Hybrid robust and stochastic optimization for closed-loop supply chain network design using accelerated Benders decomposition

Environmental, social and economic concerns motivate the operation of closed-loop supply chain networks (CLSCN) in many industries. We propose a novel profit maximization model for CLSCN design as a mixed-integer linear program in which there is flexibility in covering the proportions of demand satisfied and returns collected based on the firm\u27s policies. Our major contribution is to develop a novel hybrid robust-stochastic programming (HRSP) approach to simultaneously model two different types of uncertainties by including stochastic scenarios for transportation costs and polyhedral uncertainty sets for demands and returns. Transportation cost scenarios are generated using a Latin Hypercube Sampling method and scenario reduction is applied to consolidate them. An accelerated stochastic Benders decomposition algorithm is proposed for solving this model. To speed up the convergence of this algorithm, valid inequalities are introduced to improve the lower bound quality, and also a Pareto-optimal cut generation scheme is used to strengthen the Benders optimality cuts. Numerical studies are performed to verify our mathematical formulation and also demonstrate the benefits of the HRSP approach. The performance improvements achieved by the valid inequalities and Pareto-optimal cuts are demonstrated in randomly generated instances

Hybrid robust and stochastic optimization for closed-loop supply chain network design using accelerated Benders decomposition

Environmental, social and economic concerns motivate the operation of closed- loop supply chain networks (CLSCN) in many industries. We propose a novel profit maximization model for CLSCN design as a mixed-integer linear program in which there is flexibility in covering the proportions of demand satisfied and returns collected based on the firm\u27s policies. Our major contribution is to develop a novel hybrid robust-stochastic programming (HRSP) approach to simultaneously model two different types of uncertainties by including stochastic scenarios for transportation costs and polyhedral uncertainty sets for demands and returns. Transportation cost scenarios are generated using a Latin Hypercube Sampling method and scenario reduction is applied to consolidate them. An accelerated stochastic Benders decomposition algorithm is proposed for solving this model. To speed up the convergence of this algorithm, valid inequalities are introduced to improve the quality of lower bound, and also a Pareto-optimal cut generation scheme is used to strengthen the Benders optimality cuts. Numerical studies are performed to verify our mathematical formulation and also demonstrate the benefits of the HRSP approach. The performance improvements achieved by the valid inequalities and Pareto-optimal cuts are demonstrated in randomly generated instances

An optimum closed loop supply chain network model in a stochastic product life cycle context

Nowadays, closed loop supply chain network (CLSCN) receives considerable attention due to the growing awareness of the environmental destruction and depletion of natural resources. The establishment of a CLSCN is considered as a strategic decision that requires a lot of effort and intensive capital resources. Therefore, it is very crucial to make CLSCN design decisions taking into account multiple facets of uncertainties. Literature reviews to date reveals that uncertainties in product life cycle (PLC) or what has been called “product diffusion” have been vastly ignored. Particularly, the deterministic nature of the proposed diffusion models is a severe defect that can hinder the involvement of real-world uncertainties in design of a CLSCN. This study is an attempt to fill this gap by developing a costefficient CLSCN model for a product with dynamic and stochastic diffusion into the market that leads to an optimum design of the targeted CLSCN. Firstly, a geometric Brownian motion (GBM)-based diffusion forecast method was proposed and validated using a conventional approach namely, Holt’s method. Then, a two-stage stochastic programming mathematical model for optimum design of the targeted CLSCN was developed. The developed stochastic CLSCN model provides the optimum design of the targeted CLSCN utilizing the values predicted for the product diffusion through the PLC based on the proposed forecast method. The developed mathematical model addresses two types of decisions namely, “here and now” and “wait and see” decisions within the PLC. The “here and now” decisions were made in the first stage. The results show optimum values for decisions concerning configuration of the CLSCN as well as dynamic capacity allocation and expansion decisions through the PLC. However, the “wait and see” decisions are made in the second stage within the frame provided by the first-stage solutions. Here, the results portray optimum values for decisions concerning with the flow quantities between the CLSCN facilities, backorder and inventory levels, and recovery of returns through the PLC. In order to test the applicability of the developed CLSCN model, the mathematical model was coded by CPLEX software and solved for secondary data from the case study from previous case study in literature. Finally, a sensitivity analysis was performed to investigate the effect of diffusion uncertainty on the total cost of the CLSCN, its configuration, and production capacity allocations and expansions. The results of the sensitivity analysis revealed that, for the higher levels of diffusion uncertainty, the total cost imposed to the supply chain increases due to the increase in the allocated production capacity as well as the increase in the number of involved facilities

Closed-loop supply chain network design: Case of durable products

Closed loop supply chains comprise, in addition to the conventional forward flows from suppliers to end-users, a reverse flow of products, components, and materials from end-users to the manufacturers and secondary markets. Designing a closed-loop supply chain is a strategic level planning which considerably impacts on tactical and operational performance of the supply chain. It refers to the decisions taken on the location of facilities involved in the supply chain network along with the management of the physical flows associated with forward and product recovery channels. Our problem of interest is mainly motivated by the case of durable products including but not limited to large household appliances, computers, photocopying equipment, and aircraft engines. Such category of products has a modular structure, composed of independent components. As opposed to simple structured products, e.g., printer cartridges, that can only be recycled, each of the components in the reverse bill of materials of durable products can be recovered by a particular type of recovery process. Besides, durable products share a long life cycle characteristic which indeed makes designing their CLSC networks more complicated. In this thesis, in keeping with the abovementioned motivation, we focus on designing closed-loop and reverse supply chains in the context of durable products that are of various quality conditions. The recovery decisions for product return include remanufacturing, part harvesting, bulk recycling, material recycling, and landfilling/incineration. Moreover, we take into account environmental concerns regarding the harmful impacts of used products in the closed-loop supply chain planning. As the closed-loop supply chains typically encounter uncertainty in quality and quantity of the profitable return stream, we further aim to consider the impact of uncertainty in designing the recovery network. For such purposes, in the first phase, we address a closed-loop supply chain planning problem in the context of durable products with generic modular structures. The problem is formulated as a mixed-integer programming model which is then solved by an accelerated Benders decomposition-based algorithm. The performance of the proposed decomposition approach is enhanced through incorporating algorithmic features including valid inequalities, non-dominated optimality cuts, and local branching strategies. Next, in the second phase, we propose a precise approach to model the uncertain quality status of returns, in which the availability of each component in the reverse bill of materials is modeled as discrete scenarios. We propose a two-stage stochastic programming model to address this problem setting. Then, since the cardinality of the scenario set grows exponentially with the number of involved components, we detail on a scenario reduction scheme to alleviate the computational burden of the proposed model. The stochastic problem is solved by a L-shaped algorithm enhanced through valid inequalities and Pareto-optimal cuts. Finally, we investigate designing a dynamic reverse supply chain where the quantity of the return flows is uncertain. We introduce a multi-stage stochastic programming model and develop a heuristic inspired by scenario clustering decomposition scheme as the solution method. It revolves around decomposing the scenario tree into smaller sub-trees which consequently yields a number of sub-models in accordance with sub-trees. The resulting sub-models are then coordinated by Lagrangian penalty terms. On account of the fact that each sub-model per se is a hard to solve problem, a Benders decomposition-based algorithm is proposed to solve sub-models

Insight into the Sustainable Integration of Bio- and Petroleum Refineries for the Production of Fuels and Chemicals.

A petroleum refinery heavily depends on crude oil as its main feedstock to produce liquid fuels and chemicals. In the long term, this unyielding dependency is threatened by the depletion of the crude oil reserve. However, in the short term, its price highly fluctuates due to various factors, such as regional and global security instability causing additional complexity on refinery production planning. The petroleum refining industries are also drawing criticism and pressure due to their direct and indirect impacts on the environment. The exhaust gas emission of automobiles apart from the industrial and power plant emission has been viewed as the cause of global warming. In this sense, there is a need for a feasible, sustainable, and environmentally friendly generation process of fuels and chemicals. The attention turns to the utilization of biomass as a potential feedstock to produce substitutes for petroleum-derived fuels and building blocks for biochemicals. Biomass is abundant and currently is still low in utilization. The biorefinery, a facility to convert biomass into biofuels and biochemicals, is still lacking in competitiveness to a petroleum refinery. An attractive solution that addresses both is by the integration of bio- and petroleum refineries. In this context, the right decision making in the process selection and technologies can lower the investment and operational costs and assure optimum yield. Process optimization based on mathematical programming has been extensively used to conduct techno-economic and sustainability analysis for bio-, petroleum, and the integration of both refineries. This paper provides insights into the context of crude oil and biomass as potential refinery feedstocks. The current optimization status of either bio- or petroleum refineries and their integration is reviewed with the focus on the methods to solve the multi-objective optimization problems. Internal and external uncertain parameters are important aspects in process optimization. The nature of these uncertain parameters and their representation methods in process optimization are also discussed

Inexact Stabilized Benders' Decomposition Approaches, with Application to Chance-Constrained Problems with Finite Support

We explore modifications of the standard cutting-plane approach for minimizing a convex nondifferentiable function, given by an oracle, over a combinatorial set, which is the basis of the celebrated (generalized) Benders' decomposition approach. Specifically, we combine stabilization—in two ways: via a trust region in the L1 norm, or via a level constraint—and inexact function computation (solution of the subproblems). Managing both features simultaneously requires a nontrivial convergence analysis; we provide it under very weak assumptions on the handling of the two parameters (target and accuracy) controlling the informative on-demand inexact oracle corresponding to the subproblem, strengthening earlier know results. This yields new versions of Benders' decomposition, whose numerical performance are assessed on a class of hybrid robust and chance-constrained problems that involve a random variable with an underlying discrete distribution, are convex in the decision variable, but have neither separable nor linear probabilistic constraints. The numerical results show that the approach has potential, especially for instances that are difficult to solve with standard techniques

Bio-energy Logistics Network Design Under Price-based Supply and Yield Uncertainty

In this dissertation, we study the design and planning of bio-energy supply chain networks. This dissertation consists of 3 studies that focus on different aspects of bio-energy supply chain systems. In the first study, we consider planning and design of an extended supply chain for bio-energy networks in an integrated fashion while simultaneously addressing strategic and tactical decisions pertaining to location, production, inventory, and distribution in a multi-period planning horizon setting. For an efficient solution of our model, we suggest a Benders Decomposition based algorithm that can handle realistic size problems for design and analysis purposes. We provide computational results that demonstrate the efficiency of the solution approach on a wide ranging set of problem instances. Furthermore, we develop a realistic case by utilizing data pertaining to the state of Texas and conduct an extensive analysis on the effects of varying input parameters on the design outcomes for a bio-energy supply chain network. In the second study, we consider a two-stage stochastic problem to model farm-to-biorefinery biomass logistics while designing a policy that encourages farmers to plant biomass energy crops by offering them a unit wholesale price. In the first-stage, the model determines the supply chain network structure as well as the policy parameter, which is the biomass wholesale price offered to farmers. Second-stage problem is to determine the logistical decisions such as transportation, salvaging and out-sourcing. To solve this problem, we propose a solution framework that uses an algorithm based on the L-shaped method along with a Sample Average Approximation (SAA) approach. An extensive case study by varying some of the problem input parameters is conducted in Texas and the effects on the policy parameter (wholesale price), supply chain network design and expected total system cost are observed. In the last study, we propose a two-stage stochastic program to model a multi-period biomass-biofuel supply chain system to maximize the expected total system profit. We utilize a similar policy used in the second study to stimulate biomass energy crop production. Our model determines the policy parameter and the supply chain network structure in the first-stage and the tactical decisions for every time period in the second-stage. To solve this problem efficiently, we propose a solution algorithm based on the L-shaped method. Moreover, we also employ SAA approach in our solution methodology to statistically justify our solution quality. A case study is conducted in Texas for different biofuel prices and we analyze changes in the expected system profit the policy parameter and the supply chain network structure. Our case study results indicate that biofuel price needs to be at least $2.62/gal for the system to have a profit

Robust optimisation of dry port network design in the container shipping industry under uncertainty

PhD ThesisThe concept of dry port has attracted the attention of many researchers in the field of containerised transport industry over the past few decades. Previous research on dry port container network design has dealt with decision-making at different levels in an isolated manner. The purpose of this research is to develop a decision-making tool based on mathematical programming models to integrate strategic level decisions with operational level decisions. In this context, the strategic level decision making comprises the number and location of dry ports, the allocation of customers demand, and the provision of arcs between dry ports and customers within the network. On the other hand, the operational level decision making consists of containers flow, the selection of transportation modes, empty container repositioning, and empty containers inventory control. The containers flow decision involves the forward and backward flow of both laden and empty containers. Several mathematical models are developed for the optimal design of dry port networks while integrating all these decisions. One of the key aspects that has been incorporated in this study is the inherent uncertainty of container demands from end customers. Besides, a dynamic setting has to be adopted to consider the inevitable periodic fluctuation of demands. In order to incorporate the abovementioned decision-making integration with uncertain demands, several models are developed based on twostage stochastic programming approach. In the developed models, the strategic decisions are made in the first stage while the second-stage deals with operational decisions. The models are then solved through a robust sample average approximation approach, which is improved with the Benders Decomposition method. Moreover, several acceleration algorithms including multi-cut framework, knapsack inequalities, and Pareto-optimal cut scheme are applied to enhance the solution computational time. The proposed models are applied to a hypothetical case of dry port container network design in North Carolina, USA. Extensive numerical experiments are conducted to validate the dry port network design models. A large number of problem instances are employed in the numerical experiments to certify the capability of models. The quality of generated solutions is examined via a statistical validation procedure. The results reveal that the proposed approach can produce a reliable dry port container network under uncertain environment. Moreover, the experimental results underline the sensitivity of the configuration of the network to the inventory holding costs iii and the value of coefficients relating to model robustness and solution robustness. In addition, a number of managerial insights are provided that may be widely used in container shipping industry: that the optimal number of dry ports is inversely proportional to the empty container holding costs; that multiple sourcing is preferable when there are high levels of uncertainty; that rail tends to be better for transporting laden containers directly from seaports to customers with road being used for empty container repositioning; service level and fill rate improve when the design targets more robust solutions; and inventory turnover increases with high levels of holding cost; and inventory turnover decreases with increasing robustness