scholarly journals A Heuristic for a Mixed Integer Program using the Characteristic Equation Approach

2017 ◽  
Vol 2 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Philimon Nyamugure ◽  
Elias Munapo ◽  
‘Maseka Lesaoana ◽  
Santosh Kumar
Keyword(s):  
Characteristic Equation ◽  
Polynomial Time ◽  
Feasible Solution ◽  
Integer Program ◽  
Mixed Integer ◽  
Mixed Integer Program ◽  
Np Hard ◽  
Equation Approach ◽  
Mip Model ◽  
Polynomial Time Algorithms

While most linear programming (LP) problems can be solved in polynomial time, pure and mixed integer problems are NP-hard and there are no known polynomial time algorithms to solve these problems. A characteristic equation (CE) was developed to solve a pure integer program (PIP). This paper presents a heuristic that generates a feasible solution along with the bounds for the NP-hard mixed integer program (MIP) model by solving the LP relaxation and the PIP, using the CE.

AN INTEGRATED DP-MIP MODEL FOR OPTIMAL CROP MIX SELECTION WITH DEFICIT IRRIGATION

2008 ◽  
Vol 25 (05) ◽  
pp. 625-647 ◽  
Author(s):  
M. N. AZAIEZ
Keyword(s):  
Crop Yield ◽  
Irrigation Water ◽  
Deficit Irrigation ◽  
Programming Model ◽  
Integer Program ◽  
Mixed Integer ◽  
Mixed Integer Program ◽  
Mip Model ◽  
One Year ◽  
The Right

We consider a region suffering from irrigation water scarcity. Candidate crops differ widely in their growth cycles, economic values and water consumption. We develop an integrated dynamic programming-mixed integer programming model to solve for optimal land exploitation over a one year horizon for multiple crops. The model applies deficit irrigation in order to increase the irrigated area at the expense of reducing crop yield per unit area. The dynamic program (DP) guarantees that deficit irrigation is only considered when it is economically efficient. Moreover, it provides optimal combinations of irrigation levels for each growth stage of candidate crops, accounting for the varying impact of water stress over time and the seasonal supply of irrigation water. The output of the DP serves as input to the mixed integer program (MIP). The MIP selects the most profitable crops in the right sequence to benefit the most from the crop-yield dependence on crop predecessor and allocates water and land optimally to maximize total profit. The objective function accounts for the attitude of the decision-maker toward risk by incorporating in its expression a risk-aversion coefficient.

A stochastic mixed integer program to model spatial wildfire behavior and suppression placement decisions with uncertain weather

2016 ◽  
Vol 46 (2) ◽  
pp. 234-248 ◽  
Author(s):  
Erin J. Belval ◽  
Yu Wei ◽  
Michael Bevers
Keyword(s):  
Wildland Fire ◽  
Fire Behavior ◽  
Integer Program ◽  
Mixed Integer ◽  
Mixed Integer Program ◽  
Weather Forecasts ◽  
Placement Decisions ◽  
Weather Events ◽  
Weather Changes ◽  
Nonanticipativity Constraints

Wildfire behavior is a complex and stochastic phenomenon that can present unique tactical management challenges. This paper investigates a multistage stochastic mixed integer program with full recourse to model spatially explicit fire behavior and to select suppression locations for a wildland fire. Simplified suppression decisions take the form of “suppression nodes”, which are placed on a raster landscape for multiple decision stages. Weather scenarios are used to represent a distribution of probable changes in fire behavior in response to random weather changes, modeled using probabilistic weather trees. Multistage suppression decisions and fire behavior respond to these weather events and to each other. Nonanticipativity constraints ensure that suppression decisions account for uncertainty in weather forecasts. Test cases for this model provide examples of fire behavior interacting with suppression to achieve a minimum expected area impacted by fire and suppression.

A Note on a Conceptual Framework for Dynamic Location—Allocation Analysis

1976 ◽  
Vol 8 (4) ◽  
pp. 443-446
Author(s):  
W G Truscott
Keyword(s):  
Conceptual Framework ◽  
Integer Program ◽  
Original Version ◽  
Mixed Integer ◽  
Mixed Integer Program ◽  
Location Allocation ◽  
Dynamic Location

This note examines a previously published model for dynamic location—allocation analysis. The usefulness of this model is enhanced by reformulating the problem as an operational zero-one, mixed-integer program while retaining the intent of the original version.

scholarly journals A GRASP Approach for Solving Large-Scale Electric Bus Scheduling Problems

Energies ◽  
2021 ◽  
Vol 14 (20) ◽  
pp. 6610
Author(s):  
Raka Jovanovic ◽  
Islam Safak Bayram ◽  
Sertac Bayhan ◽  
Stefan Voß
Keyword(s):  
Public Transport ◽  
High Speed ◽  
Large Scale ◽  
Integer Program ◽  
Mixed Integer ◽  
Transport Systems ◽  
Mixed Integer Program ◽  
Scheduling Problems ◽  
Electric Bus ◽  
Battery Capacity

Electrifying public bus transportation is a critical step in reaching net-zero goals. In this paper, the focus is on the problem of optimal scheduling of an electric bus (EB) fleet to cover a public transport timetable. The problem is modelled using a mixed integer program (MIP) in which the charging time of an EB is pertinent to the battery’s state-of-charge level. To be able to solve large problem instances corresponding to real-world applications of the model, a metaheuristic approach is investigated. To be more precise, a greedy randomized adaptive search procedure (GRASP) algorithm is developed and its performance is evaluated against optimal solutions acquired using the MIP. The GRASP algorithm is used for case studies on several public transport systems having various properties and sizes. The analysis focuses on the relation between EB ranges (battery capacity) and required charging rates (in kW) on the size of the fleet needed to cover a public transport timetable. The results of the conducted computational experiments indicate that an increase in infrastructure investment through high speed chargers can significantly decrease the size of the necessary fleets. The results also show that high speed chargers have a more significant impact than an increase in battery sizes of the EBs.

Multi-Objective Territory Design for Sales Managers of a Direct Sales Company

2019 ◽  
pp. 160-178 ◽  
Author(s):  
Elias Olivares-Benitez ◽  
Pilar Novo Ibarra ◽  
Samuel Nucamendi-Guillén ◽  
Omar G. Rojas
Keyword(s):  
Integer Program ◽  
National Average ◽  
Mixed Integer ◽  
Mixed Integer Program ◽  
Sales Managers ◽  
Territory Design ◽  
Direct Sales ◽  
Sales Company ◽  
A Company

This chapter presents a case study to organize the sales territories for a company with 11 sales managers to be assigned to 111 sales coverage units in Mexico. The assignment problem is modeled as a mathematical program with two objective functions. One objective minimizes the maximum distance traveled by the manager, and the other objective minimizes the variation of the sales growth goals with respect to the national average. To solve the bi-objective non-linear mixed-integer program, a weights method is selected. Some instances are solved using commercial software with long computational times. Also, a heuristic and a metaheuristic based on simulated annealing were developed. The design of the heuristic generates good solutions for the distance objective. The metaheuristic produces better results than the heuristic, with a better balance between the objectives. The heuristic and the metaheuristic are capable of providing good results with short computational times.

scholarly journals A game-theoretic optimisation approach to fair customer allocation in oligopolies

2020 ◽  
Vol 21 (4) ◽  
pp. 1459-1486
Author(s):  
Vassilis M. Charitopoulos ◽  
Vivek Dua ◽  
Jose M. Pinto ◽  
Lazaros G. Papageorgiou
Keyword(s):  
Integer Program ◽  
Linear Constraints ◽  
Nash Bargaining ◽  
Mixed Integer ◽  
Mixed Integer Program ◽  
Programming Approach ◽  
Theoretic Approach ◽  
Separable Programming ◽  
Process Industries ◽  
Game Theoretic

Abstract Under the ever-increasing capital intensive environment that contemporary process industries face, oligopolies begin to form in mature markets where a small number of companies regulate and serve the customer base. Strategic and operational decisions are highly dependent on the firms’ customer portfolio and conventional modelling approaches neglect the rational behaviour of the decision makers, with regards to the problem of customer allocation, by assuming either static competition or a leader-follower structure. In this article, we address the fair customer allocation within oligopolies by employing the Nash bargaining approach. The overall problem is formulated as mixed integer program with linear constraints and a nonlinear objective function which is further linearised following a separable programming approach. Case studies from the industrial liquid market highlight the importance and benefits of the proposed game theoretic approach.

scholarly journals Clearing Kidney Exchanges via Graph Neural Network Guided Tree Search (Student Abstract)

2020 ◽  
Vol 34 (10) ◽  
pp. 13989-13990
Author(s):  
Zeyu Zhao ◽  
John P. Dickerson
Keyword(s):  
Neural Network ◽  
Independent Set ◽  
Integer Program ◽  
Mixed Integer ◽  
Mixed Integer Program ◽  
Tree Search ◽  
Scalable Algorithms ◽  
Monte Carlo Tree Search ◽  
Kidney Exchange ◽  
Stage Renal Disease

Kidney exchange is an organized barter market that allows patients with end-stage renal disease to trade willing donors—and thus kidneys—with other patient-donor pairs. The central clearing problem is to find an arrangement of swaps that maximizes the number of transplants. It is known to be NP-hard in almost all cases. Most existing approaches have modeled this problem as a mixed integer program (MIP), using classical branch-and-price-based tree search techniques to optimize. In this paper, we frame the clearing problem as a Maximum Weighted Independent Set (MWIS) problem, and use a Graph Neural Network guided Monte Carlo Tree Search to find a solution. Our initial results show that this approach outperforms baseline (non-optimal but scalable) algorithms. We believe that a learning-based optimization algorithm can improve upon existing approaches to the kidney exchange clearing problem.

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.

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