Mixed Integer Linear Programming Formulation Techniques

A wide range of problems can be modeled as Mixed Integer Linear Programming (MIP) problems using standard formulation techniques. However, in some cases the resulting MIP can be either too weak or too large to be effectively solved by state of the art solvers. In this survey we review advanced MIP formulation techniques that result in stronger and/or smaller formulations for a wide class of problems

Computer assisted modelling of linear, integer and separable programming problems

For mathematical programming (MP) to have greater impact upon the decision making process, MP software systems must offer suitable support in terms of model communication and modelling techniques . In this paper modelling techniques that allow logical restrictions to be modelled in integer programming terms are described and their implications discussed. In addition it is demonstrated that many classes of non-linearities which are not variable separable may be reformulated in piecewise linear form. It is shown that analysis of bounds is necessary in the following three important contexts: model reduction, formulation of logical restrictions as 0-1 mixed integer programs and reformulation of nonlinear programs as variable separable programs, It is observed that as well as incorporating an interface between the modeller and the optimiser there is a need to make available to the modeller software facilities which support the modelling techniques described here

Path planning algorithm for a car like robot based on MILP method

This project is presents an algorithm for path planning optimal routes mobile robot “like a car” to a target in unknown environment. The proposed algorithm allows a mobile robot to navigate through static obstacles and finding the path in order to reach the target without collision. This algorithm provides the robot the possibility to move from the initial position to the final position (target). The proposed path finding strategy is to use mathematical programming techniques to find the optimal path between to state for mobile robot designed in unknown environment with stationary obstacles. Formulation of the basic problems is to have the vehicle moved from the initial dynamic state to a state without colliding with each other, while at the same time avoiding other stationary obstacles. It is shown that this problem can be rewritten as a linear program with mixed integer / linear constraints that account for the collision avoidance. This approach is that the path optimization can be easily solved using the CPLEX optimization software with AMPL interface / MATLAB. The final phases are the design and build coalitions of linear programs and binary constraints to avoid collision with obstacles by Integer Mixed Linear Program (MILP). The findings of this research have shown that the MILP method can be used in the path planning problem in terms of finding a safe and shortest path. This has been combined with collision avoidance constraints to form a mixed integer linear program, which can be solved by a commercial software package

Optimal Algorithms for Near-Hitless Network Restoration via Diversity Coding

Diversity coding is a network restoration technique which offers near-hitless restoration, while other state-of-the art techniques are significantly slower. Furthermore, the extra spare capacity requirement of diversity coding is competitive with the others. Previously, we developed heuristic algorithms to employ diversity coding structures in networks with arbitrary topology. This paper presents two algorithms to solve the network design problems using diversity coding in an optimal manner. The first technique pre-provisions static traffic whereas the second technique carries out the dynamic provisioning of the traffic on-demand. In both cases, diversity coding results in smaller restoration time, simpler synchronization, and much reduced signaling complexity than the existing techniques in the literature. A Mixed Integer Programming (MIP) formulation and an algorithm based on Integer Linear Programming (ILP) are developed for pre-provisioning and dynamic provisioning, respectively. Simulation results indicate that diversity coding has significantly higher restoration speed than Shared Path Protection (SPP) and p-cycle techniques. It requires more extra capacity than the p-cycle technique and SPP. However, the increase in the total capacity is negligible compared to the increase in the restoration speed.Comment: An old version of this paper is submitted to IEEE Globecom 2012 conferenc

A Bilevel Approach to Frequency Optimization in Public Transportation Systems

We consider the problem of frequency optimization in transit systems, whose objective is to determine the time interval between subsequent buses for a set of public transportation lines. We extend an existing single level model by adding a constraint on bus capacities, while maintaining user choice on routes by means of an assignment sub-model. The resulting formulation is bilevel, and is transformed into a mixed integer linear programming formulation (MILP) that can be solved to optimality for small-sized problem instances, using standard MILP techniques. We study different variants of the same formulation to better understand the bilevel nature of the model and its application to real settings

Simultaneous mixed-integer disjunctive optimization for synthesis of petroleum refinery topology Processing Alternatives for Naphtha Produced from Atmospheric Distillation Unit

In this work, we propose a logic-based modeling technique within a mixed-integer disjunctive superstructure optimization framework on the topological optimization problem for determining the optimal petroleum refinery configuration. We are interested to investigate the use of logic cuts that are linear inequality/equality constraints to the conceptual process synthesis problem of the design of a refinery configuration. The logic cuts are employed in two ways using 0-l variables: ( l) to enforce certain design specifications based on past design experience, engineering knowledge, and heuristics; and (2) to enforce certain structural specifications on the interconnections of the process units. The overall modeling framework conventionally gives rise to a mixedinteger optimization framework, in this case, a mixed-integer linear programming model (because of the linearity of the constraints). But in this work, we elect to adopt a disjunctive programming framework, specifically generalized disjunctive programming (GDP) proposed by Grossmann and co-workers (Grossmann, l. E. (2002). Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques. Optimization & Engineering, 3, 227.) The proposed GOP-based modeling technique is illustrated on a case study to determine the optimal processing route of naphtha in a refinery using the GAMS/LogMIP platform, which yields practically-acceptable solution. The use of LogMIP obviates the need to reformulate the logic propositions and the overall disjunctive problem into algebraic representations, hence reducing the time involved in the typically time-consuming problem formulation. LogMIP typically leads to less computational time and number of iterations in its computational effort because the associated GDP formulation involves less equations and variables compared to MILP. From the computational experiments, it is found that logical constraints of design specifications and structural specifications potentially play an important role to determine the optimal selection of process units and streams. Hence, in general, the GDP formulation can be improved by adding or eliminating constraits that can accelerate or slow-down the problem solution respectively

A Comprehensive Approach for an Approximative Integration of Nonlinear-Bivariate Functions in Mixed-Integer Linear Programming Models

As decentralized energy supply units, microgrids can make a decisive contribution to achieving climate targets. In this context, it is particularly important to determine the optimal size of the energy components contained in the microgrids and their optimal operating schedule. Hence, mathematical optimization methods are often used in association with such tasks. In particular, mixed-integer linear programming (MILP) has proven to be a useful tool. Due to the versatility of the different energetic components (e.g., storages, solar modules) and their special technical characteristics, linear relationships can often only inadequately describe the real processes. In order to take advantage of linear solution techniques but at the same time better represent these real-world processes, accurate and efficient approximation techniques need to be applied in system modeling. In particular, nonlinear-bivariate functions represent a major challenge, which is why this paper derives and implements a method that addresses this issue. The advantage of this method is that any bivariate mixed-integer nonlinear programming (MINLP) formulation can be transformed into a MILP formulation using this comprehensive method. For a performance comparison, a mixed-integer quadratic constrained programming (MIQCP) model—as an MINLP special case—is applied and transformed into a MILP, and the solution of the transformed problem is compared with the one of the MIQCP. Since there are good off-the-shelf solvers for MIQCP problems available, the comparison is conservative. The results for an exemplary microgrid sizing task show that the method delivers a strong performance, both in terms of approximation error (0.08%) and computation time. The method and its implementation can serve as a general user-tool but also as a basis for further methodological developments and research