# Linear Programming Case Study – Answered

#### Linear Programming Case Study

Your instructor will assign a linear programming project for this assignment according to the following specifications.

It will be a problem with at least three (3) constraints and at least two (2) decision variables. The problem will be bounded and feasible. It will also have a single optimum solution (in other words, it won’t have alternate optimal solutions). The problem will also include a component that involves sensitivity analysis and the use of the shadow price.

You will be turning in two (2) deliverables, a short writeup of the project and the spreadsheet showing your work.

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#### Writeup.

Your writeup should introduce your solution to the project by describing the problem. Correctly identify what type of problem this is. For example, you should note if the problem is a maximization or minimization problem, as well as identify the resources that constrain the solution. Identify each variable and explain the criteria involved in setting up the model. This should be encapsulated in one (1) or two (2) succinct paragraphs.

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After the introductory paragraph, write out the L.P. model for the problem. Include the objective function and all constraints, including any non-negativity constraints. Then, you should present the optimal solution, based on your work in Excel. Explain what the results mean.

Finally, write a paragraph addressing the part of the problem pertaining to sensitivity analysis and shadow price.

#### Excel.

As previously noted, please set up your problem in Excel and find the solution using Solver. Clearly label the cells in your spreadsheet. You will turn in the entire spreadsheet, showing the setup of the model, and the results.

Complete the “Julia’s Food Booth” case problem on page 109 of the text. Address each of the issues A – D according the instructions given.

(A) Formulate and solve an L.P. model for this case.

(B) Evaluate the prospect of borrowing money before the first game.

(C) Evaluate the prospect of paying a friend \$100/game to assist.

(D) Analyze the impact of uncertainties on the model.

## Sample Answer For Linear Programming Case Study

#### Introduction

The problem involves maximizing the profit made from selling four types or colors of nail polish by a beauty salon. The Salon focuses on establishing how much they should stock from each color to be able to maximize the profit. Each type occupies different space and can remain in the display for a particular time period. The question focuses on establishing the number of fire red, bright red, basil green, and basic pink bottles that can be sold to maximize the profit of the beauty salon. It will also focus on establishing the unused space or the idle time that will be recorded while trying to maximize the profit. This is determined using Excel Solver.

#### Linear Programing

Fire red =x, bright red = y, basil green = z, and basic pink =k

For display space: x + 2y + 2z +2k < = 108

For time to set up a display = 3x + 5y + k < = 120

Maximum demand: x + z =25

Minimum demand: y + z + k > 50

Profit Maximization = 100x + 120y + 150z + 125k

Quantity Sold form each Category

• Fire red = 0
• Bright red = 24
• Basil green = 30
• Basic pink = 0

#### Maximum Profit

The salon maximum profit is: 100x + 120y + 150z + 125k = 0 + 2880 + 4500 + 0 =\$7380

#### Unused Space

The maximum available space was provided as 108 and the computed used space was found to be 108. In this regard, there was no any unused space. The entire space was utilized

#### Idle Remaining Time

The maximum available time to be set in the display was said to be 120 minutes. The computed time where the products were set on the display was 120 and thus, there was no any idle time during the entire selling process.

#### Solution Change

Based on the current solution, the salon will not need to sell any fire red bottle to maximize the profit. In this regard, there are no negative change in the fire red that can influence the outcome of the equations.

Nevertheless, the salon depend highly in the sale of basil green and bright red nail polish, the change of profit of basil green bottle sales by even \$1 will greatly change the profit of the company greatly.

#### Profit increase

Based on the evaluation, the salon has utilized all the available space and all the available time for display. Thus, there is no any space remaining or idle time that can be utilized to expand the profitability of the salon.

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