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Transcript of 2 Modeling Graphical-solution
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IES 210804:LINEAR PROGRAMMING
Modeling and Graphical solution
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Mathematics in Operation
Mathematical Solution Method (Algorithm)
Real Practical Problem
Mathematical (Optimization) Problemx2
Computer Algorithm
Human Decision-Maker
Decision Support Software System
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General Optimization Model
Problem(1):
Min f(x)
s.t. g(x)0 --------(1)
x 0
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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!
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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
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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
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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.
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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.
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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.
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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
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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.
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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
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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
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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
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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.
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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
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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
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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
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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?
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Complete LP Model
Max z =
s.t21
53 xx
0,0
1823
122
4
21
21
2
1
xx
xx
x
x
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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)
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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
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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)
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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
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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.
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The Answers
Produce 2 of product 1 and 6 of product 2
The maximum profit is $36,000
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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
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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
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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
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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
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No Optimal Solution(2)
Unbounded: the constraints do not
prevent improving the value of the
objective function indefinitely in the
favorable direction ( positively or
negatively ),
i.e. + , or -
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No Optimal Solution(2)
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(