Linear Programming using graphical method.

1. Solving Optimization Problems
BY:
DR. ABHISHEK KUMAR SINGH
M. TECH., PHD
IIT BHU VARANASI
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2. 2
 Formulate linear programming models including an objective function and constraints
 Graphical solution of a LP problem
 Understanding of sensitivity analysis
Learning Objectives

3. 3
Introduction
 Developed by Leonid Kantorovich in 1939 at the time of WWII
 A Linear Programming model seeks to maximize or minimize a linear function,
subject to a set of linear constraints.
 The linear programming model consists:
 A set of decision variables.
 An objective function (maximize or minimize)
 A set of constraints (equality or inequality)

4. LPApplications
 There are well-known successful applications in:
 Manufacturing
 Marketing
 Finance (investment)
 Transport
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5. 5
The Galaxy Industries Production Problem
 Galaxy manufactures two toy:
1. TRUCK
2. CAR
 Resources are limited to
 1000 pounds of special plastic.
 40 labor hours per week.
 Technological input
 TRUCK requires 2 pounds of plastic and 3 minutes of labor per unit.
 CAR requires 1 pound of plastic and 4 minutes of labor per unit.
 Marketing requirement
 Total production cannot exceed 700 units of both the products.
 Number of units of TRUCK cannot exceed number of units of CAR by more than 350.

6. 6
The Galaxy Industries Production Problem
Truck
(per unit)
x1
Car
(per unit)
x2
Resource
Available
Plastic
(in pound )
2 1 1000
Production Time
(Per week)
3 minute 4 minute 2400 Minute
Total Production 1 1 700 units
Profit $ 8 $ 5
Only
TRUCK
500
800
700
Only
CAR
1000
600
700
4000 3000

7. 7
8(450) + 5(100)
The Galaxy Industries Production Problem
 The current production plan:
 Produce maximum possible TRUCK as the profit is higher.
 Use remaining resources to produce CAR
 The current production plan consists of:
TRUCK = 450 units
CAR = 100 units
Profit = $4100 per week

8. 8
Management is seeking a production
level that will increase the company’s
profit.

9. 9
Decisions variables:
 X1 = Number of TRUCKS (per week)
 X2 = Number of CARS (per week)
Objective Function:
 Maximize total weekly profit
Linear Programming Model

10. 10
Max 8X1 + 5X2 (Weekly profit)
subject to
2X1 + 1X2  1000 (Plastic)
3X1 + 4X2  2400 (Production Time)
X1 + X2  700 (Production capacity)
X1 - X2  350 (Mix)
X1> = 0, X2> = 0 (Non negativity)
Linear Programming Model

11. 11
The Graphical Analysis of
Linear Programming

12. 12
The non-negativity constraints
X1> = 0, X2> = 0
CARS (X2)
TRUCKS (X1)
Graphical Analysis – the Feasible Region

13. 13
1000
800
Feasible
CARS
(X2)
Infeasible
Production
Time
3X1+4X2  2400
Total production constraint:
X1+X2  700 (redundant)
600
700
The Plastic constraint
2X1+X2  1000
TRUCKS (X1 )
700
Graphical Analysis – the Feasible Region
500

14. 14
1000
500
Feasible
X2
Infeasible
Production
Time
3X1+4X2 2400
Total production constraint:
X1+X2 700 (redundant)
500
700
Production mix
constraint:
X1-X2  350
The Plastic constraint
2X1+X2 1000
X1
700
• There are three types of feasible points
Interior points. Boundary points. Corner points.
Graphical Analysis – the Feasible Region

15. 15
The search for an optimal solution
Start at some arbitrary profit, say profit = $2,000...
Then increase the profit, if possible...
...and continue until it becomes infeasible
Profit =$4360
600
700
1000
500
X2
X1
(320, 360)

16. (0, 600)
(320, 360)
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 If a linear programming problem has an optimal solution, it always lies at one
of the corner points.
Corner points and optimal solution
(450, 100)
Profit = 8X1 + 5X2
(0, 0)
(350, 0)

17. 17
Sensitivity Analysis

18. 18
Range of Optimality
 The optimal solution will remain same as long as
An objective function coefficient lies within its range of optimality
There are no changes in any other input parameters.
 The value of the objective function will change if the coefficient multiplies a
variable whose value is nonzero.
Sensitivity Analysis of
Objective Function Coefficients

19. 19
600
1000
400 600 800
X2
X1
Range of optimality: [3.75, 10]
Sensitivity Analysis of
Objective Function Coefficients
(320, 360)

20. 20
 We are interested in the following questions:
 Keeping all other factors the same, how much would the optimal value of the
objective function (for example, the profit) change if the right-hand side of a
constraint changed by one unit?
 For how many additional or fewer units will this per unit change be valid?
 Any change to the right hand side of a binding constraint will change the optimal
solution.
 Any change to the right-hand side of a non-binding constraint that is less than its slack
or surplus, will cause no change in the optimal solution.
Sensitivity Analysis of Right-Hand Side
Values

21. 21
Shadow Prices
 Assuming there are no other changes to the input parameters, the change to the objective
function value per unit increase to a right hand side of a constraint is called the “Shadow Price”

22. 22
1000
500
X2
X1
600
When more plastic becomes available (the plastic
constraint is relaxed), the right hand side of the plastic
constraint increases.
Production time
constraint
Maximum profit = $4360
Maximum profit = $4363.4
Shadow price = 4363.40 – 4360.00 = 3.40
Shadow Price – graphical demonstration
The Plastic
constraint
800

23. Thank You
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