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Linear Programming - Operations Research

——   Created by Madhusudan Sohani

Graphical and Simplex Methods

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This course is intended to provide you with an opportunity to learn the manual process of solving simple linear programming problems, involving all types of constraints, and including unrestricted variables. It also introduces you to solve the problems by formulating the same in MS Excel, using the solver tool. To this extent, a formulated question in linear programming has been included, for solving, using the excel solver tool.

Starting with the first lecture, you will be exposed to the popular method of solving a linear programming problem (LPP as hereafter called) involving all constraints of the type ‘less than or equal to’; which is the easiest type of constraint to deal with. The questions involving other type of constraints namely ‘greater than or equal to’ or ‘equal to’ are more difficult to manually solve. They will be

gradually introduced from the second lecture onwards. Right in the second lecture, you will get used to having ‘greater than or equal to’ constraint. In fact, the second video deals with a problem having two constraints and two variables, both the constraints being ‘greater than or equal to’. In the third video, mixed type of constraints are covered, which requires to be treated differently for each constraints. The fourth video will cover a problem which has constraints of all types, <=, >= and =. Additional twist in the problem is, it has a multiple solution. You will learn how to detect a multiple solution. In lecture 05, you will see a direct application of LPP to a business problem. On simplification, it will be possible to solve the problem, which has been solved in the lecture.(It has got a slight twist in the end though).

Lecture 6 will be quite a surprise for the unsuspecting student, since you will discover the utility of artificial variables in its true sense. You will learn that at times none of the slack, or surplus or artificial variables are needed to solve the problem. The seventh lecture will confirm the use of artificial variables, through two problems. The lecture at serial number 8 will demonstrate to you how a problem containing 72 numbers in the table could be reduced to a problem containing only 18 numbers in the table. Of course, this can be done, only if minimum values of variables are given.

Degeneracy is a different class of problems. After this video lecture at serial number 9, youvwill be able to deal with a degenerate problem. Also, in this lecture is included a problemvhaving unrestricted variable. This type of a problem cannot be solved by the conventional simplex method since the simplex algorithm is based on the assumption that all variables are non-negative. For the first time you will deal with a variable which can take both, positive and negative values.

Lecture number 10 deals with a unique method of two phases. Here in this method the problem is solved in two parts. The first part or phase one, as it is called, will revolve around removing the artificial variables from the problem, by allotting them a simple contribution like + or–1. After artificial variables are removed from the basis, the phase two starts, where in the decision variables are re-allotted their original contribution, the solution checked for optimality and improved if necessary, till it is optimal. This is a very interesting process, and actually offers an improvement over the conventional ‘big M’ method.

Lecture number 11 deals with a very important aspect of LPP. The situations in business are not static, in the sense that if you obtain the solution to a problem it cannot be taken as sacrosanct, and you must offer the possibility of changes in the structure of the problem. It is very interesting to see how the optimal tableau offers answers to small changes in the structure of the problem. After viewing this video, you will be able to appreciate the importance of carrying out changes in the right-hand side of the constraints, even consider changes in the contribution of products. You can examine the introduction of new products within the given situation of existing products.

Lecture number 12 is actually taken from one my other courses available elsewhere. It will strengthen the concepts of carrying out post-optimality analysis or sensitivity analysis of a given LPP. Lecture 13 deals with viewing any primal problem through it’s dual. It’s actually like seeing the two sides of a coin. If ‘head’ is maximisation, the ‘tail’ is minimisation and vice-versa. You will be taken through simple steps of writing the dual of any problem from the given primal. The next lecture i.e., 14 will teach you the procedure of writing the dual if the primal is not in the standard form or has ‘=’ constraints. This lecture will delve on the uses of dual in practical situation when it’s more convenient to write and solve the dual of a problem than solving the original problem.

Lecture number 15 is an important lecture in the sense that once you know how to solve the LPP manually, you can get into tools to solve them automatically. To this effect, this lecture deals with solving a problem through excel solver tool It’s shown by taking a problem from start to finish through MS Excel Solver. Right in the beginning explanation has been given as to installation of Solver tool. Once you master this tool. It’s very easy to solve an LPP of any size. For many of you, this may be the most important lecture in the whole course. The lecture number 16 summarises how to deal with a simple production planning problem. You need to decide how much production you should do with normal time or with over time being paid to the workers. Lecture number 17 showcases how to detect an infeasible problem of LPP, and its characteristics.

Lecture 18 will showcase use of some android applications to solve the LPP. Actually, twoapplications have been discussed right from the stage of downloading the same and using them. A demonstration has been given in this lecture by actually solving a problem using these two applications. In fact there are several applications floating on Google Play Store, and it’s difficult to make a choice. An attempt has been made to provide you the guidance for choosing the correct app. As far as I am aware, there are no apps based on the iOS platform for solving an LPP. (This statement is valid as on 15 the March 2022).

The next in the lecture is a demonstration of using live web-pages for finding the solution to LPPs. A problem each has been solved using these web-pages. In the last lecture, a problem has been solved to showcase what is known as blending problems. Particularly useful in petroleum industry.

I hope that you will get the satisfaction of learning something new, and quite thoroughly. Enjoy the course and give your feedback to me by writing a review of my course. Thanks a lot for registering for this course. Best wishes

19 Lessons

Introductory Section
85 mins
free preview
48 mins
Constraints like >=
60 mins
Mixed Type of Constraints
37 mins
Mixed Constraints, Multiple Solutions
27 mins
Mixed Constraints Again, a Good Practice Problem
13 mins
All Constraints of ‘=’ Type. No Slack, Surplus or Artificial Variables!
free preview
48 mins
Constraints Like >=, but Great Simplification
29 mins
The Lecture Will Introduce the Student to the Concept of Simplifying the Problem
45 mins
Degeneracy Explained With an Example, Problem of ‘Unrestricted’ Variable Solved
34 mins
Two Phase Method
61 mins
Sensitivity Analysis Explained With an Example
42 mins
Sensitivity Analysis Explained With an Example
15 mins
Duality in Linear Programming
View more lessons +

About the instructor

Madhusudan  Sohani

Having completed a Master's degree in Physics, and an MBA in Financial Management, I worked in the industry from 1977 to 2002. Simultaneously to working, …

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