scholarly journals An Innovative Formulation Tightening Approach for Job-Shop Scheduling

2021 ◽  
Author(s):  
Bing Yan ◽  
Mikhail Bragin ◽  
Peter Luh
Keyword(s):  
Linear Programming ◽  
Convex Hull ◽  
Integer Linear Programming ◽  
Data Preprocessing ◽  
Mixed Integer ◽  
Continuous Variables ◽  
Job Shops ◽  
Production Environment ◽  
Binary Variables ◽  
Integer Variables

<p></p><p>Job shops are an important production environment for low-volume high-variety manufacturing.<i> </i>Its scheduling has recently been formulated as an Integer Linear Programming (ILP) problem to take advantages of popular Mixed-Integer Linear Programming (MILP) methods, e.g., branch-and-cut. When considering a large number of parts, MILP methods may experience difficulties. To address this, a critical but much overlooked issue is formulation tightening. The idea is that if problem constraints can be transformed to directly delineate the problem convex hull in the data preprocessing stage, then a solution can be obtained by using linear programming methods without much difficulty. The tightening process, however, is fundamentally challenging because of the existence of integer variables. In this paper, an innovative and systematic approach is established for the first time to tighten the formulations of individual parts, each with multiple operations, in the data preprocessing stage. It is a major advancement of our previous work on problems with binary and continuous variables to integer variables. The idea is to first link integer variables to binary variables by innovatively combining constraints so that the integer variables are uniquely determined by the binary variables. With binary and continuous variables only, it is proved that the vertices of the convex hull can be obtained based on vertices of the linear problem after relaxing binary requirements. These vertices are then converted to tight constraints for general use. This approach significantly improves our previous results on tightening individual operations. Numerical results demonstrate significant benefits on solution quality and computational efficiency. This approach also applies to other ILP problems with similar characteristics and fundamentally changes the way how such problems are formulated and solved. </p><p></p>

scholarly journals An Innovative Formulation Tightening Approach for Job-Shop Scheduling

2020 ◽  
Author(s):  
Bing Yan ◽  
Mikhail Bragin ◽  
Peter Luh
Keyword(s):  
Linear Programming ◽  
Convex Hull ◽  
Integer Linear Programming ◽  
Branch And Cut ◽  
Mixed Integer ◽  
Job Shops ◽  
Production Environment ◽  
Processing Stage ◽  
Binary Variables ◽  
Integer Variables

Job shops are an important production environment for low-volume high-variety manufacturing.<i> </i>Its scheduling has recently been formulated as an Integer Linear Programming (ILP) problem to take advantages of popular Mixed-Integer Linear Programming (MILP) methods, e.g., branch-and-cut. When considering a large number of parts, MILP methods may experience difficulties. To address this, a critical but much overlooked issue is formulation tightening. The idea is that if problem constraints can be transformed to directly delineate the problem convex hull in the data pre-processing stage, then a solution can be obtained by using linear programming methods without much difficulty. The tightening process, however, is NP hard because of the existence of integer variables. In this paper, an innovative and systematic approach is established for the first time to tighten the formulations of individual parts, each with multiple operations, in the data pre-processing stage. It is a major extension from our previous work on problems with binary and continuous variables to integer variables. The idea is to first link integer variables to binary variables by innovatively combining constraints so that the integer variables are uniquely determined by binary variables. With binary variables and continuous only, the vertices of the convex hull can be obtained based on the vertices of the linear problem after relaxing binary requirements with proved tightness. These vertices are then converted to tight constraints for general use. This approach significantly improves and extends our previous results on tightening single-operation parts without actually achieving tightness. Numerical results demonstrate significant benefits on solution quality and computational efficiency. This approach also applies to other ILP problems with similar characteristics and fundamentally changes the way how such problems are formulated and solved.

scholarly journals An Innovative Formulation Tightening Approach for Job-Shop Scheduling

2020 ◽  
Author(s):  
Bing Yan ◽  
Mikhail Bragin ◽  
Peter Luh
Keyword(s):  
Linear Programming ◽  
Convex Hull ◽  
Integer Linear Programming ◽  
Branch And Cut ◽  
Mixed Integer ◽  
Job Shops ◽  
Production Environment ◽  
Processing Stage ◽  
Binary Variables ◽  
Integer Variables

Job shops are an important production environment for low-volume high-variety manufacturing.<i> </i>Its scheduling has recently been formulated as an Integer Linear Programming (ILP) problem to take advantages of popular Mixed-Integer Linear Programming (MILP) methods, e.g., branch-and-cut. When considering a large number of parts, MILP methods may experience difficulties. To address this, a critical but much overlooked issue is formulation tightening. The idea is that if problem constraints can be transformed to directly delineate the problem convex hull in the data pre-processing stage, then a solution can be obtained by using linear programming methods without much difficulty. The tightening process, however, is NP hard because of the existence of integer variables. In this paper, an innovative and systematic approach is established for the first time to tighten the formulations of individual parts, each with multiple operations, in the data pre-processing stage. It is a major extension from our previous work on problems with binary and continuous variables to integer variables. The idea is to first link integer variables to binary variables by innovatively combining constraints so that the integer variables are uniquely determined by binary variables. With binary variables and continuous only, the vertices of the convex hull can be obtained based on the vertices of the linear problem after relaxing binary requirements with proved tightness. These vertices are then converted to tight constraints for general use. This approach significantly improves and extends our previous results on tightening single-operation parts without actually achieving tightness. Numerical results demonstrate significant benefits on solution quality and computational efficiency. This approach also applies to other ILP problems with similar characteristics and fundamentally changes the way how such problems are formulated and solved.

A Comparison of Two Mixed-Integer Linear Programs for Piecewise Linear Function Fitting

2021 ◽  
Author(s):  
John Alasdair Warwicker ◽  
Steffen Rebennack
Keyword(s):  
Linear Programming ◽  
Linear Function ◽  
Integer Linear Programming ◽  
Mixed Integer Linear Programming ◽  
Piecewise Linear ◽  
Piecewise Linear Function ◽  
Mixed Integer ◽  
Data Sets ◽  
Binary Variables ◽  
Function Fitting

The problem of fitting continuous piecewise linear (PWL) functions to discrete data has applications in pattern recognition and engineering, amongst many other fields. To find an optimal PWL function, the positioning of the breakpoints connecting adjacent linear segments must not be constrained and should be allowed to be placed freely. Although the univariate PWL fitting problem has often been approached from a global optimisation perspective, recently, two mixed-integer linear programming approaches have been presented that solve for optimal PWL functions. In this paper, we compare the two approaches: the first was presented by Rebennack and Krasko [Rebennack S, Krasko V (2020) Piecewise linear function fitting via mixed-integer linear programming. INFORMS J. Comput. 32(2):507–530] and the second by Kong and Maravelias [Kong L, Maravelias CT (2020) On the derivation of continuous piecewise linear approximating functions. INFORMS J. Comput. 32(3):531–546]. Both formulations are similar in that they use binary variables and logical implications modelled by big-[Formula: see text] constructs to ensure the continuity of the PWL function, yet the former model uses fewer binary variables. We present experimental results comparing the time taken to find optimal PWL functions with differing numbers of breakpoints across 10 data sets for three different objective functions. Although neither of the two formulations is superior on all data sets, the presented computational results suggest that the formulation presented by Rebennack and Krasko is faster. This might be explained by the fact that it contains fewer complicating binary variables and sparser constraints. Summary of Contribution: This paper presents a comparison of the mixed-integer linear programming models presented in two recent studies published in the INFORMS Journal on Computing. Because of the similarity of the formulations of the two models, it is not clear which one is preferable. We present a detailed comparison of the two formulations, including a series of comparative experimental results across 10 data sets that appeared across both papers. We hope that our results will allow readers to take an objective view as to which implementation they should use.

scholarly journals Global solutions of nonconvex standard quadratic programs via mixed integer linear programming reformulations

2021 ◽  
Author(s):  
Jacek Gondzio ◽  
E. Alper Yıldırım
Keyword(s):  
Linear Programming ◽  
Integer Linear Programming ◽  
Mixed Integer Linear Programming ◽  
Valid Inequalities ◽  
Global Minimizer ◽  
Mixed Integer ◽  
Quadratic Program ◽  
Complementarity Constraints ◽  
Binary Variables ◽  
Standard Quadratic Program

AbstractA standard quadratic program is an optimization problem that consists of minimizing a (nonconvex) quadratic form over the unit simplex. We focus on reformulating a standard quadratic program as a mixed integer linear programming problem. We propose two alternative formulations. Our first formulation is based on casting a standard quadratic program as a linear program with complementarity constraints. We then employ binary variables to linearize the complementarity constraints. For the second formulation, we first derive an overestimating function of the objective function and establish its tightness at any global minimizer. We then linearize the overestimating function using binary variables and obtain our second formulation. For both formulations, we propose a set of valid inequalities. Our extensive computational results illustrate that the proposed mixed integer linear programming reformulations significantly outperform other global solution approaches. On larger instances, we usually observe improvements of several orders of magnitude.

scholarly journals Minimum Spillage from Reservoirs using Mixed Integer Linear Programming

2021 ◽  
Vol 9 (ICRIE) ◽  
Author(s):  
Kamel A. Almohseen ◽  
Keyword(s):  
Linear Programming ◽  
Integer Linear Programming ◽  
Simplex Method ◽  
Mixed Integer Linear Programming ◽  
Optimal Solution ◽  
Mixed Integer ◽  
Integer Variables

The use of the traditional linear programming is not possible when an if-condition is to be imposed on the model unless some modifications are made. The difficulty arises due to the fact that the inclusion of if-condition to the generic formulation of the linear programming and its mechanism called "simplex method" is not a trivial task. The mixed integer linear programming seems to be a good candidate to achieve this goal. However, two issues should be satisfied beforehand if one would like to minimize the spill. 1. the reservoir should be full up to the spillway crest level in order for the spillage to occur. 2. the next state of the reservoir after the spill has been occurred should be full as well. Adding binary integer variables to the model helps in achieving the optimal solution in terms of minimum sum of spillage without violating any of the underlying constraints. When the input to the model being altered, the results showed that the model can cope with the uncertainty inherent in any natural inflow process in terms of spillage minimization.

Mixed-Integer Linear Optimization

1995 ◽  
Author(s):  
Christodoulos A. Floudas
Keyword(s):  
Linear Programming ◽  
Heat Exchanger ◽  
Integer Linear Programming ◽  
Mixed Integer Linear Programming ◽  
Linear Optimization ◽  
Batch Processes ◽  
Mixed Integer ◽  
Integer Variables ◽  
Mixed Integer Linear Optimization ◽  
Integer Linear Optimization

This chapter provides an introduction to the basic notions in Mixed-Integer Linear Optimization. Sections 5.1 and 5.2 present the motivation, formulation, and outline of methods. Section 5.3 discusses the key ideas in a branch and bound framework for mixed-integer linear programming problems. A large number of optimization models have continuous and integer variables which appear linearly, and hence separably, in the objective function and constraints. These mathematical models are denoted as Mixed-Integer Linear Programming MILP problems. In many applications of MILP models the integer variables are 0 — 1 variables (i.e., binary variables), and in this chapter we will focus on this sub-class of MILP problems. A wide range of applications can be modeled as mixed-integer linear programming MILP problems. These applications have attracted a lot of attention in the field of Operations Research and include facility location and allocation problems, scheduling problems, and fixed-charge network problems. The excellent books of Nemhauser and Wolsey (1988), and Parker and Rardin (1988) provide not only an exposition to such applications but also very thorough presentation of the theory of discrete optimization. Applications of MILP models in Chemical Engineering have also received significant attention particularly in the areas of Process Synthesis, Design, and Control. These applications include (i) the minimum number of matches in heat exchanger synthesis (Papoulias and Grossmann, 1983; see also chapter 8) (ii) heat integration of sharp distillation sequences (Andrecovich and Westerberg, 1985); (iii) multicomponent multiproduct distillation column synthesis (Floudas and Anastasiadis, 1988); (iv) multiperiod heat exchanger network, and distillation system synthesis (Floudas and Grossmann, 1986; Paules and Floudas, 1988); flexibility analysis of chemical processes (Grossmann and Floudas, 1987); (v) structural properties of control systems (Georgiou and Floudas, 1989, 1990); (vi) scheduling of batch processes (e.g., Rich and Prokapakis, 1986, 1986; Kondili et al., 1993; Shah et al, 1993; Voudouris and Grossmann, 1992, 1993); and (vii) planning and scheduling of batch processes (Shah and Pantelides, 1991, Sahinidis et al, 1989, Sahinidis and Grossmann, 1991). In addition to the above applications, MILP models are employed as subproblems in the mixed-integer nonlinear optimization approaches which we will discuss in the next chapter. In this section, we will present the formulation of Mixed-Integer Linear Programming MILP problems, discuss the complexity issues, and provide a brief overview of the solution methodologies proposed for MILP models.

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