Environmentally Conscious Design & Manufacturing (ME592) Date: May 3, 2000 Slide:1 Environmentally Conscious Design & Manufacturing Class 24: Optimization

Environmentally Conscious Design & Manufacturing (ME592)

Date: May 3, 2000 Slide:1

Environmentally Conscious Design & Manufacturing

Class 24: Optimization- Operations Research

Prof. S. M. Pandit



Linear Programming

• Problem statement

• Developing the objective function

• What are the constraints?

• Mathematical model

Example 1 & 3 from Hamdy A. Taha, “Operations

Research”

and 2 from Hillier & Lieberman, “Introduction to

Operations Research”



Example 1 of LP

Given:• A small paint company produces both interior and

exterior house paints for wholesale distribution.• Two raw materials, A & B are used.• The price per ton is $3,000 for exterior paint and

$ 2,000 for interior paint.

Ask:How much interior and exterior paints should the company produce daily to maximize gross income?



Example 1 of LP



Construction of the Math. Model

• What are the variables (unknowns ) of the problem?

• What constraints must be imposed on the variables

to satisfy the limitations of the modeled system?

• What is the objective (goal) that need to be achieved

to determine the optimum (best) solution from

among all the feasible values of the variables?



Variables & Objective Function

Variables:

xE=tons produced daily of exterior paint

xI =tons produced daily of interior paint

Objective function:

z=3 xE+2 xI



Constraints

62 IE xx (raw material A)

82 IE xx (raw material B)

Usage restriction:

1 EI xx (excess of interior over exterior paint)2Ix

Demand restriction:

0Ix0Ex

Nonnegativity restriction:

(maximum demand for interior paint)



Model

IE xxz 23 Maximize:

Subject to62 IE xx

82 IE xx

1 EI xx

2Ix

0Ix

0Ex

(constraints)

(objective function)



Graphical Solution of LP Models



Graphical Solution of LP Models



“Amalgamation”

(spend $)4 types of solid waste

3 types of products (earn $)

Requires % of specific materials (Consume $)

Reclamation Center

Example 2



grade specification amalgation cost ($/pound) selling price ($/pound)A <=30% mat.1 3.00 8.50

>=40% mat.2<=50% mat.3

B <=50% mat.1 2.50 7.00>=10% mat.2

C <=70% mat.3 2.00 5.50



Material pounds/week treatment cost available per pound

1 3,000 $ 3 2 2,000 $ 6 3 4,000 $ 4 4 1,000 $ 5



Formulation

Q: What are the decision variables?

- What information is needed?

- What kind of decision variable(s) would best

provide the information needed?



Decision

Amount of each product grade

yi=pounds of product grade “i”

Zij=proportion of material “j” in product “i”

In terms of weight,

Quantity of material “j” used =ZAjyA+ZBjyB+ZCjyC

(product of variablesnonlinear function )



Replacing ‘product terms,’ so that

xij=Zij yi {i=A,B,C j=1, 2, 3, 4}

xij-decision variablesNumber of pounds of material j allocated to product grade i per week



We want:

Total amount of product grade “i”:

Proportion of material “j” in product grade “i”:

Total profit:

4

1iijx

4

1iij

ij

x

x

)(5)(4)(6

)(3)(5.3

)(5.4)(5.5

444333222

1114321

43214321

cBAcBAcBA

cBACCCC

BBBBAAAA

xxxxxxxxx

xxxxxxx

xxxxxxxxZ



Model

Model:

4321

4321

4321

5.15.05.25.0

5.05.05.15.1

5.05.15.05.2

CCCC

BBBB

AAAA

xxxx

xxxx

xxxxZ

Constraints:

• Availability• Mixture specifications• Non-negativity



Example 3: Goal ProgrammingGiven:• Two products are manufactured by passing sequentially

through two different machines• The time available for the two products on each

machine is limited to 8 hours daily but may be exceeded by up to 4 hours on an overtime basis

• Each overtime hour will cost an additional $ 5

Require• To determine the production level for each product that

will maximize the net profit.





Goal Programming

The constraints of the model without the overtime option86/5/ 21 xx

88/4/ 21 xx

(machine1)

(machine2)

The constraints of the model with the overtime option

86/5/ 121 yxx

88/4/ 221 yxx

(machine1)

(machine2)



Mathematical Representation

To determine the number of units of each product

(variables) that maximizes net profit (objective )

provided that the maximum allowable machine hours

are exceeded only on an overtime basis (constraints)



Complete Model

}),0max{},0(max{546 2121 yyxxz

41 y

Maximize

Subject to 86/5/ 121 yxx

88/4/ 221 yxx

42 y

0, 21 xx

signinedunrestrictyy 21,



Goal Programming

},0max{ ii yw

To convert the model to a linear program, we use thesubstitution

which is equivalent to

ii yw

0iw



LP Model

)(546 2121 wwxxz

011 wy

maximize

Subject to 86/5/ 121 yxx

88/4/ 221 yxx

0,,, 2121 wwxx

signinedunrestrictyy 21,

022 wy

41 y

42 y

Documents

Environmentally Conscious Design & Manufacturing (ME592) Date: May 3, 2000 Slide:1 Environmentally Conscious Design & Manufacturing Class 24: Optimization