Linear Programming Problem. Introduction. Linear Programming was developed by George B Dantzing in 1947 for solving military logistic operations. Introduction. Meaning of Linear Programming
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Linear Programming Problem
– The word Linear refers to linear relationship among variables. i.e. a given change in one variable will always cause a resulting proportional change in another variable. For example, doubling the investment on a certain project will exactly double the rate of return.
of study. Thus, the profit per unit of the product, requirements of material and labour per unit, availability of material etc. are given and known in the problem. The LP is obviously deterministic in nature.
be equal to the sum of the profits earned separately from A & B. Similarly, the amount of a resource consumed by A & B must be equal to the sum of resources used for A & B individually.
production of a one unit of a product uses 5 hrs of a particular resource, then making 3 units of that product uses 3*5=15 hrs of that resource.
integer values. When it is necessary to have integer variables, the integer programming problem is considered to attain the desired values.
The term formulation referred to the process of converting the verbal description and numerical data into mathematical expressions which represents the relationship among relevant decision variable (factors), objective & restrictions on the use of resources.
The XYZ garment company manufactures men's shirts and women’s t-shirts for ABC Discount stores. ABC will accept all the production supplied by the company. The production process includes cutting, sewing and packaging. XYZ employs 25 workers in the cutting department, 35 in the sewing department and 5 in the packaging department. The factory works one 8-hour shift, 5 days a week.
The following table gives the time requirements and the profits per unit for the two garments:
Minutes per unit
Determine the optimal weekly production schedule for XYZ.
Assume that XYZ produces x1 shirts and x2 t-shirts per week.
Profit got =
8 x1 + 12 x2
Time spent on cutting =
20 x1 + 60 x2 mts
Time spent on sewing =
70 x1 + 60 x2 mts
Time spent on packaging =
12 x1 + 4 x2 mts
The objective is to find x1, x2 so as to
maximize the profit z = 8 x1 + 12 x2
satisfying the constraints:
20 x1 + 60 x2≤ 25 40 60
70 x1 + 60 x2 ≤ 35 40 60
12 x1 + 4 x2 ≤ 5 40 60
x1, x2≥ 0, integers
This is a typical optimization problem.
Any values of x1, x2 that satisfy all the constraints of the model is called a feasible solution. We are interested in finding the optimumfeasible solution that gives the maximum profit while satisfying all the constraints.
More generally, an optimization problem looks as follows:
Determine the decision variablesx1, x2, …, xn so as to optimize an objectivefunctionf (x1, x2, …, xn) satisfying the constraints
gi (x1, x2, …, xn) ≤ bi (i=1, 2, …, m).
An optimization problem is called a Linear Programming Problem (LPP) when the objective function and all the constraints are linear functions of the decision variables, x1, x2, …, xn. We also include the “non-negativity restrictions”, namely xj ≥ 0 for all j=1, 2, …, n. Thus a typical LPP is of the form:
Optimize (i.e. Maximize or Minimize)
z = c1 x1 + c2 x2+ …+ cn xn
subject to the constraints:
a11 x1 + a12 x2 + … + a1n xn ≤ b1
a21 x1 + a22 x2 + … + a2n xn ≤ b2
. . .
am1 x1 + am2 x2 + … + amn xn ≤ bm
x1, x2, …, xn 0
Application Areas of Linear Programming
The general LPP with n decision variables and m constraints can be stated as:
Find the values of decision variables…..