The Morton Supply Company produces clothing, footwear, and accessories for dancing and gymnastics. They produce three models of pointe shoes used by ballerinas to balance on the tips of their toes. The shoes are produced from four materials: cardstock, satin, plain fabric, and leather. The number of square inches of each type of material used in each model of shoe, the amount of material available, and the profit/model are shown below:
Material (measured in square inches) |
Model 1 | Model 2 | Model 3 | Material Available |
Cardstock | 12 | 10 | 14 | 1,200 |
Satin | 24 | 20 | 15 | 2,000 |
Plain fabric | 40 | 40 | 30 | 7,500 |
Leather | 11 | 11 | 10 | 1,000 |
Profit per model | $50 | $44 | $40 |
Please help answer the following:
a. Identify the decision variables, objective function, and constraints in simple verbal statements. |
b. Mathematically formulate a linear optimization model. c. Implement the linear optimization model that you developed on a spreadsheet and use Solver to find an optimal solution. Interpret the Solver Answer Report and identify the binding constraints. please show how you entered information in solver for (c) |
Distance variable determines how much shoes of a particular model to produce.
Let the number of shoes produce of model 1, model 2 and model 3 be x1, x2 and x3 respectively.
these are our decision variables. because we have to make a decision on how much shoes to produce a particular kind.
The objective function is determined by the objective of the problem. the objective is to maximize the profit. So, let us form the equation for the profit.
50x1 + 44x2 + 40x3
We have to maximize the above function
Constraints are the amount of material available to us. As the production is restricted by these values.
For example, for card stock equation becomes.
12x1 + 10x2 + 14x3 <=1200
For satin,
24x1 + 20x2 + 15x3 <=2000
Plain fabric
40x1 + 40x2 + 30x3 <= 7500
Leather
11x1 + 11x2 + 10x3 <=1000
another constraint is that all x1,x2 and x3 must be non- negative (>=0) because we cannot produce negative number of models.
Now let's see how to implement this in a spreadsheet.
i have entered 0, 0, 0 for x1, x2 and x3. Solver will automatically give us correct values.
First enter the desired variables and the equations as shown below:
first enter the desired variables and the equations as shown below
Now go to the data tab and click on solver. in case you don't see the solver option you need to enable it from the Excel add in. in the solver dialogue box enter the values as shown below.
to enter the constant you have to click on add button and then choose the respective cells and then the inequality.
make sure that “make unconstrained variables non-negative is ticked
Now click on solve click on Answer in the reports box as shown below and click Ok.
you will get a report as shown below in a new sheet
Upper part show the setting used to solve the equation:
Lower part of the report gives us the values.
now let us interpret the answer report.
As you can see that the original values are 0 0 0 as we had entered.
We are interested in the final values.
so the net profit of the maximum profit that can be there is $4400
number of shoes of model 1 model 2 and model 3 to be produced are 66.67, 0 and 26.67 respectively.
In the constraints table we see how much of the resources are used and how much are left.
Binding means resources that are completely used.
Not binding means we have resources left with us. Slack tells us the number of resources left with us after the production of according to the maximum profit.
If it helps, kindly upvote
comment below for doubt.
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