Column enumeration based decomposition techniques for a class of non-convex MINLP problems

We propose a decomposition algorithm for a special class of nonconvex mixed integer nonlinear programming problems which have an assignment constraint. If the assignment decisions are decoupled from the remaining constraints of the optimization problem, we propose to use a column enumeration approach. The master problem is a partitioning problem whose objective function coefficients are computed via subproblems. These problems can be linear, mixed integer linear, (non-)convex nonlinear, or mixed integer nonlinear. However, the important property of the subproblems is that we can compute their exact global optimum quickly. The proposed technique will be illustrated solving a cutting problem with optimum nonlinear programming subproblems.     

An integrated model for lot sizing with supplier selection and quantity discounts

Good inventory management is essential for a firm to be cost competitive and to acquire decent profit in the market, and how to achieve an outstanding inventory management has been a popular topic in both the academic field and in real practice for decades. As the production environment getting increasingly complex, various kinds of mathematical models have been developed, such as linear programming, nonlinear programming, mixed integer programming, geometric programming, gradient-based nonlinear programming and dynamic programming, to name a few. However, when the problem becomes NP-hard, heuristics tools may be necessary to solve the problem. In this paper, a mixed integer programming (MIP) model is constructed first to solve the lot-sizing problem with multiple suppliers, multiple periods and quantity discounts. An efficient Genetic Algorithm (GA) is proposed next to tackle the problem when it becomes too complicated. The objectives are to minimize total costs, where the costs include ordering cost, holding cost, purchase cost and transportation cost, under the requirement that no inventory shortage is allowed in the system, and to determine an appropriate inventory level for each planning period. The results demonstrate that the proposed GA model is an effective and accurate tool for determining the replenishment for a manufacturer for multi-periods.     

MIP-based decomposition strategies for large-scale scheduling problems in multiproduct multistage batch plants: A benchmark scheduling problem of the pharmaceutical industry

An efficient systematic iterative solution strategy for solving real-world scheduling problems in multiproduct multistage batch plants is presented. Since the proposed method has its core a mathematical model, two alternative MIP scheduling formulations are suggested. The MIP-based solution strategy consists of a constructive step, wherein a feasible and initial solution is rapidly generated by following an iterative insertion procedure, and an improvement step, wherein the initial solution is systematically enhanced by implementing iteratively several rescheduling techniques, based on the mathematical model. A salient feature of our approach is that the scheduler can maintain the number of decisions at a reasonable level thus reducing appropriately the search space. A fact that usually results in manageable model sizes that often guarantees a more stable and predictable optimization model behavior. The proposed strategy performance is tested on several complicated problem instances of a multiproduct multistage pharmaceuticals scheduling problem. On average, high quality solutions are reported with relatively low computational effort. Authors encourage other researchers to adopt the large-scale pharmaceutical scheduling problem to test on it their solution techniques, and use it as a challenging comparison reference.     

The plant location and flexible technology acquisition problem

In many industries, production–distribution networks have become more complex due to globalization. In particular, increasing interdependencies among structural decisions call for the development of integrated models. In this paper, we present a mathematical model for simultaneous optimization of the plant location, capacity acquisition and technology selection decisions in a multi-commodity environment. The proposed model represents the possible scale and scope economies associated with manufacturing technology alternatives. The problem is formulated as a mixed integer nonlinear program with concave costs. We developed an exact and three heuristic solution procedures. Using these procedures, we are able to solve fairly large facility design problems with reasonable computational effort.     

A decomposition approach for optimal control problems with integer innerpoint state constraints

A large class of optimal control problems for hybrid dynamic systems can be formulated as mixed‐integer optimal control problems (MIOCPs). A decomposition approach is suggested to solve a special subclass of MIOCPs with mixed integer inner point state constraints. It is the intrinsic combinatorial complexity of the discrete variables in addition to the high nonlinearity of the continuous optimal control problem that forms the challenges in the theoretical and numerical solution of MIOCPs. During the solution procedure the problem is decomposed at the inner time points into a multiphase problem with mixed integer boundary constraints and phase transitions at unknown switching points. Due to a discretization of the state space at the switching points the problem can be decoupled into a family of continuous optimal control problems (OCPs) and a problem similar to the asymmetric group traveling salesman problem (AGTSP). The OCPs are transcribed by direct collocation to large‐scale nonlinear programming problems, which are solved efficiently by an advanced SQP method. The results are used as weights for the edges of the graph of the corresponding TSP‐like problem, which is solved by a Branch‐and‐Cut‐and‐Price (BCP) algorithm. The proposed approach is applied to a hybrid optimal control benchmark problem for a motorized traveling salesman. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

安全约束条件下的经济调度优化模型研究

发电侧放开竞争的电力系统需要更加有效、准确的决策工具对有限的资源进行调度规划。短期经济调度优化问题是一个混合整数非线性规划问题,很难得到有效最优解,尤其是对于大规模电力系统。为了提高求解效率,本文提出了一个考虑安全约束的经济调度优化模型(Security-Constrained Economics Dispatch,SCED),主要采用线性化思想处理经济调度优化问题的模型以及各种约束,采用基于校正的交替求解方法,使得调度优化结果在运行成本最小化的前提下满足系统的安全稳定约束。同时,将本文方法运用到IEEE 30节点系统进行测试,从而验证本文方法有效性。     

ZIPMAP — A Zero-One Integer Programming Model for Market Segmentation and Product Positioning

An optimization model is proposed to aid marketing managers to search for and develop new product brand ideas. The model, which is founded on individual consumer behaviour constructs, determines a position for a new brand, in the consumers' perceptual space of product attributes, that maximizes company sales. The resulting mathematical model is stated as a large-scale, mixed, zero-one, integer, non-linear mathematical programming problem whose solution is sought through a two-stage optimization approach.     

On relaxations of the max k-cut problem formulations

A tight continuous relaxation is a crucial factor in solving mixed integer formulations of many NP-hard combinatorial optimization problems. The (weighted) max k-cut problem is a fundamental combinatorial optimization problem with multiple notorious mixed integer optimization formulations. In this paper, we explore four existing mixed integer optimization formulations of the max k-cut problem. Specifically, we show that the continuous relaxation of a binary quadratic optimization formulation of the problem is: (i) stronger than the continuous relaxation of two mixed integer linear optimization formulations and (ii) at least as strong as the continuous relaxation of a mixed integer semidefinite optimization formulation. We also conduct a set of experiments on multiple sets of instances of the max k-cut problem using state-of-the-art solvers that empirically confirm the theoretical results in item (i). Furthermore, these numerical results illustrate the advances in the efficiency of global non-convex quadratic optimization solvers and more general mixed integer nonlinear optimization solvers. As a result, these solvers provide a promising option to solve combinatorial optimization problems. Our codes and data are available on GitHub.     

An efficient computational method for a stochastic dynamic lot-sizing problem under service-level constraints

We provide an efficient computational approach to solve the mixed integer programming (MIP) model developed by Tarim and Kingsman [8] for solving a stochastic lot-sizing problem with service level constraints under the static-dynamic uncertainty strategy. The effectiveness of the proposed method hinges on three novelties: (i) the proposed relaxation is computationally efficient and provides an optimal solution most of the time, (ii) if the relaxation produces an infeasible solution, then this solution yields a tight lower bound for the optimal cost, and (iii) it can be modified easily to obtain a feasible solution, which yields an upper bound. In case of infeasibility, the relaxation approach is implemented at each node of the search tree in a branch-and-bound procedure to efficiently search for an optimal solution. Extensive numerical tests show that our method dominates the MIP solution approach and can handle real-life size problems in trivial time.     

On the inventory model with variable lead time and price–quantity discount

In this paper, we propose a mixed integer optimization approach for solving the inventory problem with variable lead time, crashing cost, and price–quantity discount. A linear programming relaxation based on piecewise linearization techniques is derived for the problem. It first converts non-linear terms into the sum of absolute terms, which are then linearized by goal programming techniques and linearization approaches. The proposed method can eliminate the complicated multiple-step solution process used in the traditional inventory models. In addition, the proposed model allows constraints to be added by the inventory decision-maker as deemed appropriate in real-world situations.     

Mathematical programming based heuristics for the 0–1 MIP: a survey

The 0–1 mixed integer programming problem is used for modeling many combinatorial problems, ranging from logical design to scheduling and routing as well as encompassing graph theory models for resource allocation and financial planning. This paper provides a survey of heuristics based on mathematical programming for solving 0–1 mixed integer programs (MIP). More precisely, we focus on the stand-alone heuristics for 0–1 MIP as well as those heuristics that use linear programming techniques or solve a series of linear programming models or reduced problems, deduced from the initial one, in order to produce a high quality solution of a considered problem. Our emphasis will be on how mathematical programming techniques can be used for approximate problem solving, rather than on comparing performances of heuristics.     

A MIP-based framework and its application on a lot sizing problem with setup carryover

In this paper we present a framework to tackle mixed integer programming problems based upon a “constrained” black box approach. Given a MIP formulation, a black-box solver, and a set of incumbent solutions, we iteratively build corridors around such solutions by adding exogenous constraints to the original MIP formulation. Such corridors, or neighborhoods, are then explored, possibly to optimality, with a standard MIP solver. An iterative approach in the spirit of a hill climbing scheme is thus used to explore subportions of the solution space. While the exploration of the corridor relies on a standard MIP solver, the way in which such corridors are built around the incumbent solutions is influenced by a set of factors, such as the distance metric adopted, or the type of method used to explore the neighborhood. The proposed framework has been tested on a challenging variation of the lot sizing problem, the multi-level lot sizing problem with setups and carryovers. When tested on 1920 benchmark instances of such problem, the algorithm was able to solve to near optimality every instance of the benchmark library and, on the most challenging instances, was able to find high quality solutions very early in the search process. The algorithm was effective, in terms of solution quality as well as computational time, when compared with a commercial MIP solver and the best algorithm from the literature.     

A trajectory-based method for mixed integer nonlinear programming problems

A local trajectory-based method for solving mixed integer nonlinear programming problems is proposed. The method is based on the trajectory-based method for continuous optimization problems. The method has three phases, each of which performs continuous minimizations via the solution of systems of differential equations. A number of novel contributions, such as an adaptive step size strategy for numerical integration and a strategy for updating the penalty parameter, are introduced. We have shown that the optimal value obtained by the proposed method is at least as good as the minimizer predicted by a recent definition of a mixed integer local minimizer. Computational results are presented, showing the effectiveness of the method.     

Scheduling a hybrid assembly-differentiation flowshop to minimize total flow time

This study considers a hybrid assembly-differentiation flowshop scheduling problem (HADFSP), in which there are three production stages, including components manufacturing, assembly, and differentiation. All the components of a job are processed on different machines at the first stage. Subsequently, they are assembled together on a common single machine at the second stage. At the third stage, each job of a particular type is processed on a dedicated machine. The objective is to find a job schedule to minimize total flow time (TFT). At first, a mixed integer programming (MIP) model is formulated and then some properties of the optimal solution are presented. Since the NP-hardness of the problem, two fast heuristics (SPT-based heuristic and NEH-based heuristic) and three hybrid meta-heuristics (HGA-VNS, HDDE-VNS and HEDA-VNS) are developed for solving medium- and large-size problems. In order to evaluate the performances of the proposed algorithms, a lower bound for the HADFSP with TFT criteria (HADFSP-TFT) is established. The MIP model and the proposed algorithms are compared on randomly generated problems. Computational results show the effectiveness of the MIP model and the proposed algorithms. The computational analysis indicates that, in average, the HDDE-VNS performs better and more robustly than the other two meta-heuristics, whereas the NEH heuristic consume little time and could reach reasonable solutions.     

A mathematical programming model and solution for scheduling production orders in Shanghai Baoshan Iron and Steel Complex

In this paper, we investigate the production order scheduling problem derived from the production of steel sheets in Shanghai Baoshan Iron and Steel Complex (Baosteel). A deterministic mixed integer programming (MIP) model for scheduling production orders on some critical and bottleneck operations in Baosteel is presented in which practical technological constraints have been considered. The objective is to determine the starting and ending times of production orders on corresponding operations under capacity constraints for minimizing the sum of weighted completion times of all orders. Due to large numbers of variables and constraints in the model, a decomposition solution methodology based on a synergistic combination of Lagrangian relaxation, linear programming and heuristics is developed. Unlike the commonly used method of relaxing capacity constraints, this methodology alternatively relaxes constraints coupling integer variables with continuous variables which are introduced to the objective function by Lagrangian multipliers. The Lagrangian relaxed problem can be decomposed into two sub-problems by separating continuous variables from integer ones. The sub-problem that relates to continuous variables is a linear programming problem which can be solved using standard software package OSL, while the other sub-problem is an integer programming problem which can be solved optimally by further decomposition. The subgradient optimization method is used to update Lagrangian multipliers. A production order scheduling simulation system for Baosteel is developed by embedding the above Lagrangian heuristics. Computational results for problems with up to 100 orders show that the proposed Lagrangian relaxation method is stable and can find good solutions within a reasonable time.     

Optimization of power production through coordinated use of hydroelectric and conventional power units

In the present work a methodology to tackle the problem of simultaneous utilization of hydroelectric and conventional power units with the goal of optimizing power production operations over the short term is presented. Most problem formulations found in the literature result in the development of nonlinear optimization programs, which are solved with stochastic methods. The methodology presented in this paper leads to the development of a convex mixed integer quadratic programming (MIQP) model, which is a special type of nonlinear model that enables reaching the global optimum solution in short computational time. The efficiency of the proposed approach is demonstrated by its application to a realistic power production system.     

New convergent heuristics for 0–1 mixed integer programming

Several hybrid methods have recently been proposed for solving 0–1 mixed integer programming problems. Some of these methods are based on the complete exploration of small neighborhoods. In this paper, we present several convergent algorithms that solve a series of small sub-problems generated by exploiting information obtained from a series of relaxations. These algorithms generate a sequence of upper bounds and a sequence of lower bounds around the optimal value. First, the principle of a linear programming-based algorithm is summarized, and several enhancements of this algorithm are presented. Next, new hybrid heuristics that use linear programming and/or mixed integer programming relaxations are proposed. The mixed integer programming (MIP) relaxation diversifies the search process and introduces new constraints in the problem. This MIP relaxation also helps to reduce the gap between the final upper bound and lower bound. Our algorithms improved 14 best-known solutions from a set of 108 available and correlated instances of the 0–1 multidimensional Knapsack problem. Other encouraging results obtained for 0–1 MIP problems are also presented.     

Stochastic multi-site capacity planning of TFT-LCD manufacturing using expected shadow-price based decomposition

This paper presents a stochastic optimization model and efficient decomposition algorithm for multi-site capacity planning under the uncertainty of the TFT-LCD industry. The objective of the stochastic capacity planning is to determine a robust capacity allocation and expansion policy hedged against demand uncertainties because the demand forecasts faced by TFT-LCD manufacturers are usually inaccurate and vary rapidly over time. A two-stage scenario-based stochastic mixed integer programming model that extends the deterministic multi-site capacity planning model proposed by Chen et al. (2010) [1] is developed to discuss the multi-site capacity planning problem in the face of uncertain demands. In addition a three-step methodology is proposed to generate discrete demand scenarios within the stochastic optimization model by approximating the stochastic continuous demand process fitted from the historical data. An expected shadow-price based decomposition, a novel algorithm for the stage decomposition approach, is developed to obtain a near-optimal solution efficiently through iterative procedures and parallel computing. Preliminary computational study shows that the proposed decomposition algorithm successfully addresses the large-scale stochastic capacity planning model in terms of solution quality and computation time. The proposed algorithm also outperforms the plain use of the CPLEX MIP solver as the problem size becomes larger and the number of demand scenarios increases.     

A Dynamic Programming Approach for Product Selection and Supermarket Shelf-Space Allocation

A dynamic programming approach is proposed to select optimally among a given set of products and allocate integer shelf-space units to the selected products in supermarkets. The approach is designed to consider general objective-function specifications that account for space elasticity, costs of sales, and potential demand-related marketing variables. The optimization is subject to constraints due to product supply availability, 'block' product allocation and operational requirements. A primary focus is on the development of a tractable model approach that can effectively be implemented on a microcomputer. A discussion of applications and computational experience on a microcomputer is provided to support the practical applicability of the optimization approach.     

An improved local branching approach for train formation planning

The train formation plan (TFP) determines routing and frequency of trains, and assigns the demands to trains. In this paper, an improved local branching algorithm is proposed for the TFP model in Iranian railway. This solution strategy is exact in nature, although it is designed to improve the heuristic behavior of the mixed integer programming (MIP) solver at hand. In the local branching algorithm, additional constraints are built in the model for the binary variables, but in the improved local branching algorithm, the additional constraints are built in the model for integer variables. A state-of-the-art method is applied for parameter tuning using design of experiments approach. To evaluate the proposed solution method, we have simulated and solved twenty test problems. The results show the efficiency and effectiveness of the proposed approach. The proposed algorithm is implemented for Iranian Railway network as a case study.