LP Formulation: Product Mix
-- Structure of Problem:
 One or several limited resources shared by several different but similar
products
 Find the best mix of products to maximize total profit
-- LP Formulation:
max
s.t
total profit
resource used < resource available
other constraints
-- Decision Variables: Amount of each product to be produced
Example 1: Product Manufacturing
A certain plant can manufacture four products A, B, C, and D in any combination. Each
product requires time on each of three machines in the following manner (figure in
minutes/unit):
Machine
1
2
3
Product
A
12
8
5
B
7
9
10
C
8
4
7
D
10
0
3
Machine 1,2, and 3 are available 20, 40, and 10 hours per week, respectively. Material
costs are $2 each for products A and B and $1 each for C and D. Find the best product
mix in order to maximize the total profit of the week.
1) Assume all products are competitive and any amounts made may be sold at respective
prices of $5, $4, $5, and $5. Variable labor costs are $4/hour for machines 1 and 2 and
$3/hour for machine 3.
-- Excel Data Organization:
 Production plan and gross profit
 Machine hours needed for each machine
 Machine hours used and capacity requirement
 Labor cost and net profit
2) Assume products are not purely competitive and the prices in (1) apply to only first 20
units. The prices of additional products A, B, C, and D are $4, $3, $4, and $4,
respectively. Additional machine hour, up to 10 hours, is available for each machine and
the corresponding labor costs are $5/hour for machine 1 and 2 and $4/hour for machine 3.
ABCD Product Manufacturing Excel files
-- Modeling Tricks
 Treat first 20 products and the products beyond 20 as different products.
-- Excel Data Organization
 Production plan for products under 20
 Production plan for products above 20
 Machine hours needed for each product
 Machine hours used
 Regular time and overtime available and unit cost
 Labor cost and net profit.
-- LP Formulations:
Example 2: Farm Planning
Buster Sod operates an 800 acre irrigated farm in the Red River Valley of Arizona. Sod’s
principal activities are raising wheat, alfalfa, and beef. The Red River Valley Water
Authority has just given its water allotments for next year (Sod was allotted 1000 acrefeet), and Sod is busy preparing his production plan for next year. He figures that beef
prices will hold at around $500 per ton and that wheat will sell at $2 per bushel. Best
guesses are that he will be able to sell alfalfa at $22 per ton, but if he needs more alfalfa
to feed his beef than he can raise, he will have to pay $28 per ton to get the alfalfa to his
feedlot.
Some technological features of Sod’s operation are wheat yield, 70 bushels per acre;
alfalfa yield, 4 tons per acre. Other features are given in the following table:
Activity
Labor, Machine
Cost ($)
1 acre of wheat
1 acre of alfalfa
1 ton of beef
20
28
50
Water
Requirements
(Acre-Ft)
2
3
0.05
Land
Requirements
(Acres)
1
1
0.1
Help Sod to come up with a good production for the coming year.
-- LP Formulation:
Decision Variables:





W:
B:
AR:
AS:
AB:
Remark: define more variables to make LP formulation easier to read.
Alfalfa
Requirements
(Tons)
5
MAX
SUBJECT TO
2)
3)
4)
END
LP OPTIMUM FOUND AT STEP
5
OBJECTIVE FUNCTION VALUE
1)
VARIABLE
W
AR
B
AB
AS
ROW
2)
3)
4)
2480000.0
VALUE
.000000
.000000
8000.000000
40000.000000
.000000
REDUCED COST
2980.000000
754.000000
.000000
.000000
6.000000
SLACK OR SURPLUS
.000000
600.000000
.000000
DUAL PRICES
3100.000000
.000000
28.000000
NO. ITERATIONS=
5
RANGES IN WHICH THE BASIS IS UNCHANGED:
VARIABLE
W
AR
B
AB
AS
ROW
2
3
4
CURRENT
COEF
120.000000
-7.000000
450.000000
-28.000000
22.000000
OBJ COEFFICIENT RANGES
ALLOWABLE
ALLOWABLE
INCREASE
DECREASE
2980.000000
INFINITY
754.000000
INFINITY
INFINITY
298.000000
6.000000
55.851850
6.000000
INFINITY
CURRENT
RHS
800.000000
1000.000000
.000000
RIGHTHAND SIDE RANGES
ALLOWABLE
ALLOWABLE
INCREASE
DECREASE
1200.000000
800.000000
INFINITY
600.000000
40000.000000
INFINITY
-- Questions based on the LP output:
1. How much water is being used?
2. How much beef is being produced?
3. Does Sod buy or sell alfalfa?
4. How much profit will Sod receive from the optimal operation of his farm?
5. Among all possible activities Sod might take, which is most attractive and which least
attractive?
6. How much should Sod pay to acquire another acre-ft of water? another acre of land?
What will he use the additional resource for?
7. What happens to the optimal planting policy if the price of wheat triples and the labor
and machinery costs for wheat planting decreases by 50%?
8. What happens to the profit if both purchasing and selling prices of alfalfa increase by
$2?
9. The labor and machinery costs listed may vary by up to 20% in both directions.
Discuss the potential change in the current planting policy.
10. Do you suggest Sod to plant soybean, which currently sells for $5/bushel and each
acre of land will produce 100 bushel of soybean. Other features of soybean is given
below:
Activity
1 acre of
soybean
Labor, Machine
Water
Land
Alfalfa
Cost ($)
Requirements
Requirements
Requirements
(Acre-Ft)
(Acres)
(Tons)
1
1
30
Farm Planning Excel files: Edition1
Edition2
MAX
120 W - 7 AR + 450 B - 28 AB + 22 AS
SUBJECT TO
2)
W + 0.25 AR + 0.1 B <=
800
3)
2 W + 0.75 AR + 0.05 B <=
1000
4) - AR + 5 B - AB + AS =
0
END
Homework:
Homework for Product Mix
( Solutions )
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