Question

Suipersport Footballs, Inc. has the problem of determining the best number of All-Pro (x1), College (x2), and High School (x3) models of footballs to produce in order to maximize profits. Constraints include production capacity in each of three departments (cutting & dyeing, sewing, and inspection & packaging) as well as a constraint that requires production of at least 1000 All-Pro footballs. The linear programming model of Supersport's problem is shown below Supersport's Problem Supersport's Problem All-Pro 1000 X2 College HS 200 Decision Variables Total obj fn Maximize Profit Cutting & dyeing, Minutes 12 Sewing, Minutes Inspection & pack, Minutes 3 Model 14000 18000 3800 1000 18000 18000 9000 1000 12 4. Target Cell (Max) Cell SF$6 Original Final Value Value Name Profit Total obj fn 4000 Adjustable Cells Original Final Value Value SC$4 SD$4 SE$4 All-Pro College High School 1000 Constraints Cell Value Formula Name All-Pro Model Cutting & Status Slack SF$11 SF$8 SFS9 SFS10 1000SF$11 SH$11 Binding 14000SF$8-SHS8 Not Binding 18000SF$9 SH$9 Binding 3800SF$10 SH$10 Not Binding 4000 dyeing Sewing Inspection & pack 5200 Adjustable Cells Final Reduced Objective Allowable Allowable Coefficient ncrease Value Cost SC$4 SD$4 SE$4 Ali-Pro College High School 1000 200 1E+30 0 1E+30 Constraints Final Value Shadow Constraint Allowable R.H. Side ncrease 1000 18000 18000 Allowable Cell SF$11 Al-Pro Model SF$8 $F$9 SF$10 Inspection & pack Name 1000 14000 18000 3800 1000 200 1E+30 6000 Cutting & dyeing 0.333333333 5200

1. How many footballs of each type should Supersport produce in order to increase to maximize the profit?
2. How much maximum profit it would make?
3. Overtime rates in the sewing department are $12 per hour. Which of the following is correct?
   1. Get overtime up to 100 hours as it increases net profit by $8 per hour.
   2. Get as much overtime as possible as it increases net profit by $8 per hour
   3. Do not get overtime because it would not help in incrementing the profit
   4. Sewing time should be reduced by 15000 min because shadow price is $0.33
4. If College football profit was $3 more (i.e. $8), which of the following is correct.
   1. produce 200 college footballs and no change in maximum profit
   2. produce 200 college footballs and $600 increase in maximum profit
   3. produce more than 200 college footballs for maximum profits
   4. it can not be determined from the given information
5. All-Pro football constraint has negative $2 shadow price. What does it mean?
   1. an increase of one unit in All-Pro requirement will increase profit by $2
   2. an increase of one in All-Pro requirement will reduce profit by $2
   3. a decrease of one unit in All-Pro requirement will reduce profit by $2
   4. it is a mis-print, Shadow price can not be negative.

Answer 1

This is a simple problem related to linear optimization modeliing.

From the excel sheet model provided to us, we can infer that

1. The objective function is 3X1+5X2+4X3 which is needed to maximized.

2. The constraint functions are

1. for cutting and sewing :12X1+10X2+8X3<=18000

2. For sewing : 15X1+15X2+12X3<=18000

3. For inspection and packaging : 3X1+4X2+2X3<=9000

4 . For all pro model s: X1>=1000

Aslo all variables should be non negative

X1,X2,X3>=0

Now we are sorted . We have the subjective function and the constraints given as well.

Now we shall try to find the optimal values of each of X1,X2, and X3.

I seeked solution of the given LP model with the help of simplex method and got the following results.

There are infinitely many values of X1, X2, X3 for the optimal value Z = 4000, which are contained in the part of the plane 3 X1 + 5 X2 + 4 X3 = 4000 that satisfy the constraints of this problem. One of them is:
X1 = 1000
X2 = 200
X3 = 0

1. Thus the maximized profit will be $4000 and to make it happen we must have 1000 All pros ,200 collage and nil HS balls.

Q2. Overtime rates in the sewing department are $12 per hour. Which of the following is correct?

Look at the constraints associated with overtime on sewing machines.

In the sewing department all of the production time is used .Thus If the production time at sewing department is increased by one minute, the profits will increase by $0.333. If the production time at sewing department is increased by one hour, the profits will increase by $0.333(60) = $20. So, management would be willing to pay $12 per hour for overtime work at the sewing department.

Hence overall profit per hour will be $(20-12)=$8

Hence we can say that option c is correct.

Q3. If College football profit was $3 more (i.e. $8), which of the following is correct.

Reflect back on the objective function.

3X1+5X2+4X3

Here what we are essentially doing is that we are increasing the profit by 3$ to $8 for collage balls.

This will actually change hte coeffifcnet of college balls (X2) from 5 to 8.

We run the optimal solution and get

There is any possible solution for the problem, so we can continue to Phase II to calculate it.

The optimal solution value is Z = 4600
X1 = 1000
X2 = 200
X3 = 0

The overall profit increases by $600 while the number of balls of each type remain the same. (no change on optimality)

Hence option (b).

Q4. All-Pro football constraint has negative $2 shadow price. What does it mean?

This means that there is a reverse relationship between the ball requirmenets and profit.

Hence

an increase of one in All-Pro requirement will reduce profit by $2

Hence option (b).