LP Formulation LP Applications - PowerPoint Presentation

Slide1

LP Formulation

LP Applications

Product Mix

Diet / Blending

Scheduling

Transportation / Distribution

Assignment

Portfolio Selection (Quadratic)

The Quality Furniture Corporation produces benches and tables. The firm has two main resourcesResourceslabor and redwood for use in the furniture. During the next production period1200 labor hours are available under a union agreement.A stock of 5000 pounds of quality redwood is also available.

Product mix problem : Narrative representation

Consumption and profitEach bench that Quality Furniture produces requires 4 labor hours and 10 pounds of redwoodEach picnic table takes 7 labor hours and 35 pounds of redwood. Total available 1200, 5000Completed benches yield a profit of $9 each, and tables a profit of $20 each. Formulate the problem to maximize the total profit.

Product mix problem : Narrative representation

x1 = number of benches to producex2 = number of tables to produceMaximize Profit = ($9) x1 +($20) x2subject to Labor: 4 x1 + 7 x2  1200 hours Wood: 10 x1 + 35 x2  5000 pounds and x1  0, x2  0.We will now solve this LP model using the Excel Solver.

Product Mix : Formulation

Product Mix : Excel solution

Electro-Poly is a leading maker of slip-rings.A new order has just been received.

Model 1 Model 2 Model 3Number ordered 3,000 2,000 900Hours of wiring/unit 2 1.5 3Hours of harnessing/unit 1 2 1Cost to Make $50 $83 $130Cost to Buy $61 $97 $145

The company has 10,000 hours of wiring capacity and 5,000 hours of harnessing capacity.

Make / buy decision : Narrative representation

x1 = Number of model 1 slip rings to makex2 = Number of model 2 slip rings to make x3 = Number of model 3 slip rings to make y1 = Number of model 1 slip rings to buy y2 = Number of model 2 slip rings to buy y3 = Number of model 3 slip rings to buyThe Objective FunctionMinimize the total cost of filling the order.MIN: 50x1 + 83x2 + 130x3 + 61y1 + 97y2 + 145y3

Make / buy decision : decision variables

Demand Constraintsx1 + y1 = 3,000 } model 1x2 + y2 = 2,000 } model 2x3 + y3 = 900 } model 3Resource Constraints2x1 + 1.5x2 + 3x3 <= 10,000 } wiring1x1 + 2.0x2 + 1x3 <= 5,000 } harnessingNonnegativity Conditionsx1, x2, x3, y1, y2, y3 >= 0

Make / buy decision : Constraints

Make / buy decision : Excel

Do we really need 6 variables? x1 + y1 = 3,000 ===> y1 = 3,000 - x1 x2 + y2 = 2,000 ===> y2 = 2,000 - x2x3 + y3 = 900 ===> y3 = 900 - x3 The objective function was MIN: 50x1 + 83x2 + 130x3 + 61y1 + 97y2 + 145y3Just replace the valuesMIN: 50x1 + 83x2 + 130x3 + 61 (3,000 - x1 ) + 97 ( 2,000 - x2) + 145 (900 - x3 )MIN: 507500 - 11x1 -14x2 -15x3We can even forget 507500, and change the the O.F. into MIN - 11x1 -14x2 -15x3 or MAX + 11x1 +14x2 +15x3

Make / buy decision : Constraints

Resource Constraints2x1 + 1.5x2 + 3x3 <= 10,000 } wiring1x1 + 2.0x2 + 1x3 <= 5,000 } harnessingDemand Constraintsx1 <= 3,000 } model 1x2 <= 2,000 } model 2x3 <= 900 } model 3Nonnegativity Conditionsx1, x2, x3 >= 0

Make / buy decision : Constraints

MAX

+ 11x

1

+14x

2

+15x

3

MIN: 50x1 + 83x2 + 130x3 + 61y1 + 97y2 + 145y3Demand Constraintsx1 + y1 = 3,000 } model 1x2 + y2 = 2,000 } model 2x3 + y3 = 900 } model 3Resource Constraints2x1 + 1.5x2 + 3x3 <= 10,000 } wiring1x1 + 2.0x2 + 1x3 <= 5,000 } harnessingNonnegativity Conditionsx1, x2, x3, y1, y2, y3 >= 0

Make / buy decision : Constraints

y1 = 3,000- x1 y2 = 2,000-x2 y3 = 900-x3

MIN: 50x1 + 83x2 + 130x3 + 61(3,000- x1) + 97(2,000-x2) + 145(900-x3)

y1 = 3,000- x1>=0 y2 = 2,000-x2>=0 y3 = 900-x3>=0

x1 <= 3,000

x2 <= 2,000

x3 <= 900

Marketing : narrative

A department store want to maximize exposure.

There are 3 media; TV, Radio, Newspaper

each ad will have the following impact

Media Exposure (people / ad) Cost

TV 20000

15000

Radio 12000

6000

News paper 9000 4000

Additional information

1-Total budget is $100,000.

2-The maximum number of ads in T, R, and N are limited to

4

, 10, 7 ads respectively.

3-The total number of ads is limited to 15.

Marketing : formulation

Decision variables

x

1

= Number of ads in TV

x

2

= Number of ads in R

x

3

= Number of ads in N

Max

Z = 20

x

1

+

12x

2

+9x

3

15

x

1

+

6x

2

+ 4x

3

 100

x

1

 4

x

2

 10

x

3

 7

x

1

+

x

2

+ x

3

 15

x

1

,

x

2

, x

3



0

Problem ( From Hillier and Hillier)

Men, women, and children

gloves.

Material and labor requirements for each type and the corresponding profit are given below.

Glove Material (

sq

-feet) Labor (

hrs

) Profit

Men 2 .5 8

Women 1.5 .75 10

Children 1 .67 6

Total

available material is

5000

sq

-feet.

We

can have full time and part time workers.

Full time workers work

40

hrs

/w

and are paid

$13/

hr

Part time workers work

20

hrs

/w

and are paid

$10/

hr

We should have at least

20 full time

workers.

The number of full time workers must be

at least twice

of that of part times.

Decision variables

X

1

: Volume of production of Men’s gloves

X

2

: Volume of production of Women’s gloves

X

3

: Volume of production of Children’s gloves

Y

1

: Number of full time employees

Y

2

: Number of part time employees

Constraints

Row material constraint

2X1 + 1.5X2 + X3

 5000

Full

time employees

Y1

 20

Relationship

between the number of Full and Part time employees

Y1

 2

Y2

Labor

Required

.5X

1

+ .75X

2

+ .67X

3

 40

Y

1

+

20

Y

2

Objective

Function

Max Z =

8X

1

+ 10X

2

+ 6X

3

- 520

Y

1

- 200

Y

2

Non-negativity

X

1

, X

2

, X

3

,

Y

1

,

Y

2



0

Excel Solution