Question

A firm, DLS Apps, Inc. decides to enter the market for apps made for teenagers and children. Their Production Supervisor analyzed each of the operations and concluded that if the company produces apps for children, each app would require 8 hours in the theme creation department, 12 hours in the graphics and imaging department, 4 hours in the educational content department, 6 hours in the coding department, and 5 hours in the testing and launching department. On the other hand, each app for teenagers would require 4 hours in the theme creation department, 15 hours in the graphics and imaging department, 15 hours in the educational content department, 10 hours in the coding department, and 2 hours in the testing and launching department. After studying the departmental workload projections, the Production Supervisor estimates that 100 hours for theme creation, 250 hours for graphics and imaging, 200 hours for the educational content creation, 150 hours of coding and 50 hours for the testing and launching are available for the production of the new apps during the next four months.

The accounting department analyzed the costs based on the production data and overheads and arrived at prices for the apps that will result in a profit contribution of $10,000 per children’s app and $15,000 per teenager app produced. The CEO asserts that to be successful in the long run, at least 5 children’s apps must be launched in the next four months. The management is trying to identify how many apps of each type should they launch in the next four months and have hired you to help them figure out what the optimal solution would be and what the expected profits would be.

1. Write out the Linear Program (equations)
2. Provide the optimal solution, either graphically or by using Excel Solver. What is the total expected profit? How many apps of each type would you recommend that they make?
3. Identify which constraints are binding.
4. Which department has the highest slack? How much?
5. If the company was able to obtain 5 more hours in the coding department, would the profit change? Explain why or why not.

Answer 1

The data can be summarized as follows:

	Children App	Teenager app
Profit	$10,000	$15,000
	Hours required per app	Hours available in each department
Theme creation dept	8	4	100
Graphics and Imaging Dept	12	15	250
Educational content dept	4	15	200
Coding dept	6	10	150
Testing and Launching dept.	5	2	50

LPP formulation:

Decision variable:

Let,

X = number of children apps to be launched within 4 months

Y = number of teenager apps to be launched within 4 months

Objective Function:

The objective of the company is to maximize the profit form the launching of the mix of apps.

Total Profit = Total profit form children app + Total profit from teenager app

Total Profit = ($10,000)X + ($15,000)Y

Te objective function is given as:

Maximize Z = ($10,000)X + ($15,000)Y

Subject to:

Sr. No	Constraint	Equation
1	Time required and available in Theme creation dept	Time required <= 100 hours 8X + 4Y <= 100
2	Time required and available in Graphics and Imaging Dept	Time required <= 250hours 12X + 15Y <= 250
3	Time required and available in Educational content dept	Time required <= 200 hours 4X + 15Y <= 200
4	Time required and available in Coding dept	Time required <= 150 hours 6X + 10Y <= 150
5	Time required and available in Testing and Launching dept.	Time required <= 50 hours 5X + 2Y <= 50
6	At least 5 children’s app should be launched	X >= 5
7	Non-negativity and integer constraint	X and Y >= 0 and integer variable

Excel Model and Solver Solution:

The optimal Solution:

X = 5.26 Children’s apps

Y = 11.84 teenager’s apps

The expected profit = $230,263.16

Binding constraints:

Binding constraints are those constraints which are completely utilized in case of “<=” type constraint. According to optimal solution, the hours of coding department and testing & Launching department is completely utilized, there are no hours available after optimals solution mix apps produced.

Thus, Binding Constraints are - Coding department and testing & Launching department

Slack available:

The slack is given as the difference of available time and time require d for the optimal solution. It shown in column G. The highest slack is 10.53 hours for the Theme creation depts..

Additional hours at coding department

The Coding dept. hours are completely utilized, thus this resource limits the optimal solution, but if additional hour of coding is made available, then more profit can be generated.