scholarly journals A branch-and-cut algorithm for Mixed-Integer Bilinear Programming

2020 ◽  
Vol 282 (2) ◽  
pp. 506-514 ◽  
Author(s):  
Matteo Fischetti ◽  
Michele Monaci
Keyword(s):  
Branch And Cut ◽  
Bilinear Programming ◽  
Mixed Integer

A Criterion Space Branch-and-Cut Algorithm for Mixed Integer Bilinear Maximum Multiplicative Programs

2022 ◽  
Author(s):  
Vahid Mahmoodian ◽  
Iman Dayarian ◽  
Payman Ghasemi Saghand ◽  
Yu Zhang ◽  
Hadi Charkhgard
Keyword(s):  
Objective Function ◽  
Branch And Bound ◽  
Capacity Allocation ◽  
Branch And Cut ◽  
Nash Bargaining ◽  
Mixed Integer ◽  
Bargaining Solution ◽  
Reliability Optimization ◽  
Criterion Space ◽  
Multiplicative Programs

This study introduces a branch-and-bound algorithm to solve mixed-integer bilinear maximum multiplicative programs (MIBL-MMPs). This class of optimization problems arises in many applications, such as finding a Nash bargaining solution (Nash social welfare optimization), capacity allocation markets, reliability optimization, etc. The proposed algorithm applies multiobjective optimization principles to solve MIBL-MMPs exploiting a special characteristic in these problems. That is, taking each multiplicative term in the objective function as a dummy objective function, the projection of an optimal solution of MIBL-MMPs is a nondominated point in the space of dummy objectives. Moreover, several enhancements are applied and adjusted to tighten the bounds and improve the performance of the algorithm. The performance of the algorithm is investigated by 400 randomly generated sample instances of MIBL-MMPs. The obtained result is compared against the outputs of the mixed-integer second order cone programming (SOCP) solver in CPLEX and a state-of-the-art algorithm in the literature for this problem. Our analysis on this comparison shows that the proposed algorithm outperforms the fastest existing method, that is, the SOCP solver, by a factor of 6.54 on average. Summary of Contribution: The scope of this paper is defined over a class of mixed-integer programs, the so-called mixed-integer bilinear maximum multiplicative programs (MIBL-MMPs). The importance of MIBL-MMPs is highlighted by the fact that they are encountered in applications, such as Nash bargaining, capacity allocation markets, reliability optimization, etc. The mission of the paper is to introduce a novel and effective criterion space branch-and-cut algorithm to solve MIBL-MMPs by solving a finite number of single-objective mixed-integer linear programs. Starting with an initial set of primal and dual bounds, our proposed approach explores the efficient set of the multiobjective problem counterpart of the MIBL-MMP through a criterion space–based branch-and-cut paradigm and iteratively improves the bounds using a branch-and-bound scheme. The bounds are obtained using novel operations developed based on Chebyshev distance and piecewise McCormick envelopes. An extensive computational study demonstrates the efficacy of the proposed algorithm.

A New Formulation for the Dial-a-Ride Problem

2021 ◽  
Author(s):  
Yannik Rist ◽  
Michael A. Forbes
Keyword(s):  
Time Window ◽  
Valid Inequalities ◽  
Route Planning ◽  
Branch And Cut ◽  
Mixed Integer ◽  
Planning Problem ◽  
Current State ◽  
New Formulation ◽  
First Time ◽  
Delivery Location

This paper proposes a new mixed integer programming formulation and branch and cut (BC) algorithm to solve the dial-a-ride problem (DARP). The DARP is a route-planning problem where several vehicles must serve a set of customers, each of which has a pickup and delivery location, and includes time window and ride time constraints. We develop “restricted fragments,” which are select segments of routes that can represent any DARP route. We show how to enumerate these restricted fragments and prove results on domination between them. The formulation we propose is solved with a BC algorithm, which includes new valid inequalities specific to our restricted fragment formulation. The algorithm is benchmarked on existing and new instances, solving nine existing instances to optimality for the first time. In comparison with current state-of-the-art methods, run times are reduced between one and two orders of magnitude on large instances.

scholarly journals Polyhedral Results and Branch-and-Cut for the Resource Loading Problem

2020 ◽  
Author(s):  
Guopeng Song ◽  
Tamás Kis ◽  
Roel Leus
Keyword(s):  
Resource Use ◽  
Capacity Planning ◽  
Mixed Integer Linear Programming ◽  
Branch And Cut ◽  
Mixed Integer ◽  
The Novel ◽  
Due Dates ◽  
Capacity Limits ◽  
Resource Loading ◽  
Loading Problem

We study the resource loading problem, which arises in tactical capacity planning. In this problem, one has to plan the intensity of execution of a set of orders to minimize a cost function that penalizes the resource use above given capacity limits and the completion of the orders after their due dates. Our main contributions include a novel mixed-integer linear-programming (MIP)‐based formulation, the investigation of the polyhedra associated with the feasible intensity assignments of individual orders, and a comparison of our branch-and-cut algorithm based on the novel formulation and the related polyhedral results with other MIP formulations. The computational results demonstrate the superiority of our approach. In our formulation and in one of the proofs, we use fundamental results of Egon Balas on disjunctive programming.

Branch and Price for Chance-Constrained Bin Packing

2020 ◽  
Vol 32 (3) ◽  
pp. 547-564
Author(s):  
Zheng Zhang ◽  
Brian T. Denton ◽  
Xiaolan Xie
Keyword(s):  
Bin Packing ◽  
Real Data ◽  
Integer Program ◽  
Chance Constraints ◽  
Branch And Cut ◽  
Mixed Integer ◽  
Closed Form Expression ◽  
Mixed Integer Program ◽  
Branch And Price ◽  
Item Size

This article describes two versions of the chance-constrained stochastic bin-packing (CCSBP) problem that consider item-to-bin allocation decisions in the context of chance constraints on the total item size within the bins. The first version is a stochastic CCSBP (SP-CCSBP) problem, which assumes that the distributions of item sizes are known. We present a two-stage stochastic mixed-integer program (SMIP) for this problem and a Dantzig–Wolfe formulation suited to a branch-and-price (B&P) algorithm. We further enhance the formulation using coefficient strengthening and reformulations based on probabilistic packs and covers. The second version is a distributionally robust CCSBP (DR-CCSBP) problem, which assumes that the distributions of item sizes are ambiguous. Based on a closed-form expression for the DR chance constraints, we approximate the DR-CCSBP problem as a mixed-integer program that has significantly fewer integer variables than the SMIP of the SP-CCSBP problem, and our proposed B&P algorithm can directly solve its Dantzig–Wolfe formulation. We also show that the approach for the DR-CCSBP problem, in addition to providing robust solutions, can obtain near-optimal solutions to the SP-CCSBP problem. We implement a series of numerical experiments based on real data in the context of surgery scheduling, and the results demonstrate that our proposed B&P algorithm is computationally more efficient than a standard branch-and-cut algorithm, and it significantly improves upon the performance of a well-known bin-packing heuristic.

Mixed-integer bilinear programming problems

1993 ◽  
Vol 59 (1-3) ◽  
pp. 279-305 ◽  
Author(s):  
Warren P. Adams ◽  
Hanif D. Sherali
Keyword(s):  
Bilinear Programming ◽  
Mixed Integer

scholarly journals A stochastic MPEC approach for grid tariff design with demand-side flexibility

2020 ◽  
Author(s):  
Magnus Askeland ◽  
Thorsten Burandt ◽  
Steven A. Gabriel
Keyword(s):  
Decision Making ◽  
Power System ◽  
Distribution System ◽  
End Users ◽  
Branch And Cut ◽  
Mixed Integer ◽  
Electricity Grid ◽  
Decentralized Decision Making ◽  
System Operator ◽  
Aggregate Network

Abstract As the end-users increasingly can provide flexibility to the power system, it is important to consider how this flexibility can be activated as a resource for the grid. Electricity network tariffs is one option that can be used to activate this flexibility. Therefore, by designing efficient grid tariffs, it might be possible to reduce the total costs in the power system by incentivizing a change in consumption patterns. This paper provides a methodology for optimal grid tariff design under decentralized decision-making and uncertainty in demand, power prices, and renewable generation. A bilevel model is formulated to adequately describe the interaction between the end-users and a distribution system operator. In addition, a centralized decision-making model is provided for benchmarking purposes. The bilevel model is reformulated as a mixed-integer linear problem solvable by branch-and-cut techniques. Results based on both deterministic and stochastic settings are presented and discussed. The findings suggest how electricity grid tariffs should be designed to provide an efficient price signal for reducing aggregate network peaks.

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