Download Linear Programming: A Comprehensive Guide to Problem Formulation and Graphical Solutions and more Study notes Linear Programming in PDF only on Docsity!
LINEAR PROGRAMING
LECTURE THREE
LECTURE OBJECTIVES
By the end of the lecture, the learner should be able to:
๏ท Determine the components of a linear programming problem. ๏ท Formulate a Linear Programming Problem.
๏ท Solve Linear Programming Problems with the graphical method.
INTRODUCTION
It is one of the techniques of optimization.
Optimization deals with maximizing or minimizing of some functions over some restrictions.
The word linear is used because all the functions are expressed in a linear form.
The word programming denotes the procedure used to solve the problem; therefore, a linear programming problem involves the optional allocation of scarce resource between alternative uses within an all linear framework.
COMPONENTS OF LINEAR PROGRAMMING PROBLEM
1) Objective function
This is a mathematical statement of the organizations objective. It is expressed in terms of decision variables. A decision variables is one in which the decision maker law some control
They are the variables that are to be manipulated in order to achieve the objectives. Also referred to instruments.
An objective function is generally stated as;
๐๐๐ฅ ๐๐ ๐๐๐ ๐(๐ฅ) = ๐ถ 1 ๐ 1 + ๐ถ 2 ๐ 2 + ๐ถ 3 ๐ 3 +... +๐ถ๐๐๐ Where ๐ถ๐= Parameters to be estimated
They measure the impact on the objective function resulting from in change in the decision variables
๐๐=Decision variables
2) Constraints;
These are the restrictions that characterize the environment of the objective. They are also expressed in terms of decision problems variables but expressed as linear inequalities where the โค is used for a maximizing objective and โฅ for a minimizing objective.
The general expression of the constraints is:
๐ 11 ๐ฅ 1 + ๐ 12 ๐ฅ 2 + โฏ + ๐1๐๐ฅ๐ โค ๐๐ โฅ ๐ 1
๐ 21 ๐ฅ 1 + ๐ 22 ๐ฅ 2 + โฏ + ๐2๐๐ฅ๐ โค ๐๐ โฅ ๐ 2 ๐ 31 ๐ฅ 1 + ๐ 32 ๐ฅ 2 + โฏ + ๐3๐๐ฅ๐ โค ๐๐ โฅ ๐ 3
Where
๐๐๐ - Represents the amount of resource is used to the production of a unit of ๐๐.
๐๐ - Represents the total availability of the resources.
For instance ๐๐๐ represents the amount of resources 1 used to produce a unit of amount of resource X 2.
3) Non-negativity constraint
This is a restriction placed on all decision variables that their values should be non-negative.
The general statement of a linear programming model therefore becomes;
Manpower Unit Profit Products A B C Tables 10 7 2 12 Chairs 2 3 4 3 Total availability of manpower(hrs)
Where. A= Roughing, B= Furnishing. C= Painting
NB: The number of tables and chairs are the variables within the companies control and they are the elements of profit.
Let T= tables produced and C = Chairs produced
The unit profits of the two products are 12 and 3 respectively and therefore, the companies profit may be expressed as
๐ = 12๐ + 3๐ถ.
This unit profits of the companyโs objective functions.
To produce in table and chair requires 10hrs and 2 hrs of roughing respectively.
The total availability of manpower for roughing is 100hrs. Therefore the total usage of manpower for the two products in this section MUST not exceed 100hrs. i.e
10๐ + 2๐ถ 100
Similarly the total manpower required for furnishing all the tables and chairs must not exceed 77 hrs and 80 hrs pointing time i.e.
7๐ + 3๐ถ 77
The overall model may be stated as
๐๐๐ฅ ๐ = 12๐ + 3๐ถ ๐ . ๐ก
10๐ + 2๐ถ 100
7๐ + 3๐ถ โค 77 2๐ + 4๐ถ โค 80 ๐, ๐ถ, > 0
Example 2
A firm is engaged in breeding pigs. The pigs are fed on variables products grown on the firm. Because of the need to ensure certain nutrient requirements, it is necessary to buy two additional products are summarized in the following table
Constraints of nutrient in products Nutrients A B Minimum amount of nutrients required 1 36 6 108 2 3 12 36 3 20 10 100
Additional info:
Vertexes= (0,0) (0,7) (2,6) (5,3) (6,0)
Trying them in the objectives function
(0,0) = 5(0) + 3(0) = 0
(^06 )
18
8 7
R
Therefore 5 units of X 1 and 3 units of X 2 are produced to maximize profit.
WEAKNESSES OF THE GRAPHICAL METHOD
- Useful only when we have two division variables.
- Very economic with information, gives minimal information on the decision variables.
