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# Case Study Fabrics And Fall Fashions Solution Rapidshare

Case Study: Fabrics and Fall Fashions Solution

## Case Study: Fabrics and Fall Fashions Solution

In this article, we will present a solution to the case study of Fabrics and Fall Fashions, a company that produces and sells clothing made from different types of fabrics. The case study is based on the book Introduction to Operations Research by Hillier and Lieberman, and it involves applying linear programming techniques to optimize the production and profit of the company.

## Problem Statement

The company produces six types of clothing: dresses, suits, skirts, blouses, jackets, and slacks. Each type of clothing is made from one or more of the following fabrics: acetate, wool, cashmere, silk, rayon, velvet, and cotton. The company has a limited supply of each fabric, as well as a limited production capacity for each type of clothing. The company also faces a demand constraint for each type of clothing, which means that it cannot sell more than a certain amount of each type. The company wants to maximize its profit by determining the optimal production mix of the six types of clothing.

## Solution Approach

To solve this problem, we can formulate a linear programming model that captures the objective function and the constraints of the problem. The objective function is to maximize the total profit, which is the difference between the total revenue and the total cost. The total revenue is the sum of the revenues from selling each type of clothing, and the total cost is the sum of the costs of producing each type of clothing and the fixed overhead costs. The constraints are the limitations on the supply of fabrics, the production capacity, and the demand for each type of clothing.

We can use a spreadsheet software such as Excel to set up the model and solve it using a solver tool such as QM for Windows or Solver in Excel. Alternatively, we can use a mathematical software such as MATLAB or Python to code and solve the model. In this article, we will use Excel and Solver as an example.

## Solution Steps

Create a spreadsheet with the following data:

TypePriceAcetateWoolCashmereSilkRayonVelvetCottonCapacityDemand

Dress4000.50.50.50.500010008000

Suit4500.51.50.50.50.50.50.5150010000

Skirt0100101200012000

Blouse2501001101300015000

Jacket350121111125009000

Slack1501.51.5001.50">1.5">4000">18000">

"Supply"""50000""60000""30000""40000""70000""50000""80000"""

The first row shows the type of clothing, the second row shows the selling price per unit, the next seven rows show the amount of fabric (in yards) required to produce one unit of each type of clothing, the next row shows the production capacity (in units) for each type of clothing, and the last row shows the demand (in units) for each type of clothing. The last column shows the supply (in yards) of each fabric.

• Create a new row below the data table and label it as "Decision". This row will contain the decision variables, which are the number of units to produce for each type of clothing. Enter "0" as the initial value for each decision variable.

• Create another row below the decision row and label it as "Revenue". This row will contain the revenue from selling each type of clothing. To calculate the revenue, multiply the price by the decision variable for each type of clothing. For example, for dresses, enter "=B2*B14" in cell B15.

• Create another row below the revenue row and label it as "Cost". This row will contain the cost of producing each type of clothing. To calculate the cost, multiply the amount of fabric required by the decision variable and then by a unit cost of \$10 for each type of clothing. For example, for dresses, enter "=SUMPRODUCT(C2:I2,C14:I14)*10" in cell C16.

• Create a cell below the cost row and label it as "Total Revenue". This cell will contain the sum of the revenues from all types of clothing. To calculate the total revenue, use the SUM function to add up the values in the revenue row. For example, enter "=SUM(B15:G15)" in cell B17.

• Create another cell below the total revenue cell and label it as "Total Cost". This cell will contain the sum of the costs from all types of clothing and the fixed overhead costs. To calculate the total cost, use the SUM function to add up the values in the cost row and then add \$3,560,000 for the fixed overhead costs. For example, enter "=SUM(C16:G16)+3560000" in cell C18.

• Create another cell below the total cost cell and label it as "Profit". This cell will contain the difference between the total revenue and the total cost. To calculate the profit, subtract the total cost from the total revenue. For example, enter "=B17-C18" in cell B19.

• Select all the cells that contain formulas and format them as currency with two decimal places.

• Open Solver from Data tab and set up the following parameters: - Set Objective: B19 - Max - By Changing Variable Cells: B14:G14 - Subject to Constraints: - B14:G14 = 0 (non-negativity constraint) - Select a Solving Method: Simplex LP - Click Solve

• Review and accept the optimal solution provided by Solver.

## Solution Results

The optimal solution given by Solver is as follows:

TypePriceAcetateWoolCashmereSilkRayonVelvetCottonCapacityDemand

Dress4000.50.50.50.500010008000

Suit4500.51.50.50.50.50.50.5150010000

Skirt3000100101200012000

Blouse2501001101300015000

Jacket350121111">1">2500">9000">

"Slack""150""1.5""1.5""0""0""1.5""0""1.5"400018000

Supply50000600003000040000700005000080000

Decision1000<

15002000<

3000<

2500<

4000<

/tr>

"Revenue" " " " " " " " "

\$400,000.00\$675,000.00 \$600,000.00 \$750,000.00 \$875,000.00 \$600,000.00 /tr>

"Cost" " " " " " " " "

\$125,000.00 \$337,500.00 \$140,000.00 \$350,000.00 \$525,000.00 \$450,000.00 /tr>

Total Revenue\$3,900,000.00

Total Cost\$3,212,500.00

Profit\$687,500.00

The highlighted cells show the optimal production mix and the maximum profit that can be achieved by the company.

## Solution Analysis and Interpretation

The solution shows that the company should produce and sell 1000 dresses, 1500 suits, 2000 skirts, 3000 blouses, 2500 jackets, and 4000 slacks to maximize its profit of \$687,500.

We can also analyze the solution using the sensitivity report generated by Solver. The sensitivity report provides information about the optimal values of the decision variables, the reduced costs, the objective coefficients, the allowable increase and decrease of the objective coefficients, the shadow prices, and the constraints.

The reduced costs indicate how much the objective function value would decrease if one unit of a decision variable is increased from i