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
Decision100015002000300025004000
"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 its optimal value, while keeping the other decision variables at their optimal values and satisfying the constraints. A reduced cost of zero means that the decision variable is at its optimal value and increasing it would not affect the objective function value. A positive reduced cost means that the decision variable is at its lower bound (zero) and increasing it would decrease the objective function value. A negative reduced cost means that the decision variable is at its upper bound (capacity or demand) and increasing it would increase the objective function value.
The objective coefficients indicate how much the objective function value would increase if one unit of a decision variable is increased from its optimal value, while keeping the other decision variables at their optimal values and satisfying the constraints. The objective coefficients are equal to the selling prices of each type of clothing.
The allowable increase and decrease of the objective coefficients indicate how much the objective coefficients can change without changing the optimal solution. If the objective coefficient of a decision variable increases by more than its allowable increase or decreases by more than its allowable decrease, then the optimal solution will change.
The shadow prices indicate how much the objective function value would increase if one unit of a right-hand side value of a constraint is increased, while keeping the other right-hand side values at their original values and satisfying the constraints. A shadow price of zero means that the constraint is not binding and increasing its right-hand side value would not affect the objective function value. A positive shadow price means that the constraint is binding and increasing its right-hand side value would increase the objective function value. A negative shadow price means that the constraint is binding and decreasing its right-hand side value would increase the objective function value.
The constraints indicate the left-hand side values, the right-hand side values, and the slack or surplus values of each constraint. The slack or surplus value is the difference between the right-hand side value and the left-hand side value of a constraint. A slack or surplus value of zero means that the constraint is binding and equal to its right-hand side value. A positive slack or surplus value means that the constraint is not binding and less than its right-hand side value. A negative slack or surplus value means that the constraint is not binding and greater than its right-hand side value.
The sensitivity report for this problem is shown below:
VariableCellsFinal ValueReduced CostObjective CoefficientAllowable IncreaseAllowable Decrease
DressB14100004005050
SuitC14150004505050
SkirtD14200003005050
BlouseE14300002505050
JacketF14250003505050
SlackG14400001505050
ConstraintsNameCelFina ValuSado PricRigh-Han SidAllwabl IncreasAllwabl DecreasCacit DresB13=1000-=1000==
"Capacity Suit" "C13" "=1500" "-" "=1500" "=" "="
"Capacity Skirt" "D13" "=2000" "-" "=2000" "=" "="
"Capacity Blouse" "E13" "=3000" "-" ==3000==
Capacity JacketF13=2500-=2500==
Capacity SlackG13=4000-=4000==
Demand DressB12=100050=800070001000
Demand SuitC12=150050=1000085001500
Demand SkirtD12=200050100002000
Demand BlouseE12=300050=15000120003000
Demand JacketF12=250050=900065002500
Demand SlackG12=400050=18000140004000
Acetate SupplyC11=50000-=50000==
"Wool Supply" "D11" "=60000" "-" "=60000" "=" "="
"Cashmere Supply" "E11" "=30000" "-" "=30000" "=" "="
"Silk Supply" "F11" "=40000" "-" ==
Rayon SupplyG11=70000-=70000==
Velvet SupplyH11=50000-=50000==
Cotton SupplyI11=80000-=80000==
The sensitivity report shows that all the decision variables have a reduced cost of zero, which means that they are at their optimal values and increasing them would not affect the profit. The objective coefficients are equal to the selling prices of each type of clothing, and they have an allowable increase and decrease of 50, which means that the selling prices can change by up to 50 without changing the optimal solution.
The sensitivity report also shows that all the capacity and supply constraints have a shadow price of zero, which means that they are not binding and increasing their right-hand side values would not affect the profit. The demand constraints have a shadow price of 50, which means that they are binding and increasing their right-hand side values by one unit would increase the profit by 50. The demand constraints also have an allowable increase and decrease that indicate how much the demand can change without changing the optimal solution.
The sensitivity report can help the company to analyze the effects of changing the parameters of the problem on the optimal solution and the profit. For example, the company can use the sensitivity report to answer questions such as:
How much would the profit increase if the demand for dresses increases by 500 units?
How much would the profit decrease if the supply of wool decreases by 10000 yards?
How much can the selling price of jackets increase or decrease without changing the optimal production mix?
How much slack or surplus is there in each constraint?
Which constraints are more critical and which are more flexible?
Solution Conclusion and Recommendations
In conclusion, we have solved the case study of Fabrics and Fall Fashions using linear programming techniques and spreadsheet software. We have found the optimal production mix and the maximum profit that can be achieved by the company. We have also generated a sensitivity report that provides useful information for analyzing and interpreting the solution.
Based on our solution, we can make some recommendations for the company to improve its performance and profitability. Some of these recommendations are:
The company should focus on producing and selling more suits, skirts, blouses, and jackets, as they have higher profit margins and higher demand than dresses and slacks.
The company should try to increase its production capacity for suits, skirts, blouses, and jackets, as they are currently limited by their demand constraints.
The company should try to reduce its fixed overhead costs, as they account for a large portion of its total cost.
The company should monitor the changes in the market conditions, such as the selling prices, the demand, and the supply of fabrics, and use the sensitivity report to adjust its production plan accordingly.
The company should explore other types of fabrics and clothing that might have higher profitability and customer preference.
We hope that this article has provided a clear and comprehensive solution to the case study of Fabrics and Fall Fashions. We also hope that this article has demonstrated how linear programming can be applied to real-world problems in operations research.
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