Spring 2015 Mathematics in Management Science Linear Prog & Mix Problems Two products Two resources...
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Transcript of Spring 2015 Mathematics in Management Science Linear Prog & Mix Problems Two products Two resources...
Spring 2015Mathematics in
Management Science
Linear Prog & Mix Problems
Two products
Two resources
Minimum constraints
Mixture Problem Algorithm
Display all info in mixture chart.
Write down
resource constraints (RC, MC),
profit formula (PF).
Draw feasible region & mark corner pts.
Evaluate PF at each corner point.
State OPP.
Solving Mixture Problems1. Identify products, resources, constraints (both
resource and minimums). Assign production variables (x, y).
2. Construct mixture chart.3. Find resource inequalities and Profit Formula.4. Sketch Feasible Region:
Draw boundary lines.Find intersections of these lines.Draw minimum constraint lines.Mark all corner points.
5. Apply corner-point principle: evaluate Profit Formula at each corner point.
6. State Optimal Production Policy.
Bikes & Wagons
Bill’s Toy Shop manufactures bikes and wagons for profits of $12 per bike and $10 per wagon.
Each bike requires 2 hours of machine time and 4 hours of painting time.
Each wagon takes 3 hours of machine time and 2 hours of painting time.
Bikes & Wagons
Bill’s Toy factory manufactures bikes and wagons for profits of $12 per bike and $10 per wagon.
Each bike requires 2 hours of machine time and 4 hours of painting time.
Each wagon takes 3 hours of machine time and 2 hours of painting time.
Each day have 12 hours of machine time and 16 hours of painting time.
How many bikes/wagons to make?
Bikes & Wagons
Bill’s Toy factory manufactures bikes and wagons for profits of $12 per bike and $10 per wagon.
Each bike requires 2 hours of machine time and 4 hours of painting time.
Each wagon takes 3 hours of machine time and 2 hours of painting time.
There are 12 hours of machine time and 16 hours of painting time available per day.
How many bikes/wagons to make?
What if must make at least 2 of each?
Mixture Chart
RC 2x + 3y ≤ 12 ,
4x + 2y ≤ 16 ,
x ≥ 0, y ≥ 0
PF P = 12 x + 10 y
Products Resourcesmachine time painting time
12 16
Profit
bikes (x) 2 4 12
wagons (y) 3 2 10
4x+2y=16
2x+3y=12
(0,8)
(4,0) (6,0)
(0,4)
(3,2)
Corner Pts are
(0,0),(0,4),(3,2),(6,0)
Feasible Region
Feasible
Region
Corner Point Principle
Have corner points
(0,0), (0,4), (3,2),(4,0).
Evaluate profit P=12x+10y at each
P=0 at (0,0) P=40 at (0,4)
P=56 at (3,2) P=48 at (4,0)
Optimal production policy is to make
3 bikes & 2 wagons.
Same OPP when make 2 of each
JuiceCompany makes & sells two fruit juices. 1 gallon of cranapple made from 3 quarts of cranberry juice and 1 quart of apple juice;
1 gallon of appleberry made from 2 quarts of cranberry juice and 2 quarts of apple juice.
Profit:2 cents profit on a gallon of cranapple.
5 cents on a gallon of appleberry.
Have 200 quarts of cranberry juice and 100 quarts of apple juice available.
Want at least 20 gallons of cranapple and 10 gallons of appleberry.
How much of each should they produce in order to maximize profit?
Mixture chart
Cranberry: 3x + 2y ≤ 200
Apple: 1x + 2y ≤ 100
Minimums: x ≥ 20 , y ≥ 10
Constraint Inequalities
Graphing the feasible region
Intercepts for 3x + 2y = 200:
x-intercept: x = 200/3 = 66.7
y-intercept: y = 200/2 = 100
Intercepts for x + 2y = 100:
x-intercept: x = 100
y-intercept: y = 100/2 = 50
Point of Intersection of 2 Lines
Point of intersection of 3x + 2y = 200 & x + 2y = 100
3x + 2y = 200x + 2y = 1002x + 0 = 100
x = 50
Plug in to get 50 + 2y = 100y = 25
Intersection point is (50,25)