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

(0, 100)

(100,0)

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)

Now add Min Constraints

Using Corner Point Principle

Conclusion: For maximum profit, company should produce 20 gallons of cranapple and 40 gallons of appleberry.