2 Modeling Graphical-solution

8/3/2019 2 Modeling Graphical-solution

1/32

IES 210804:LINEAR PROGRAMMING

Modeling and Graphical solution


2/32

Mathematics in Operation

Mathematical Solution Method (Algorithm)

Real Practical Problem

Mathematical (Optimization) Problemx2

Computer Algorithm

Human Decision-Maker

Decision Support Software System


3/32

General Optimization Model

Problem(1):

Min f(x)

s.t. g(x)0 --------(1)

x 0


4/32

Example: The Burroughs garment company manufactures men's

shirts and womens blouses for Walmark Discount stores.

Walmark will accept all the production supplied by Burroughs.

The production process includes cutting, sewing and packaging.Burroughs 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:

Garment Cutting Sewing Packaging Unitprofit($)

Shirts 20 70 12 8.00

Blouses 60 60 4 12.00

Determine the optimal weekly production schedule for

Burroughs!


5/32

Solution: Define decision variables

x1: shirts produce per week andx2 : blouses produce per week.

8x1 + 12x2Time spent on cutting =

Profit got =

Time spent on sewing = 70x1

+ 60x2

mts

Time spent on packaging =12x1 + 4x2 mts

20x1 + 60x2mts


6/32

The objective is to findx1,x2 so as to

Max z = 8x1 + 12x2

st:

20x1 + 60x2 25 40 60

70x1 + 60x2 35 40 60

12x1 + 4x2 5 40 60

x1,x2 0, integers


7/32

This is a typical optimization problem.

Any values ofx1, x2 that satisfy all

the constraints of the model is called

a feasible solution. We areinterested in finding the optimum

feasible solution that gives the

maximum profit while satisfying allthe constraints.


8/32

Wild West produces two types of cowboy hats.

Type I hat requires twice as much labor as a

Type II. If all the available labor time is

dedicated to Type II alone, the company can

produce a total of 400 Type II hats a day. Therespective market limits for the two types of

hats are 150 and 200 hats per day. The profit is

$8 per Type I hat and $5 per Type II hat.

Formulate the problem as an LPP so as to

maximize the profit.


9/32

Solution: Decision Variablesx1 produces Type

I hats andx2 Type II hats per day.

8x1 + 5x2

Labour Time spent is (2x1 + x2) c minutes

Per day Profit got =

Assume the time spent in producing onetype II hat is c minutes.


10/32

The objective is to findx1,x2 so as to

Max z = 8x1 + 5x2

st:

(2x1 +x2 ) c 400 c

x1 150

x2 200

x1,x2 0, integers


11/32

Trim Loss problem: A company has to

manufacture the circular tops of cans. Two

sizes, one of diameter 10 cm and the other

of diameter 20 cm are required. They are to

be cut from metal sheets of dimensions 20

cm by 50 cm. The requirement of smaller

size is 20,000 and of larger size is 15,000.

The problem is : how to cut the tops from

the metal sheets so that the number of

sheets used is a minimum. Formulate the

problem as a LPP.


12/32

A sheet can be cut into one of the following

three patterns:

Pattern I

Pattern II

Pattern III

10

20

20

10

10

10

20

10


13/32

Pattern I: cut into 10 pieces of size 10 by 10

so as to make 10 tops of size 1

Pattern II: cut into 2 pieces of size 20 by 20

and 2 pieces of size 10 by 10 so as to make

2 tops of size 2and 2 tops of size 1

Pattern III: cut into 1 piece of size 20 by 20

and 6 pieces of size 10 by 10 so as to make1 top of size 2 and 6 tops of size 1


14/32

Define:x1 sheets are cut according to pattern

I,x2

according to pattern II,x3

according to

pattern III

The problem is to

Minimizez =x1 +x2 +x3

Subject to 10x1 + 2x2 + 6x3 20,000

2x2 + x3 15,000

x1,x2,x3 0, integers


15/32

A Post Office requires different number of

full-time employees on different days of the

week. The number of employees required oneach day is given in the table below. Union

rules say that each full-time employee must

receive two days off after working for fiveconsecutive days. The Post Office wants to

meet its requirements using only full-time

employees. Formulate the above problem asa LPP so as to minimize the number of full-

time employees hired.


16/32

Requirements of full-time employees

day-wise

Day No. of full-timeemployees required

1 - Monday 10

2 - Tuesday 6

3 - Wednesday 8

4 - Thursday 125 - Friday 7

6 - Saturday 9

7 - Sunday 4


17/32

Solution: Letxibe the number of full-time

employees employed at the beginning of day

i (i = 1, 2, , 7). Thus our problem is to findxi so as to


18/32

Minimize 1 2 3 4 5 6 7z x x x x x x x

Subject to1 4 5 6 7

10 (Mon)x x x x x

1 2 5 6 76 (Tue)x x x x x

1 2 3 6 78 (Wed)x x x x x

1 2 3 4 712 (Thu)x x x x x

1 2 3 4 57 (Fri)x x x x x

2 3 4 5 69 (Sat)x x x x x

3 4 5 6 7 4 (Sun)x x x x x

xi 0.

integers


19/32

A company with three plants produces twoproducts. First plant that operates for 4 hours

produces only product 1. Second plantoperates for 12 hours but produces onlyproduct 2. The last plant operates for 18 hoursand produces both products. It takes one hour

to produce product 1 at plant 1 and 3 hours atplant 3 while product 2 needs 2 hour to beproduced at available facilities. If the selling

price for product 1 and 2 is $3,000 and$5,000, respectively. Find how many product 1and product 2 should be made to maximize theprofit? How much the profit?


20/32

Complete LP Model

Max z =

s.t21

53 xx

0,0

1823

122

4

21

21

2

1

xx

xx

x

x


21/32

Solution: Graphical Method

Corner point feasible solution (C.P.F.) solution:

(0,0),(0,6),(2,6),(4,3),(4,0)

.

3 2

,

1 2

1

2

1 2

1 2

Max z 3x 5x

s.t x 4

2x 12

x x 18

x x 0

feasibleregionconstraint

boundary

(0,6)

(0,9) (2,6)

(4,0)

(4,3)

(6,0)


22/32

Definitions

For any L.P problem with n decisionvariables to C.P.F. solution are adjacentto each other if they share n-1 constraint

boundaries. The two adjacent C.P.F. solutions are

connected by a line segment that lies on

these same shared constraintboundaries.

Such a line segment is referred to as

edgeof the feasible region


23/32

Example (Continued)

CPF

solution

Adjacent CPF

solutions

(0,0)

(0,6)

(2,6)

(4,3)(4,0)

(0,6) and (4,0)

(2,6) and (0,0)

(4,3) and (0,6)

(4,0) and (2,6)(0,0) and (4,3)

feasible

regionconstraint

boundary

(0,6)

(0,9) (2,6)

(4,0)

(4,3)

(6,0)


24/32

Example (Continued)

CPF

solution

Adjacent CPF

solutions

(0,0)

(0,6)

(2,6)

(4,3)(4,0)

(0,6) and (4,0)

(2,6) and (0,0)

(4,3) and (0,6)

(4,0) and (2,6)(0,0) and (4,3)Z=0 Z=12

Z=27

Z=36Z=30


25/32

Optimality Test

Optimality testConsider any L.P problemthat possesses at least one optimalsolution. If a C.P.F. solution has no

adjacent C.P.F. solution that are better,then it must be an optimal solution.

( 2, 6 ) is the optimal solution of the

Example and the optimal value z=36.


26/32

The Answers

Produce 2 of product 1 and 6 of product 2

The maximum profit is $36,000


27/32

Some Definitions

Optimal solution: An optimal solution isa feasible solution that has the most valueof the objective function. ( the largest

value or smallest value )

At least one optimal solution .

Multiple optimal solution .

No optimal solution


28/32

At Least One Optimal Solution

Max z =

s.t.

optimal solution is ( ) = (0,6),optimal value is z = 30

21 53 xx

0,0

1823122

4

21

21

2

1

xx

xx

x

x

21, xx


29/32

Multiple Optimal Solution

Max z =

s.t.

optimal solution is ( ) =(2,6),(4,3),..

optimal value is z = 36

21 46 xx

0,0

1823122

4

21

21

2

1

xx

xx

x

x

21, xx


30/32

No Optimal Solution(1)

Infeasible: no feasible solutionMax z =

s.t.

infeasible No optimal solution!

21 53 xx

0,0

1823

122

7

21

21

2

1

xx

xx

x

x

2x

1x


31/32


Unbounded: the constraints do not

prevent improving the value of the

objective function indefinitely in the

favorable direction ( positively or

negatively ),

i.e. + , or -


32/32


Max z =

s.t.

( ) = (4, ), z =

2153 xx

0,0

4

21

1

xx

x

21, xx

2x

1x

41 x

2153z xx

)0,4(

Documents

2 Modeling Graphical-solution