Solution to Toomer Sporting Goods[1]

Ishani Mukherjee is the production manager at Toomer Sporting Goods, which produces cricket ballsfor 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 each4-piece ball costs ₹286 to produce and sells for ₹442.

Yellow Jacket / Sachin Special
Revenue / ₹390 / ₹442
Cost / ₹273 / ₹286
Profit / ₹117 / ₹156

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

AmountRequired Per / Amount
Resource / Yellow Jacket / Sachin Special / Available
Leather / 4 oz / 5 oz / 6,000 oz
Nylon / 3 m / 6 m / 5,400 m
Cork / 2 oz / 4 oz / 4,000 oz
Labor / 2 min / 2.5 min / 3,500 min
Stitching / 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?

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 Variables

In 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.

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 and . We could represent that by the following point on the graph:

Step 2: The Objective

Toomermakes ₹117for 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:

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:

Step 3: The Constraints

We 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:

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:

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

This constraint also corresponds to a line on the graph:

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

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

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 and . 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 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 Solution

As 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.

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
A / 0 / 0
B / 0 / 900
C / 1000 / 400
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) / = ₹0
B / 0 / 900 / 117(0)+156(900) / = ₹140,400
C / 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 1,000Yellow 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.

Appendix: Finding the Corner Points

Using 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.

/ / (this is the line for the Nylon constraint)
/ / (we subtracted 3X from each side of the equation)
/ / (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.

/ / (this is the line for the Leather constraint)
/ / (we substituted our expression for Y)
/ / (we used the distributive property of multiplication)
/ / (we rearranged the left side of the equation)
/ / (we subtracted 4,500 from each side)
/ / (we divided each side by 1.5)

Now, use this value for X with the Nylon constraint to find the value of Y.

/ / (this is the line for the Nylon constraint)
/ / (we substituted our value for X)
/ / (we subtracted 3,000 from each side)
/ / (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.

B60.23501Prof. Juran

[1]Adapted from2-17 (p. 42) in Spreadsheet Modeling and Decision Analysis (6th ed., Cliff T. Ragsdale, South-Western). Solution by David Juran.