Example for Recitations Optimizatio Optimi zation n – Bann Banner er Chemic Chemical al Connor Makowski
ctl.mit.edu 1
Before Starting This Recitation •
You should:
Have watched all of the lessons for this week
2
Recitation Agenda •
Banner Chemical
Understanding the Formulation
Algebraic optimization model
Summation based optimization model
3
Formulating – Banner Chemicals •
Situation
•
Banner Chemicals manufactures specialty chemicals. One of their products comes in two grades, high and supreme. The capacity at the plant is 110 barrels per week. The high and supreme grade products use the same basic raw materials but require different ratios of additives. The high grade requires 3 gallons of additive A and 1 gallon of additive B per barrel while the supreme grade requires 2 gallons of additive A and 3 gallons of additive B per barrel. The supply of both of these additives is quite limited. Each week, this product line is allocated only 300 gallons of additive A per week and 280 gallons of additive B. A barrel of the high grade has a profit margin of $80 per barrel while the supreme grade has a profit margin of $200 per barrel.
Question
How many barrels of High and Supreme grade should Banner Chemicals produce each week?
Step 1. Determine Decision Variables
XS
XH = Number of High grade barrels to produce per week XS = Number of Supreme grade barrels to produce per week Bounds
XH ≥ 0
XS ≥ 0
XH 4
Formulating – Banner Chemicals •
Situation
•
Banner Chemicals manufactures specialty chemicals. One of their products comes in two grades, high and supreme. The capacity at the plant is 110 barrels per week. The high and supreme grade products use the same basic raw materials but require different ratios of additives. The high grade requires 3 gallons of additive A and 1 gallon of additive B per barrel while the supreme grade requires 2 gallons of additive A and 3 gallons of additive B per barrel. The supply of both of these additives is quite limited. Each week, this product line is allocated only 300 gallons of additive A per week and 280 gallons of additive B. A barrel of the high grade has a profit margin of $80 per barrel while the supreme grade has a profit margin of $200 per barrel.
Question
How many barrels of High and Supreme grade should Banner Chemicals produce each week?
Step 2. Formulate Objective Function Profit = 80XH + 200XS
XS (0,12) (20,4)
(0,5)
Maximize z(XH, XS) = 80XH + 200XS (12.5,0) Slides from Dr. Caplice
(30,0)
XH 5
Formulating – Banner Chemicals •
Situation
•
Banner Chemicals manufactures specialty chemicals. One of their products comes in two grades, high and supreme. The capacity at the plant is 110 barrels per week. The high and supreme grade products use the same basic raw materials but require different ratios of additives. The high grade requires 3 gallons of additive A and 1 gallon of additive B per barrel while the supreme grade requires 2 gallons of additive A and 3 gallons of additive B per barrel. The supply of both of these additives is quite limited. Each week, this product line is allocated only 300 gallons of additive A per week and 280 gallons of additive B. A barrel of the high grade has a profit margin of $80 per barrel while the supreme grade has a profit margin of $200 per barrel.
Question
How many barrels of High and Supreme grade should Banner Chemicals produce each week? XS
Step 3. Formulate Constraints Plant Capacity is 110 barrels High
XH + XS ≤ 110
Supreme Available
Additive A
3 gal
2 gal
300 gal
3XH + 2XS ≤ 300
Additive B
1 gal
3 gal
280 gal
XH + 3XS ≤ 280
Feasible Region XH Slides from Dr. Caplice
6
Algebraic Formulation – Banner Chemicals Objective Function - The thing you are trying to maximize or minimize
Max z(XH, XS) = 80XH + 200XS subject to
XH +
XS ≤ 110
3XH +
2XS ≤ 300
XH + 3XS ≤ 280 XH
≥0 XS ≥ 0
Constraints - Limits to resources or requirements of the system that must be adhered to absolutely. Consists of a Left Hand Side (LHS) function, that has some relationship (≤,=,≥) to a Right Hand Side (RHS) that must be satisfied. Bounds or Non-Negativity Conditions Decision variables typically can’t be negative.
Decision Variables
The unknowns in the problem whose values you are trying to determine. XH = Number of High grade barrels to produce per week XS = Number of Supreme grade barrels to produce per week Slides from Dr. Caplice
7
Moving to Summation Notation Max z(XH, XS) = 80XH + 200XS s.t.
XH + XS ≤ 110 3XH + 2XS ≤ 300 XH + 3XS ≤ 280 XH ≥0 XS ≥ 0
Max
z = å
i Î M
pi xi
Summation from i to M
s.t .
åi
Î M
åi
Î M
xi £ C
xi ³ 0
aij xi £ A j
for all
" j Î N
"i Î M
Indices
is a member of
Products i in M Additives j in N Input Data
pi = Profit margin for product i ($/barrel) C = Plant capacity (barrels/week) A j = Additive j available (gallons/week) aij = Quantity of additive j required per barrel of product i (gallons/barrel) Decision Variables
xi = Quantity of product i to produce (barrels)
Slides from Dr. Caplice
8
Questions, Comments, Suggestions? Use the Discussion!
“Margaret - After immersing herself in constrained optimization”
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