational experiments demonstrate that these enhancements are effective, dramatically reducing the solution time for large instances.     

One-dimensional average on spheres and approximation

In this paper, we consider the approximation of the degeneration of classic average on sphere on Hardy-Sobolev spaces I_λ(H^p) . We prove that the degenerate expression of average on sphere of an L^p function is convergent almost everywhere, and the speed of the approximation depends on λ. And when λ ≥ 2, the approximation of the average operator will be saturated. On the other hand, we also study the generalization of the average on the product of Hardy-Sobolev spaces.     

Analysis of stochastic dual dynamic programming method

In this paper we discuss statistical properties and convergence of the Stochastic Dual Dynamic Programming (SDDP) method applied to multistage linear stochastic programming problems. We assume that the underline data process is stagewise independent and consider the framework where at first a random sample from the original (true) distribution is generated and consequently the SDDP algorithm is applied to the constructed Sample Average Approximation (SAA) problem. Then we proceed to analysis of the SDDP solutions of the SAA problem and their relations to solutions of the “true” problem. Finally we discuss an extension of the SDDP method to a risk averse formulation of multistage stochastic programs. We argue that the computational complexity of the corresponding SDDP algorithm is almost the same as in the risk neutral case.     

Development of a new approach for deterministic supply chain network design

This paper proposes a mixed integer linear programming model and solution algorithm for solving supply chain network design problems in deterministic, multi-commodity, single-period contexts. The strategic level of supply chain planning and tactical level planning of supply chain are aggregated to propose an integrated model. The model integrates location and capacity choices for suppliers, plants and warehouses selection, product range assignment and production flows. The open-or-close decisions for the facilities are binary decision variables and the production and transportation flow decisions are continuous decision variables. Consequently, this problem is a binary mixed integer linear programming problem. In this paper, a modified version of Benders’ decomposition is proposed to solve the model. The most difficulty associated with the Benders’ decomposition is the solution of master problem, as in many real-life problems the model will be NP-hard and very time consuming. In the proposed procedure, the master problem will be developed using the surrogate constraints. We show that the main constraints of the master problem can be replaced by the strongest surrogate constraint. The generated problem with the strongest surrogate constraint is a valid relaxation of the main problem. Furthermore, a near-optimal initial solution is generated for a reduction in the number of iterations.     

Supply chain network equilibrium with strategic financial hedging using futures

In this paper, we develop a network equilibrium model for supply chain networks with strategic financial hedging. We consider multiple competing firms that purchase multiple materials and parts to manufacture their products. The supply chain firms’ procurement activities are exposed to commodity price risk and exchange rate risk. The firms can use futures contracts to hedge the risks. Our research studies the equilibrium of the entire network where each firm optimizes its own operation and hedging decisions. We use variational inequality theory to formulate the equilibrium model, and provide qualitative properties. We provide analytical results for a special case with duopolistic competition, and use simulations to study an oligopolistic case. The analytical and simulation studies reveals interesting managerial insights.