Large Mixed Integer Linear Programming Research Articles

Data driven robust optimization for handling uncertainty in supply chain planning models

While addressing supply chain planning under uncertainty, Robust Optimization (RO) is regarded as an efficient and tractable method. As RO involves calculation of several statistical moments or maximum / minimum values involving the objective functions under realizations of these uncertain parameters, the accuracy of this method significantly depends on the efficient techniques to sample the uncertainty parameter space with limited amount of data. Conventional sampling techniques, e.g. box/budget/ellipsoidal, work by sampling the uncertain parameter space inefficiently, often leading to inaccuracies in such estimations. This paper proposes a methodology to amalgamate machine learning and data analytics with RO, thereby making it data-driven. A novel neuro fuzzy clustering mechanism is implemented to cluster the uncertain space such that the exact regions of uncertainty are optimally identified. Subsequently, local density based boundary point detection and Delaunay triangulation based boundary construction enable intelligent Sobol based sampling to sample the uncertain parameter space more accurately. The proposed technique is utilized to explore the merits of RO towards addressing the uncertainty issues of product demand, machine uptime and production cost associated with a multiproduct, and multisite supply chain planning model. The uncertainty in supply chain model is thoroughly analysed by carefully constructing examples and its case studies leading to large scale mixed integer linear and nonlinear programming problems which were efficiently solved in GAMS® framework. Demonstration of efficacy of the proposed method over the box, budget and ellipsoidal sampling method through comprehensive analysis adds to other highlights of the current work.

A deterministic model for the transshipment problem of a fast fashion retailer under capacity constraints

In this paper we present a novel transshipment problem for a large apparel retailer that operates an extensive retail network. Our problem is inspired by the logistics operations of a very large fast fashion retailer in Turkey, LC Waikiki, with over 475 retail branches and thousands of products. The purpose of transshipments is to rebalance stocks across the retail network to better match supply with demand. We formulate this problem as a large mixed integer linear program and develop a Lagrangian relaxation with a primal–dual approach to find upper bounds and a simulated annealing based metaheuristic to find promising solutions, both of which have proven to be quite effective. While our metaheuristic does not always produce better solutions than a commercial optimizer, it has consistently produced solutions with optimality gaps lower than 7% while the commercial optimizer may produce very poor solutions with optimality gaps as high as almost 300%. We have also conducted a set of numerical experiments to uncover implications of various operational practices of LC Waikiki on its system’s performance and important managerial insights.

A Deterministic Model for the Transshipment Problem of a Fast Fashion Retailer under Capacity Constraints

In this paper we present a novel transshipment problem for a large apparel retailer that operates an extensive retail network. Our problem is inspired by the logistics operations of a very large fast fashion retailer in Turkey, LC Waikiki, with over 450 retail branches and thousands of products. The purpose of transshipments is to rebalance stocks across the retail network to better match supply with demand. We formulate this problem as a large mixed integer linear program and develop a Lagrangian relaxation with a primal-dual approach to find upper bounds and a simulated annealing based metaheuristic to find promising solutions, both of which have proven to be quite effective. While our metaheuristic does not always produce better solutions than a commercial optimizer, it has consistently produced solutions with optimality gaps lower than 7% while the commercial optimizer may produce very poor solutions with optimality gaps as high as almost 300%. We have also conducted a set of numerical experiments to uncover implications of various operational practices of LC Waikiki on its system’s performance and important managerial insights.

Long-haul vehicle routing and scheduling with idling options

This paper introduces the vehicle routing and truck driver scheduling problem with idling options, an extension of the long-haul vehicle routing and truck driver scheduling problem with a more comprehensive objective function that accounts for routing cost, driver cost and idling cost, i.e., the cost associated with energy supply used to maintain drivers’ comfort when the vehicle is not moving. For the idling cost, we consider Electrified Parking Space (EPS) and Auxiliary Power Unit (APU) usage costs. The use of EPSs or APUs avoids keeping the engine running while the vehicle is not moving. We develop a multi-start matheuristic algorithm that combines adaptive large neighborhood search and mixed integer linear programming. We present extensive computational results on instances derived from the Solomon test bed.

Design of the optimal industrial symbiosis system to improve bioethanol production

Abstract The emergence of environmental and sustainability regulations, such as Kyoto protocol, Energy Policy Act and the increasingly limited availability of fossil fuels has brought the notion of gradually substituting petroleum products with bioethanol into the limelight. Even though, bioethanol is one of the cleanest sources of energy, a major concern of bioethanol production is its economic feasibility. Industrial symbiosis is one of the sustainable strategies that can help to reduce bioethanol production and logistic costs. In industrial symbiosis, traditionally separate plants collocate in order to efficiently utilize resources, reduce wastes and increase profits for the entire industrial symbiosis and each player in the industrial symbiosis. This paper focuses on developing optimal configurations of bioenergy-based industrial symbiosis under certain constraints, such that the bioethanol production cost (or profit) is reduced (or increased). A decision framework that combines the Linear Programming models and large scale Mixed Integer Linear Programming model is proposed to determine the optimal configuration of bioenergy-based industrial symbiosis and to design the optimal network flows of various products in the bioenergy-based industrial symbiosis. A case study has been conducted to study the efficiency and effectiveness of the proposed model and the results suggest significant increase in profitability for biorefinery plant and the rest of the players in the bioenergy-based industrial symbiosis system. Sensitivity analysis is also conducted to provide deep understanding of the proposed bioenergy-based industrial symbiosis system and to identify the factors that might impact the performance of biorefinery plant in bioenergy-based industrial symbiosis.

Coordinated capacity planning in two-stage thin-film-transistor liquid-crystal-display (TFT-LCD) production networks

This paper studies capacity planning problems in two-stage Thin Film Transistor-Liquid Crystal Display (TFT-LCD) production networks. In the TFT-LCD industry, capacity planning must coordinate capacity expansion and capacity allocation decisions within the planning horizon. Capacity planning problems are extremely challenging for the TFT-LCD industry owing to the following characteristics: (1) multi-stage, multi-site, and multi-generation production network with limited configuration flexibility; (2) an economic cutting ratio that varies at different sites; (3) multiple capacity types at each production site; and (4) a complex product hierarchy and rapidly changing demand. Due to the limited configuration flexibility, capacity expansion at different production sites must be coordinated to prevent bottlenecks at other sites. Moreover, capacity allocation decisions must be coordinated with the capacity expansion decisions to generate most profitable product mixes.To address these problems, we develop a mixed integer linear programming (MILP) model to maximize profit. Since large MILP models are computationally intractable, an iterative two-phase shadow-price heuristic with reactive and conservative expansion strategies is proposed to find near-optimal solutions. Through an extensive computational study, the effectiveness and robustness of the proposed heuristic is demonstrated in terms of solution quality and computational time. In addition, by conducting the sensitivity analysis, we suggest the suitable usage timing and choice of the two expansion strategies under different price and cost situations for capacity planners of the TFT-LCD manufacturing.

Railway Rolling Stock Planning: Robustness Against Large Disruptions

In this paper we describe a two-stage optimization model for determining robust rolling stock circulations for passenger trains. Here robustness means that the rolling stock circulations can better deal with large disruptions of the railway system. The two-stage optimization model is formulated as a large mixed-integer linear programming (MILP) model. We first use Benders decomposition to determine optimal solutions for the LP-relaxation of this model. Then we use the cuts that were generated by the Benders decomposition for computing heuristic robust solutions for the two-stage optimization model. We call our method Benders heuristic. We evaluate our approach on the real-life rolling stock-planning problem of Netherlands Railways, the main operator of passenger trains in the Netherlands. The computational results show that, thanks to Benders decomposition, the LP-relaxation of the two-stage optimization problem can be solved in a short time for a representative number of disruption scenarios. In addition, they demonstrate that the robust rolling stock circulation computed heuristically has total costs that are close to the LP lower bounds. Finally, we discuss the practical effectiveness of the robust rolling stock circulation: When a large number of disruption scenarios were applied to these robust circulations and to the nonrobust optimal circulations, the former appeared to be much more easily recoverable than the latter.

A DC programming heuristic applied to the logistics network design problem

This paper proposes a new heuristic method for the logistics network design and planning problem based on linear relaxation and DC (difference of convex functions) programming. We consider a multi-period, multi-echelon, multi-commodity and multi-product problem defined as a large scale mixed integer linear programming (MILP) model. The method is experimented on data sets of various size. The numerical results validate the efficiency of the heuristic for instances with up to several dozens facilities, 18 products and 270 retailers.

Simulation of Optimal Medium-Term Hydro-Thermal System Operation by Grid Computing

This paper presents a model for the simulation of the optimal medium-term operation of a hydrothermal system. System stochastic parameters are modeled by Monte Carlo scenarios, which are solved on distributed processors. For each scenario a yearly hydro-thermal scheduling (HTS) problem with hourly time resolution is formulated and solved as a large mixed integer linear program (MILP). HTS modeling includes unit commitment, start-up costs and minimum up/down time constraints. The model is applied to the Greek power system, comprising 29 thermal units and 13 hydroplants; 100 simulation scenarios are generated and solved on 18 distributed processors. Test results include both medium-term objectives, such as reservoir water management, and short-term decisions such thermal unit start-up decisions.

Opportunistic supply chain formation from qualified partners for a new market demand

This paper addresses the problem of short-term supply chain design using the idle capacities of qualified partners in order to seize a new market opportunity. The new market opportunity is characterized by a deterministic forecast over a planning horizon. The production–distribution process is assumed to be organized in stages or echelons, and each echelon may have several qualified partners willing to participate. Partners within the echelon may differ in idle production capacity, operational cost, storage cost, etc, and we assume that idle capacity may be different from one period to another period. The objective is to design a supply chain by selecting one partner from each echelon to meet the forecasted demand without backlog and best possible production and logistics costs over the given planning horizon. The overall problem is formulated as a large mixed integer linear programming problem. We develop a decomposition-based solution approach that is capable of overcoming the complexity and dimensionality associated with the problem. Numerical results are presented to support the effectiveness of this approach.

Mixed Integer Models for the Stationary Case of Gas Network Optimization

A gas network basically consists of a set of compressors and valves that are connected by pipes. The problem of gas network optimization deals with the question of how to optimize the flow of the gas and to use the compressors cost-efficiently such that all demands of the gas network are satisfied. This problem leads to a complex mixed integer nonlinear optimization problem. We describe techniques for a piece-wise linear approximation of the nonlinearities in this model resulting in a large mixed integer linear program. We study sub-polyhedra linking these piece-wise linear approximations and show that the number of vertices is computationally tractable yielding exact separation algorithms. Suitable branching strategies complementing the separation algorithms are also presented. Our computational results demonstrate the success of this approach.

Supply chain modelling of forest fuel

We study the problem of deciding when and where forest residues are to be converted into forest fuel, and how the residues are to be transported and stored in order to satisfy demand at heating plants. Decisions also include whether or not additional harvest areas and saw-mills are to be contracted. In addition, we consider the flow of products from saw-mills and import harbors, and address the question about which terminals to use. The planning horizon is one year and monthly time periods are considered. The supply chain problem is formulated as a large mixed integer linear programming model. In order to obtain solutions within reasonable time we have developed a heuristic solution approach. Computational results from a large Swedish supplying entrepreneur are reported.

Multi-product inventory planning with downward substitution, stochastic demand and setup costs

In this paper we consider a single period multi-product inventory problem with stochastic demand, setup cost for production, and one-way product substitution in the downward direction. We model the problem as a two-stage integer stochastic program with recourse where the first stage variables determine which products to produce and how much to produce, and the second stage variables determine how the products are allocated to satisfy the realized demand. We exploit structural properties of the model and utilize a combination of optimization techniques including network flow, dynamic programming, and simulation-based optimization to develop effective heuristics. Through a computational study, we evaluate the performance of our heuristics by comparison with the corresponding optimal solution obtained from a large scale mixed integer linear program. The computational study indicates that our solution methodology can be very effective (98.8% on average) and can handle industrial-sized problems efficiently. We also provide several new qualitative insights on issues such as the effect of demand variance and cost parameters on the optimal number of products setup, the amount produced or inventoried, and the benefits of allowing substitution.

A MILP decomposition approach to large scale optimization in structural design of energy supply systems

An optimal structural design method is proposed to determine the structures of energy supply systems in consideration of their multi-period operation. The optimization problem is formulated as a large scale mixed integer linear programming (MILP) problem with binary variables for selection and on/off status of operation of equipment and continuous variables for capacities and load allocation of equipment. The dependence of the performance characteristics of equipment on their capacities and part loads, as well as that of capital costs of equipment on their capacities, are incorporated into the optimization model. A decomposition method of solving MILP problems with the block angular structure is applied to derive a suboptimal solution close to the optimal one in a short computation time. The meaning and validity of the decomposition method are clarified through an analytical study on an energy supply system with a simple structure. The validity and effectiveness of the decomposition method are also clarified in terms of solution optimality and computation time through a numerical study on another energy supply system with a complex structure.

D.C. programming approach for multicommodity network optimization problems with step increasing cost functions

We address a class of particularly hard-to-solve combinatorial optimization problems, namely that of multicommodity network optimization when the link cost functions are discontinuous step increasing. Unlike usual approaches consisting in the development of relaxations for such problems (in an equivalent form of a large scale mixed integer linear programming problem) in order to derive lower bounds, our d.c.(difference of convex functions) approach deals with the original continuous version and provides upper bounds. More precisely we approximate step increasing functions as closely as desired by differences of polyhedral convex functions and then apply DCA (difference of convex function algorithm) to the resulting approximate polyhedral d.c. programs. Preliminary computational experiments are presented on a series of test problems with structures similar to those encountered in telecommunication networks. They show that the d.c. approach and DCA provide feasible multicommodity flows xa such that the relative differences between upper bounds (computed by DCA) and simple lower bounds r:=(f(xa)-LB)/{f(xa)} lies in the range [4.2 %, 16.5 %] with an average of 11.5 %, where f is the cost function of the problem and LB is a lower bound obtained by solving the linearized program (that is built from the original problem by replacing step increasing cost functions with simple affine minorizations). It seems that for the first time so good upper bounds have been obtained.

Branch and cut methods for network optimization

Combinatorial optimization problems arising in network applications are usually computationally difficult. Typically, one needs to solve a large mixed integer linear programming problem. Over the past decade, the method of branch and cut has emerged as a powerful technique for solving such problems. In this paper, we focus on computationally difficult network optimization problems, highlighting the effectiveness of branch and cut methods.

A novel decomposition method for MILP and its application to optimal operation of a thermal storage system

A novel decomposition method for solving large scale mixed integer linear programming (MILP) problems with block angular structure is proposed to conduct operational planning of energy supply systems with storage units rationally and efficiently. The method is composed of solving two types of problems repeatedly: one is a large scale MILP master problem with the values of part of the integer variables fixed from the results of the other; the other is a small scale MILP subproblem with the values of the simplex multipliers obtained from the results of the former. The method is applied to the operational planning of a heat supply system with cold and hot water storage tanks. The validity and effectiveness of the method is evaluated in terms of solution optimality and computation time. As a result, it turns out that the method is superior to the conventional branch and bound method with the depth-first or best-bound search.

Design of multipurpose production facilities: A RTN decomposition-based algorithm

A general mathematical formulation for the design of multipurpose facilities has recently been presented by Barbosa-Póvoa and Pantelides (1997). The model proposed permits a detailed consideration of the design problem taking account of trade-offs between capital costs, revenues and operational flexibility. The optimal solution involves the selection of the required processing and storage equipment items and the required levels of provision of other production resources such as utilities, manpower, cleaning and transportation equipment. In order to guarantee solution optimality, the above design formulation has to consider a large number of equipment items, out of which it will select the ones that will actually be incorporated in the plant. This may result in large mixed-integer linear programming (MILP) problems that are expensive to solve. This paper presents a decomposition approach for the solution of large batch process design problems. The approach involves the iterative solution of a master problem (representing a relaxation of the original design problem) and a design sub-problem (in which several of the design decisions are already fixed). An example illustrating the effectiveness of the proposed decomposition approach is presented.

Optimizing the natural gas supply mix of local distribution utilities

A large mixed-integer linear program (MILP) and a much smaller nonlinear programming (NLP) approximation of the MILP, involving simulation and response surface estimation via regression analysis, are proposed to solve the problem of the optimal selection of natural gas supply contracts by local gas distribution utilities. Each potential supply source is characterized by several price and nonprice parameters. Weather variability is the basic stochastic factor that drives the demand for gas by various market segments. The model minimizes the total cost of gas supply and market curtailment, and thus determines the size of the interruptible market. A numerical application of the methodology illustrates the excellent quality of the NLP approximation and the importance of the trade-offs between contract characteristics. A multi-temporal extension of the modeling methodology is outlined.

A Continuous Time Mixed Integer Linear Programming Model for Short Term Scheduling of Multistage Batch Plants

The problem of short term scheduling of batch plants consists of determining the optimal production policy for satisfying the production demands for different products at due dates and/or at the end of a given time horizon. The objective of the work is to propose an optimization model and solution method to the short term scheduling of batch plants with multiple stages which may contain equipment in parallel. A large scale mixed integer linear programming (MILP) model with continuous time domain representation is proposed that relies on the use of parallel time axes for units and tasks. Although in principle an LP-based branch and bound method can be used to solve the problem, there is a limitation when the instances become large. The first solution strategy that is proposed consists of the use of preordering constraints. Furthermore, a second strategy relies on a decomposition scheme for large systems which is based on the solution of an MILP model that minimizes total in process time in which assignments are determined and the subsequent solution of an LP to minimize earliness and to eliminate unnecessary setups. Several examples are presented, including a large real world problem, to illustrate the performance of the model and solution method