1. Spring 2015 Mathematics in Management Science Linear Prog & Mix Problems Two products Two resources Minimum constraints

2. 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.

3. Solving Mixture Problems 1.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.

4. 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.

5. 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?

6. 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?

7. Mixture Chart RC2x + 3y ≤ 12, 4x + 2y ≤ 16, x ≥ 0, y ≥ 0 PFP = 12 x + 10 y ProductsResources machine time painting time 12 16 Profit bikes (x)2412 wagons (y) 3210

8. 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

9. 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

10. Juice Company 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?

11. Mixture chart Cranberry: 3x + 2y ≤ 200 Apple:1x + 2y ≤ 100 Minimums:x ≥ 20, y ≥ 10 Constraint Inequalities

12. 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

13. (0, 100) (100,0)

14. Point of Intersection of 2 Lines Point of intersection of 3x + 2y = 200 & x + 2y = 100 3x + 2y = 200 x + 2y = 100 2x + 0 = 100 x = 50 Plug in to get 50 + 2y = 100 y = 25 Intersection point is (50,25)

15. Now add Min Constraints

17. Using Corner Point Principle Conclusion: For maximum profit, company should produce 20 gallons of cranapple and 40 gallons of appleberry.