Section 4.1 - Finding the Optimal
Production Policy
Special Topics
Chapter 4: Linear Programming
Finding the Optimal Production Policy
Corner point principle –
States that in a linear programming
problem, the maximum value for the
profit formula always corresponds to a
corner point of the feasible region.
Chapter 4: Linear Programming
Finding the Optimal Production Policy
 Feasibility Set or Feasibility Region
 Next step is to find the optimal production
policy, a point within that region that gives a
maximum profit.
1. Find the corner points of the feasible
region.
2. Evaluate the profit at each corner point.
3. Choose the corner point with the highest
profit as the production policy.
Chapter 4: Linear Programming
Finding the Optimal Production Policy
Chapter 4: Linear Programming
Finding the Optimal Production Policy
Calculation of the Profit Formula for Skateboards and Dolls
Corner
Point
Value of the Profit Formula: $1.00x + $0.55y
(0,0)
$1.00(0)
+ $0.55(0)
=
$0.00
+ $0.00
= $0.00
(0,30)
$1.00(0)
+ $0.55(30) =
$0.00
+ $16.50
= $16.50
(12,0)
$1.00(12) + $0.55(0)
= $12.00 + $0.00
= $12.00
Optimal production policy would be the
point (0,30), which gives the maximum
profit of $16.50.
Chapter 4: Linear Programming
Finding the Optimal Production Policy
Optimal production policy –
Corresponds to a corner point of the feasible
region where the profit formula has a
maximum value.
Chapter 4: Linear Programming
Finding the Optimal Production Policy
Example
A juice manufacturer produces and sells two fruit
beverages: 1 gallon of cranapple is made from 3 quarts of
cranberry juice and 1 quart of apple juice; and 1 gallon of
appleberry juice is made from 2 quarts of apple juice and 2
quarts of cranberry juice. The manufacturer makes a profit
of 3 cents on a gallon of cranapple juice and 4 cents on a
gallon of appleberry juice. Today, there are 200 quarts of
cranberry juice and 100 quarts of apple juice available.
How many gallons of cranapple and how many gallons of
appleberry should be produced to obtain the highest profit
without exceeding available supplies?
Chapter 4: Linear Programming
Finding the Optimal Production Policy
1.
Make a mixture chart.
Chapter 4: Linear Programming
Finding the Optimal Production Policy
2.
Find the constraints.
Minimum Constraints: x ≥ 0 and y ≥ 0
Resource Constraints:
Cranberry: 3x + 2y  200
Apple: 1x + 2y  100
Maximize profit formula: 3x + 4y
Chapter 4: Linear Programming
Finding the Optimal Production Policy
3x + 2y = 200
3. Find the feasibility
region.
x + 2y = 100
Chapter 4: Linear Programming
Finding the Optimal Production Policy
Finding the Optimal Production Policy for Beverages
Corner
Point
(0, 0)
Value of the Profit Formula: 3x + 4y cents
3(0) + 4(0)
= 0 cents
(0, 50)
3(0) + 4(50) =
200 cents
(50, 25)
3(50) + 4(25) =
250 cents
(66.7, 0)
3(66.7) + 4(0)
=
200.1 cents
Optimal production policy is (50, 25) with
max profit = 250 cents.