Spreadsheet-based 0ptimization

Spreadsheet-based 0ptimization

Objective: Execute the optimization of profit functions using the Excel spreadsheet.

With modern

spreadsheets,

optimization is a snap

Problem : Maximizing profits from the sale of microchips

Our inverse demand function for microchips [1]:P = 170 – 20Q

The Revenue (R) function is given by:R = P · Q = (170 – 20Q)Q = 170Q – 20Q2

Thus marginal revenue (MR) is given by dR/dQ = 170 –40Q

The cost function (C) is given by [2]C = 100 + 38Q

Thus the marginal cost (MC) function is given by:dC/dQ = 38

The profit () function is given by:R – C = 170Q – 20Q2 –(100 + 38Q) = 132Q – 20Q2 –

100Thus the marginal profit function (M) is given by:

d /dQ = 132 – 40Q

A B C D E F G12 The Optimal Output of Microchips345 Quantity Price Revenue Cost Profit67891011121314

Step 1: Set up a spreadsheet like this

A B C D E F G12 The Optimal Output of Microchips345 Quantity Price Revenue Cost Profit67 2.0891011121314

Step 2: Type the number “2.0” in cell b7

Step 3: Move your cursor to cell c7 and type the following in the formula bar: =170-20*b7Hit “enter” or click right on the green check mark to the left of the formula bar. Now your spreadsheet should look like this:

A B C D E F G12 The Optimal Output of Microchips345 Quantity Price Revenue Cost Profit67 2.0 130891011121314

Step 4: Move your cursor to cell d7 and type the following in the formula bar: =b7*c7Hit “enter” or click right on the green check mark to the left or the formula bar. Now your spreadsheet should look like this:

A B C D E F G12 The Optimal Output of Microchips345 Quantity Price Revenue Cost Profit67 2.0 130 260891011121314

Step 5: Move your cursor to cell e7 and type the following in the formula bar: =100+38*b7Hit “enter” or click right on the green check mark to the left or the formula bar. Now your spreadsheet should look like this:

A B C D E F G12 The Optimal Output of Microchips345 Quantity Price Revenue Cost Profit67 2.0 130 260 176891011121314

Step 6: Move your cursor to cell f7 and type the following in the formula bar: =d7-e7Hit “enter” or click right on the green check mark to the left or the formula bar. Now your spreadsheet should look like this:

A B C D E F G12 The Optimal Output of Microchips345 Quantity Price Revenue Cost Profit67 2.0 130 260 176 84891011121314

3 ways to maximize profits ()

Now we will show you 3 methods of maximizing the profit function using Excel.

Method 1: Change the value of the number in cell b7 until you find the highest corresponding value in cell f7.

Example: Enter the number “3.0” in cell b7. Notice that profit increases to 116.

A B C D E F G12 The Optimal Output of Microchips345 Quantity Price Revenue Cost Profit67 3.0 110 330 214 116891011121314

Method 2: Use MR and MC as guides. Vary the numerical values in cell b7 until MR =MC (or alternatively, Mprofit = 0).Example: Enter the number “3.0” in cell b7. Notice that profit increases to 116.

A B C D E F G12 The Optimal Output of Microchips345 Quantity Price Revenue Cost Profit67 3.0 110 330 214 116891011121314

Method 2: Use MR and MC as guides. Vary the numerical values in cell b7 until MR =MC (or alternatively, Mprofit = 0).

A B C D E F G12 The Optimal Output of Microchips345 Quantity Price Revenue Cost Profit67 2.0 130 260 176 8489101112 MR MC Mprofit1314

Step 1: Type MR, MC, and Mprofit into cells d12, e12, and f12 respectively

A B C D E F G12 The Optimal Output of Microchips345 Quantity Price Revenue Cost Profit67 2.0 130 260 176 8489101112 MR MC Mprofit1314 90

Step 2: To compute MR when quantity is equal to 2 lots, place your cursor in cell d14 and type the following in the formula bar: =170-40*b7

Your spreadsheet should look like this:

A B C D E F G12 The Optimal Output of Microchips345 Quantity Price Revenue Cost Profit67 2.0 130 260 176 8489101112 MR MC Mprofit1314 90 38 52

Step 3: Note that MC = 38, so type this into cell e14.To compute marginal profit (Mprofit) move your cursor to cell f14 and type the following in the formula bar: =132-40*b7

Your spreadsheet should now look like this:

A B C D E F G12 The Optimal Output of Microchips345 Quantity Price Revenue Cost Profit67 3.0 130 390 214 17689101112 MR MC Mprofit1314 50 38 12

Step 4: Now adjust the numerical values in cell b7 until MR = MC, or Mprofit = 0.

Example: Type “3.0” in cell b7. Your spreadsheet should now look like this:

Method 3: Use the Excel “solver” function

1. Move your cursor to cell f7.

2. From the “tools” menu select “solver”. You should see a dialog box like this:

Solver Parameters

Solve

Close

Options

Set Target Cells $F$7Equal to Max Min

Add

Change

Delete

Subject to Constraints:

By Changing Cells:

1. Notice that the default is “Max”—that’s OK– we are trying a maximize a profit function.

2. In the “By Changing Cells” space type: $B$7. Remember we are seeking to find the profit maximizing output-price combination.

3. Now click on the “solve” button.

Solver Parameters

Solve

Close

Options

Set Target Cells $F$7Equal to Max Min

Add

Change

Delete

$B$7Subject to Constraints:

By Changing Cells:

A B C D E F G12 The Optimal Output of Microchips345 Quantity Price Revenue Cost Profit67 3.3 104 343.2 225.4 117.889101112 MR MC Mprofit1314 38 38 0

The solver function found the profit maximizing output (3.3 lots) and price ($104,000 per lot).

Constrained optimization

Suppose we are seeking to maximize profits subject to the constraint that our

price per lot cannot exceed $91,000—that is:

P 91

1. Move your cursor to cell f7 and access the “solver” dialog box from the tools menu.

2. Now click on the add button and you will find a dialog box (something) like this:

3. Type c7 into Cell Reference space and 91 into constraint space. Now click on OK

Add Constraint

c7

OK Cancel Add

Cell Reference:

<=

Constraint:

91

Help

Note: <= is the default, which works in our case.

A B C D E F G12 The Optimal Output of Microchips345 Quantity Price Revenue Cost Profit67 4.0 91 359.45 250.1 109.3589101112 MR MC Mprofit1314 12 38 -26

The solver function found the output (4.0 lots) that maximizing profits subject to the price constraint.