Essentials of Linear Programming for Managers

2014 ◽  
pp. 96-127
Author(s):  
Hossein Arsham ◽  
M. Bardossy ◽  
D. K. Sharma
Keyword(s):  
Linear Programming ◽  
Standard Form ◽  
Optimal Solution ◽  
Solution Algorithm ◽  
Managerial Decision ◽  
Software Packages ◽  
Time Sensitivity ◽  
Decision Variables ◽  
New Variables ◽  
Critical Overview

This chapter provides a critical overview of Linear Programming (LP) from a manager's perspective. The main objective is to provide managers with the essentials of LP as well as cautionary notes and defenses on common modeling issues and software limitations. The authors illustrate the findings by solving a simple LP directly on the original decision variables and constraints space without adding new variables or translating the model to fit a specific solution algorithm. The aims are the unification of diverse set of topics in their natural states in a manner that are easy to understand and providing useful information to the managers. The advances in computing software have brought LP tools to the desktop for a variety of applications to support managerial decision-making. However, it is already recognized that current LP tools, in ample circumstances, do not answer the managerial questions satisfactorily. For instance, there is a costly difference between the mathematical and managerial interpretations of sensitivity analysis. LP software packages provide one-change-at-a-time sensitivity results; the authors develop the largest sensitivity region, which allows for simultaneous dependent and/or independent changes, based on the optimal solution. The procedures are illustrated by numerical examples including LP in standard-form and LP in non standard-form.

Foundation of Linear Programming

2012 ◽  
Vol 3 (3) ◽  
pp. 40-60 ◽  
Author(s):  
Hossein Arsham
Keyword(s):  
Decision Making ◽  
Sensitivity Analysis ◽  
Linear Programming ◽  
Optimal Solution ◽  
Decision Makers ◽  
Shadow Prices ◽  
Managerial Decision ◽  
Managerial Decision Making ◽  
Software Packages ◽  
Decision Variables

The aim and scope of this paper are the infusion of purposeful action by decision makers with an explicit understanding of analytical linear programming (LP) tools in order that the managers will implement strategic results correctly. The advances in computing software have brought LP tools to the desktop for a variety of applications to support managerial decision-making. However, it is already recognized that current LP tools do not answer the managerial questions satisfactorily. For instance, there is a costly difference between the mathematical and managerial interpretations of sensitivity analysis. The LP software packages provide sensitivity results about the optimality of a basis and not about the optimality of the values of the decision variables, and the shadow prices that are of interest to the manager. Society has developed the largest sensitivity region based on “optimal solution,” not that which “preserves the basis” that allows for simultaneous dependent/independent changes. The aim and results of this article are found further in the work.

A One-Layer Recurrent Neural Network with a Discontinuous Activation Function for Linear Programming

2008 ◽  
Vol 20 (5) ◽  
pp. 1366-1383 ◽  
Author(s):  
Qingshan Liu ◽  
Jun Wang
Keyword(s):  
Neural Network ◽  
Linear Programming ◽  
Recurrent Neural Network ◽  
Optimal Solution ◽  
Activation Function ◽  
High Gain ◽  
Discontinuous Activation ◽  
The Neural Network ◽  
Decision Variables ◽  
Discontinuous Activation Function

A one-layer recurrent neural network with a discontinuous activation function is proposed for linear programming. The number of neurons in the neural network is equal to that of decision variables in the linear programming problem. It is proven that the neural network with a sufficiently high gain is globally convergent to the optimal solution. Its application to linear assignment is discussed to demonstrate the utility of the neural network. Several simulation examples are given to show the effectiveness and characteristics of the neural network.

scholarly journals Integer linear programming application in production results optimization using cutting plane method

2021 ◽  
Vol 4 (1) ◽  
pp. 57-66
Author(s):  
Fery Firmansah ◽  
Fitriana Wulandari
Keyword(s):  
Linear Programming ◽  
Integer Linear Programming ◽  
Raw Materials ◽  
Optimal Solution ◽  
Cutting Plane ◽  
Cutting Plane Method ◽  
Daily Production ◽  
Plane Method ◽  
Decision Variables ◽  
The Right

Integer Linear Programming is a special form of linear programming which the decision variables are in integer form. Berkah Rasa is a home industry business in the form of Jenang Ayu and Jenang Krasikan processed food.  The daily production that carried out by Berkah Rasa is based on the availability of raw materials and the number of requests. So far, Berkah Rasa has not had the right strategy in producing Jenang to get maximum profit. The purpose of this research is to apply integer linear programming to the optimization of Jenang Ayu and Jenang Krasikan production. The method used to solve this problem is the cutting plane method. The results of the research obtained is the optimal solution for Berkah Rasa, that is by producing 25 kg of Jenang Ayu and 22 kg of Jenang Krasikan every day. So that the benefits obtained by Berkah Rasa every day are IDR 727,000.00.

scholarly journals Solving Linear Programming Problems with Grey Data

2018 ◽  
Vol 3 (1) ◽  
pp. 17-23 ◽  
Author(s):  
Michael Gr. Voskoglou
Keyword(s):  
Linear Programming ◽  
Objective Function ◽  
Programming Problem ◽  
Standard Method ◽  
Linear Programming Problem ◽  
Real Life ◽  
Optimal Solution ◽  
Real Numbers ◽  
Decision Variables ◽  
Linear Programming Problems

A Grey Linear Programming problem differs from an ordinary one to the fact that the coefficients of its objective function and / or the technological coefficients and constants of its constraints are grey instead of real numbers. In this work a new method is developed for solving such kind of problems by the whitenization of the grey numbers involved and the solution of the obtained in this way ordinary Linear Programming problem with a standard method. The values of the decision variables in the optimal solution may then be converted to grey numbers to facilitate a vague expression of it, but this must be strictly checked to avoid non creditable such expressions. Examples are also presented to illustrate the applicability of our method in real life applications.

Technology Tips: Linear Programming with More Than Two Decision Variables

2014 ◽  
Vol 108 (1) ◽  
pp. 64-69
Author(s):  
Thomas G. Edwards ◽  
Kenneth R. Chelst
Keyword(s):  
Linear Programming ◽  
Real World ◽  
Mathematics Teacher ◽  
Corner Point ◽  
Optimal Solution ◽  
Linear Inequalities ◽  
Decision Variables ◽  
Linear Programming Problems ◽  
Systems Of Linear Inequalities

In a 1999 article in Mathematics Teacher, we demonstrated how graphing systems of linear inequalities could be motivated using real-world linear programming problems (Edwards and Chelst 1999). At that time, the graphs were drawn by hand, and the corner-point principle was applied to find the optimal solution. However, that approach limits the number of decision variables to two, and problems with only two decision variables are often transparent and inauthentic.

scholarly journals Solution of Fully Bipolar Fuzzy Linear Programming Models

2021 ◽  
Vol 2021 ◽  
pp. 1-31
Author(s):  
Muhammad Athar Mehmood ◽  
Muhammad Akram ◽  
Majed G. Alharbi ◽  
Shahida Bashir
Keyword(s):  
Linear Programming ◽  
Programming Problem ◽  
Linear Programming Problem ◽  
Fuzzy Numbers ◽  
Optimal Solution ◽  
Equality Constraints ◽  
Fuzzy Linear Programming ◽  
Mathematical Tool ◽  
Decision Variables ◽  
Crisp Linear Programming

The Yin-Yang bipolar fuzzy set is a powerful mathematical tool for depicting fuzziness and vagueness. We first extend the concept of crisp linear programming problem in a bipolar fuzzy environment based on bipolar fuzzy numbers. We first define arithmetic operations of unrestricted bipolar fuzzy numbers and multiplication of an unrestricted trapezoidal bipolar fuzzy number (TrBFN) with non-negative TrBFN. We then propose a method for solving fully bipolar fuzzy linear programming problems (FBFLPPs) with equality constraints in which the coefficients are unrestricted triangular bipolar fuzzy numbers and decision variables are nonnegative triangular bipolar fuzzy numbers. Furthermore, we present a method for solving FBFLPPs with equality constraints in which the coefficients and decision variables are unrestricted TrBFNs. The FBFLPP is transformed into a crisp linear programming problem, and then, it is solved to achieve the exact bipolar fuzzy optimal solution. We illustrate the proposed methodologies with several numerical examples.

Decision Making Approach to Fuzzy Linear Programming (FLP) Problems with Post Optimal Analysis

2015 ◽  
Vol 6 (4) ◽  
pp. 75-90 ◽  
Author(s):  
Monalisha Pattnaik
Keyword(s):  
Linear Programming ◽  
Fuzzy Numbers ◽  
Optimal Solution ◽  
Fuzzy Linear Programming ◽  
Fuzzy Decision ◽  
Optimal Analysis ◽  
Fuzzy Environment ◽  
Multi Objective Programming ◽  
Decision Variables ◽  
Ranking Technique

This paper finds solutions to the fuzzy linear program where some parameters are fuzzy numbers. In practice, there are many problems in which all decision parameters are fuzzy numbers, and such problems are usually solved by either probabilistic programming or multi objective programming methods. Unfortunately all these methods have shortcomings. In this paper, using the concept of comparison of fuzzy numbers, the author introduces a very effective method for solving these problems. This paper extends linear programming based problem in fuzzy environment. With the problem assumptions, the optimal solution can still be theoretically solved using the simplex based method. To handle the fuzzy decision variables can be initially generated and then solved and improved sequentially using the fuzzy decision approach by introducing robust ranking technique. The model is illustrated with an application and a post optimal analysis approach is obtained. The proposed procedure was programmed with MATLAB (R2009a) version software, the four dimensional slice diagram is represented to the application. Finally, numerical example is presented to illustrate the effectiveness of the theoretical results, and to gain additional managerial insights.

A Study of Fully Fuzzy Linear Fractional Programming Problems by Signed Distance Ranking Technique

2018 ◽  
pp. 73-115 ◽  
Author(s):  
Moumita Deb
Keyword(s):  
Linear Programming ◽  
Fractional Programming ◽  
Optimal Solution ◽  
Fuzzy Linear Programming ◽  
Linear Fractional Programming ◽  
Signed Distance ◽  
Decision Variables ◽  
Ranking Technique ◽  
Crisp Linear Programming ◽  
Fuzzy Linear Fractional Programming

The aim of this chapter is to study fully fuzzy linear fractional programming (FFLFP) problems where all coefficients of the decision variables and parameters are characterized by triangular fuzzy numbers. To deal with this, the authors have first to transform FFLFP problems to fuzzy linear programming (FLP) problems by using Charnes and Cooper method and then use signed distance ranking to convert fuzzy linear programming (FLP) problems to crisp linear programming (LP) problems. The proposed method is solved by using the simplex method to find the optimal solution of the problem. The authors have studied sensitivity analysis to determine changes in the optimal solution of the fully fuzzy linear fractional programming (FFLFP) problems resulting from changes in the parameters. To demonstrate the proposed method, one numerical example is solved.

scholarly journals Solving fully fuzzy linear programming problems by controlling the variation range of variables

2021 ◽  
Vol 103 (3) ◽  
pp. 13-24
Author(s):  
S.M. Davoodi ◽  
◽  
N.A. Abdul Rahman ◽  
Keyword(s):  
Linear Programming ◽  
Optimal Solution ◽  
Fuzzy Linear Programming ◽  
Ranking Functions ◽  
Fuzzy Variables ◽  
Decision Variables ◽  
Proposed Model ◽  
Linear Programming Problems ◽  
Left And Right ◽  
The Right

This paper deals with a fully fuzzy linear programming problem (FFLP) in which the coefficients of decision variables, the right-hand coefficients and variables are characterized by fuzzy numbers. A method of obtaining optimal fuzzy solutions is proposed by controlling the left and right sides of the fuzzy variables according to the fuzzy parameters. By using fuzzy controlled solutions, we avoid unexpected answers. Finally, two numerical examples are solved to demonstrate how the proposed model can provide a better optimal solution than that of other methods using several ranking functions.

scholarly journals A Fuzzy Optimization Model for the Berth Allocation Problem and Quay Crane Allocation Problem (BAP + QCAP) with n Quays

2021 ◽  
Vol 9 (2) ◽  
pp. 152
Author(s):  
Edwar Lujan ◽  
Edmundo Vergara ◽  
Jose Rodriguez-Melquiades ◽  
Miguel Jiménez-Carrión ◽  
Carlos Sabino-Escobar ◽  
...  
Keyword(s):  
Optimization Model ◽  
Fuzzy Optimization ◽  
Optimal Solution ◽  
Allocation Problem ◽  
Berth Allocation ◽  
Triangular Fuzzy Numbers ◽  
Berth Allocation Problem ◽  
Arrival Times ◽  
Decision Variables ◽  
Short Time

This work introduces a fuzzy optimization model, which solves in an integrated way the berth allocation problem (BAP) and the quay crane allocation problem (QCAP). The problem is solved for multiple quays, considering vessels’ imprecise arrival times. The model optimizes the use of the quays. The BAP + QCAP, is a NP-hard (Non-deterministic polynomial-time hardness) combinatorial optimization problem, where the decision to assign available quays for each vessel adds more complexity. The imprecise vessel arrival times and the decision variables—berth and departure times—are represented by triangular fuzzy numbers. The model obtains a robust berthing plan that supports early and late arrivals and also assigns cranes to each berth vessel. The model was implemented in the CPLEX solver (IBM ILOG CPLEX Optimization Studio); obtaining in a short time an optimal solution for very small instances. For medium instances, an undefined behavior was found, where a solution (optimal or not) may be found. For large instances, no solutions were found during the assigned processing time (60 min). Although the model was applied for n = 2 quays, it can be adapted to “n” quays. For medium and large instances, the model must be solved with metaheuristics.

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