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 total profit
s.t 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.
-- 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 acre-feet), 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 $28 per ton to get the alfalfa to his feedlot.
Some technological features of Sod’s operation are wheat yield, 70 bushels per acre; alfalfa yield, 4 tons per acre. Other features are given in the following table:
Activity / Labor, Machine Cost ($) / Water Requirements (Acre-Ft) / Land Requirements (Acres) / Alfalfa Requirements (Tons)1 acre of wheat / 20 / 2 / 1
1 acre of alfalfa / 28 / 3 / 1
1 ton of beef / 50 / 0.05 / 0.1 / 5
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.
MAX
SUBJECT TO
2)
3)
4)
END
LP OPTIMUM FOUND AT STEP 5
OBJECTIVE FUNCTION VALUE
1) 2480000.0
VARIABLE VALUE REDUCED COST
W .000000 2980.000000
AR .000000 754.000000
B 8000.000000 .000000
AB 40000.000000 .000000
AS .000000 6.000000
ROW SLACK OR SURPLUS DUAL PRICES
2) .000000 3100.000000
3) 600.000000 .000000
4) .000000 28.000000
NO. ITERATIONS= 5
RANGES IN WHICH THE BASIS IS UNCHANGED:
OBJ COEFFICIENT RANGES
VARIABLE CURRENT ALLOWABLE ALLOWABLE
COEF INCREASE DECREASE
W 120.000000 2980.000000 INFINITY
AR -7.000000 754.000000 INFINITY
B 450.000000 INFINITY 298.000000
AB -28.000000 6.000000 55.851850
AS 22.000000 6.000000 INFINITY
RIGHTHAND SIDE RANGES
ROW CURRENT ALLOWABLE ALLOWABLE
RHS INCREASE DECREASE
2 800.000000 1200.000000 800.000000
3 1000.000000 INFINITY 600.000000
4 .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 $5/bushel and each acre of land will produce 100 bushel of soybean. Other features of soybean is given below:
Activity / Labor, Machine Cost ($) / Water Requirements (Acre-Ft) / Land Requirements (Acres) / Alfalfa Requirements (Tons)1 acre of soybean / 30 / 1 / 1
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