Environmentally Conscious Design & Manufacturing (ME592)
Date: May 3, 2000 Slide:1
Environmentally Conscious Design & Manufacturing
Class 24: Optimization- Operations Research
Prof. S. M. Pandit
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 3, 2000 Slide:2
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”
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 3, 2000 Slide:3
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?
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 3, 2000 Slide:5
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?
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 3, 2000 Slide:6
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
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 3, 2000 Slide:7
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)
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 3, 2000 Slide:8
Model
IE xxz 23 Maximize:
Subject to62 IE xx
82 IE xx
1 EI xx
2Ix
0Ix
0Ex
(constraints)
(objective function)
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 3, 2000 Slide:9
Graphical Solution of LP Models
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 3, 2000 Slide:10
Graphical Solution of LP Models
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 3, 2000 Slide:11
“Amalgamation”
(spend $)4 types of solid waste
3 types of products (earn $)
Requires % of specific materials (Consume $)
Reclamation Center
Example 2
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 3, 2000 Slide:12
grade specification amalgation cost ($/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
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 3, 2000 Slide:13
Material pounds/week treatment cost available per pound
1 3,000 $ 3 2 2,000 $ 6 3 4,000 $ 4 4 1,000 $ 5
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 3, 2000 Slide:14
Formulation
Q: What are the decision variables?
- What information is needed?
- What kind of decision variable(s) would best
provide the information needed?
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 3, 2000 Slide:15
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 )
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 3, 2000 Slide:16
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
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 3, 2000 Slide:17
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
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 3, 2000 Slide:18
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
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 3, 2000 Slide:19
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.
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 3, 2000 Slide:21
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)
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 3, 2000 Slide:22
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)
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 3, 2000 Slide:23
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,
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 3, 2000 Slide:24
Goal Programming
},0max{ ii yw
To convert the model to a linear program, we use thesubstitution
which is equivalent to
ii yw
0iw