Solution to Toomer Sporting Goods1

Ishani Mukherjee is the production manager at Toomer Sporting Goods, which produces cricket balls for youth recreation activities and some professional cricket leagues. She makes two products, the popular 2-piece “Yellow Jacket” ball and the more expensive 4-piece “Sachin Special” used for professional play.Each 2-piece ball costs ₹273 and sells for ₹390, while each 4-piece ball costs ₹286 to produce and sells for ₹442.

Yellow Jacket

Sachin Special

Revenue ₹390 ₹442

Cost ₹273 ₹286Profit ₹117 ₹156

The material and labor required to produce each item is listed here along with the availability of each resource.

Amount Required Per AmountResource Yellow Jacket Sachin

SpecialAvailable

Leather 4 oz 5 oz 6,000 ozNylon 3 m 6 m 5,400 mCork 2 oz 4 oz 4,000 ozLabor 2 min 2.5 min 3,500 minStitching 1 min 1.6 min 2,000 min

What should Ishani do? What is the best production plan to maximize her profit while using only the resources available?

1 Adapted from 2-17 (p. 42) in Spreadsheet Modeling and Decision Analysis (6th ed., Cliff T. Ragsdale, South-Western). Solution by David Juran.

This problem is one example of a class of problem that can be solved with the technique of linear programming. We can solve problems like this by following these four steps:1. Define the choices to be made by the manager (called decision

variables).2. Find a mathematical expression for the manager's goal (called the

objective function).3. Find expressions for the things that restrict the manager's range of

choices (called constraints).4. Use algebra to find the best solution.

Step 1: Decision VariablesIn this situation, Ishani needs to decide how many Yellow Jacket (2 piece) and Sachin Special (4 piece) cricket balls to produce. In other words, she needs to assign numerical values to two variables. We can define two variables, X and Y, as follows:

Variable Name

Symbol

Units

Yellow Jacket X Cricket balls

Sachin Special

Y Cricket balls

One way to think about the problem is to draw a graph, with the number of Yellow Jackets on the horizontal axis, and the number of Sachin Specials on the vertical axis.

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Any solution to Ishani’s problem can be represented by a point on the graph. For example, if Ishani decided to produce 1,000 units of each product, then X=1000 and Y=1000 . We could represent that by the following point on the graph:

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Step 2: The ObjectiveToomer makes ₹117 for every Yellow Jacket it sells, and ₹156 for every Sachin Special. Ishani wants to make sure she chooses the right mix of the two products so as to make the most money for her company.We can make an expression for the amount of money Ishani makes in terms of the decision variables. Each unit of the variable X represents 1 Yellow Jacket, and 1 Yellow Jacket will earn a profit of ₹117. Similarly, for every unit of the variable Y, which represents 1 Sachin Special, she will earn ₹156. Therefore, Ishani's profit (the total amount of money she earns for the company) can be represented by this expression:

Profit=117X+156Y

Her task, then, is to find values for X and Y that maximize this expression.Different profit levels can be represented by lines on the graph:

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Step 3: The ConstraintsWe are given the following factors that limit Ishani's range of options:1. Each Yellow Jacket requires 4 ounces of leather, and each Sachin

Special requires 5 ounces. Ishani has 6,000 ounces available, so the total leather used for both products cannot exceed 6,000 ounces.

We can make an expression for this constraint:Total ounces of leather

used= (total ounces for Yellow Jacket) + (total ounces for Sachin Special)= (4 ounces per Yellow Jacket) + (5 ounces per Sachin Special)= 4X + 5Y

We know that there are only 6,000 ounces available, so we can write:4 X+5Y ¿6 ,000

This expression can also be represented on our graph; to stay within the constraint, Ishani needs to stay to the lower-left side of this line:

2. Ishani only has 5,400 meters of nylon. Each Yellow Jacket requires 3 meters, and each Sachin Special requires 6 meters. Therefore:

3 X+6Y ¿5 ,400

This constraint also corresponds to a line on the graph:

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3. There are only 4,000 ounces of cork available. Yellow Jackets use 2 ounces of cork each, and Sachin Specials use 4 ounces. Therefore:

2 X+4 Y≤4 ,000

4. There are only 3,500 minutes of labor available. Yellow Jackets take 2 minutes of labor each, and Sachin Specials take 2.5 minutes. Therefore:

2 X+2 .5Y≤3 ,500

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5. Finally, Ishani has only 2,000 minutes of stitching time available. Each Yellow Jacket takes 1 minute and each Sachin Special takes 1.6 minutes. Therefore:

1 X+1.6Y≤2,000

Here is the graph with all five constraints drawn:

Note: there are also two other constraints that are not obvious here. Since Ishani can't make a negative number of either product, we need to remember that X≥0 and Y≥0 . These non-negativity constraints can be represented by lines on the X and Y axes. Notice the polygon formed by the constraint lines, and in particular the four corner points. The polygon forms the shaded region that

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represents all of the possible choices Ishani could make without violating any of the constraints. This region is called the feasible region.

Step 4: Finding the SolutionAs it happens, we can find the best solution (called the optimal solution) by looking at the corners of the polygon that makes up the feasible region.

0

500

1000

1500

0 500 1000 1500 2000

Sach

in S

peci

al (Y

)

Yellow Jacket (X)

Toomer Sporting Goods

Point C

Point B

Point DPoint A

Using algebra, we can determine the X and Y values at these corner points (see the Appendix to see an example of how this is done). The four corner points are:

Point X Y

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A 0 0B 0 90

0C 100

0400

D 1500 0

Using the objective function formula, we can see what Toomer's profit would be at each of the five corner points:

Point X Y Objective Function Profit

A 0 0 117(0)+156(0) = ₹0B 0 900 117(0)+156(900) =

₹140,400C 1000 400 117(1,000)+156(

400)= ₹179,400

D 1500 0 117(1,500)+156(0)

= ₹175,500

It turns out that Point C is the best solution; it has the largest profit. Therefore, it would appear that Ishani ought to plan on producing 1,000 Yellow Jackets and 400 Sachin Specials.

Here we can see the five corner points and their corresponding profits:

In graph below, we can visually compare the corner points to the isoprofit lines. You can see that Point C comes the closest to the outer line — this means Point C will yield the greatest profit.

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0

500

1000

1500

0 500 1000 1500 2000

Sach

in S

peci

al (Y

)

Yellow Jacket (X)

Toomer Sporting Goods

C

B

DA

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Appendix: Finding the Corner PointsUsing algebra, we can determine the values for X and Y that correspond to the intersection of two lines. For example, in our problem, Point C is where the line representing the Nylon constraint intersects the line representing the Leather constraint. There is only one point that satisfies both of the equations for those lines, and we can find that point using this algebraic procedure: First, use one of the equations to express X in terms of Y.3 X+6Y =5 , 400 (this is the line for the Nylon constraint)

6Y =5 , 400−3 X (we subtracted 3X from each side of the equation)

Y =900−0 .5 X (We divided both sides of the equation by 6; now we have an expression for Y in terms of X.)

Now we use that expression for Y with the Leather constraint to figure out the value of X at Point C.

4 X+5Y =6 ,000 (this is the line for the Leather constraint)

4 X+5 (900−0 .5 X ) =6 ,000 (we substituted our expression for Y)

4 X+(4 ,500−2 .5 X ) =6 ,000 (we used the distributive property of multiplication)

1 .5 X+4 ,500 =6 ,000 (we rearranged the left side of the equation)

1 .5 X =1 ,500 (we subtracted 4,500 from each side)

X =1 ,000 (we divided each side by 1.5)

Now, use this value for X with the Nylon constraint to find the value of Y.3 X+6Y =5 , 400 (this is the line for the Nylon constraint)

3(1 ,000)+6Y =5 , 400 (we substituted our value for X)

6Y =2 ,400 (we subtracted 3,000 from each side)

Y =400 (we divided each side by 6; now we know the value of Y at Point C)

Therefore, Point C corresponds to (1000, 400), or 1,000 Yellow Jackets and and 400 Sachin Specials.

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