Business Operations Modeling: Costs, Breakeven, and Decision Analysis

Cost, Revenue, and Demand Calculations

Variable Expenses Formula:
Variable Expenses = $0.75 per Donut × Number of Donuts Produced

Example: If 285.00 units are made and demand is 261.00, your fixed expenses are $1,257.00, your variable cost per unit is $2.25, and your sales price is $6.67. What are the variable expenses for the day?
Calculation: 285.00 × $2.25 = $641.25.

Revenue Formula:
Revenue = Selling Price per Donut × Donuts Sold

Example: If 294.00 units are made and demand is 245.00, your variable cost per unit is $2.25, and your sales price is $6.27. What is the revenue for the day?
Calculation: 245 × $6.27 = $1,536.15.

Inventory Discrepancy Definitions

• Unhappy Customer Expense: Demand is greater than production.
• Leftovers: Demand is less than production.
• Variable Cost: Incurred for what you make, regardless of what you sell.

Example: If 285.00 units are made and demand is 200.00, you can sell leftovers for $2.00 per unit. Your variable expense for each unsatisfied customer is $1.12. Based on this, what will your unhappy customer expense be?
Answer: $0. If 285 units are made and demand is 200 units, there are no unsatisfied customers since supply exceeds demand.

Example: You are able to sell each unit for $14.00. If 216.00 units are made and demand is 185.00, you can sell leftovers for $2.00 per unit. Your variable expense for each unsatisfied customer is $1.12. Based on this, what is your revenue from selling the leftovers?
Step 1: 216 − 185 = 31 leftover units.
Step 2: 31 × $2.00 = $62.00.
Answer: $62.00.

Contribution Margin

Contribution Margin (Per Unit) = Sales Price (per Unit) − Variable Cost (per Unit)

Example: If 286.00 units are made and demand is 287.00, you can sell leftovers for $3.32 per unit. Your expense for each unsatisfied customer is $2.25, your fixed expenses are $1,257.00, your variable cost per unit is $2.42, and your sales price is $6.42. What is the contribution margin?
Calculation: $6.42 − $2.42 = $4.00.

Total Revenue with Leftover Inventory

Revenue = (Full Price Revenue per Unit × Number of Units Sold) + (Reduced Price Revenue per Unit × Number of Items Left Over)

Example: If 279.00 units are made and demand is 264.00, you can sell leftovers for $3.09 per unit. Your variable expense for each unsatisfied customer is $1.23, fixed expenses are $1,257.00, variable cost per unit is $2.25, and sales price is $6.01. What is the total revenue from all revenue sources?
Calculation: 279 − 264 = 15 items leftover.
(6.01 × 264) + (3.09 × 15) = 1,586.64 + 46.35 = $1,632.99.

Decision Analysis: Payoff and Regret Values

Question: What is the payoff of choosing to buy 10 units in a state of nature of "Bad"?
Answer: To find this answer, look at the row for "Buy 10 Units" and locate the value in the "Bad" column, which is −3.0.

Question: What is the regret value of "Sell 10 Units" in the state of nature of "Bad"?

1. Find the best outcome (highest payoff) in the "Bad" column.
2. Subtract the payoff for "Sell 10 Units" in the "Bad" state from this best outcome.

The payoffs in the "Bad" column are: −3.0, 8.1, 0, 5.2.
The highest value (best outcome) is 8.1.
The payoff for "Sell 10 Units" in the "Bad" state is 5.2.
Calculate the Regret: 8.1 − 5.2 = 2.9.
Therefore, the regret value of "Sell 10 Units" in the "Bad" state of nature is 2.9.

Profit, Costs, and Expenses Review

1. To calculate revenue, you need price and quantity sold.

2. Based on the following three expenses:
Rent for building: $1,000.00, Rent for machines: $250.00, Other items: $500.00.
Your total fixed expenses are: $1,750.00.

3. To calculate variable expenses, you need cost per unit and quantity produced.

4. Which expense varies directly with the number of units produced?
Variable Expenses.

5. Which expense varies directly with the number of units sold?
Neither fixed nor variable expenses.
(Explanation: Revenue varies directly with units sold, while expenses may not directly correlate strictly to sales volume when production differs from sales.)

Breakeven and Contribution Margin Analysis

1. At breakeven, Profit = 0.

2. If revenue (price) per donut is $4.00, fixed costs are $1,750, and variable costs are $0.75 per donut, what is the break-even point?
Break-even Point = Fixed Costs / (Price − Variable Cost) = 1,750 / (4 − 0.75) = 539.22 donuts (approximately 540 donuts).
What is the profit at the break-even point?
Profit at Break-even Point = $0.

3. If you graph the profit, revenue, and total expenses equations, which two lines cross at breakeven?
Revenue and Total Expenses lines cross at breakeven.
Which line goes through 0 at breakeven?
The Profit line goes through 0 at breakeven.

4. Based on the above data, what is the contribution margin?
Contribution Margin = Price − Variable Cost = $4.00 − $0.75 = $3.25.

5. What does the contribution margin tell you?
It tells you how much each unit sold contributes to covering fixed costs and generating profit.

6. What is the equation to solve for breakeven using contribution margin?
Breakeven Units = Fixed Costs / Contribution Margin per Unit.

7. A coffee shop, Moonbucks, charges $15 for each cup of coffee. The cost of milk, sugar, coffee powder, etc., is $5/cup. A rent of $500 is paid each month for the place. How many cups of coffee should the restaurant sell in a month to break even?
Breakeven = Fixed Costs / (Price − Variable Cost) = 500 / (15 − 5) = 50 cups of coffee.
How much is the vendor contributing to covering his fixed costs or generating profits (what is the contribution margin)?
Contribution Margin = Price − Variable Cost = $15 − $5 = $10 per cup.

Crossover Point Analysis

1. Explain what the crossover point tells you (helps you decide):
It helps you decide between two alternatives by showing where their costs are equal; beyond this point, one option becomes more cost-effective than the other.

2. How many things are we comparing at the crossover point?
We are comparing two things at the crossover point.

3. What equation do you graph to find the crossover point?
You graph the expense equations to find the crossover point.

4. Based on the following data, what is the crossover point?

• Cake: Machine Cost = $700, Variable Cost = $0.55, Selling Price = $4.50
• Glazed: Machine Cost = $250, Variable Cost = $0.75, Selling Price = $4.00

Crossover Point Calculation:
Set total costs equal:
700 + 0.55x = 250 + 0.75x
450 = 0.20x
x = 2,250 units.

5. Based on the above, what is the profit at the crossover point?
Revenue at Crossover Point: 2,250 × $4.00 = $9,000.00
Total Costs at Crossover Point: 700 + (2,250 × 0.55) = $1,937.50
Profit = Revenue − Total Costs = $9,000 − $1,937.50 = $7,062.50.

6. Can the crossover point be negative?

1. From the equation: Yes, mathematically it can be negative if costs exceed revenue consistently.
2. In reality: Negative production does not make sense; thus it cannot exist in practical scenarios.
3. What does it mean? A negative result indicates that one option is consistently more costly than another without any feasible break-even point.

Profit Modeling Under Unequal Production and Demand

1. What assumption that applied in previous sections no longer applies to calculations in this section?
The assumption that production equals demand no longer applies; we now consider scenarios where they differ significantly.

2. What does the cost of lost sales represent and when do we have lost sales?
Cost of lost sales represents potential revenue lost when demand exceeds supply; we have lost sales when demand exceeds what can be produced or supplied.

3. When do we have times that we have to sell at a reduced cost?
We have to sell at a reduced cost when supply exceeds demand and excess inventory needs to be cleared out quickly to avoid spoilage or obsolescence.

4. Kiana sells donuts for $4.00. Kiana produces 100 donuts one day; however, she forgets it is spring break and therefore demand is down; she only sells 80 donuts. At the end of the day, she brings the extra donuts to a day-old donut shop that pays her $0.25 per donut. If Kiana has lost sales, she charges herself an expense of $0.50. Her monthly fixed expenses are $1,750, and her variable expenses are $0.75 per unit.

1. Since we are calculating daily profits, what should she use for her fixed expenses (assuming there are 25 days in a month)?
   Daily Fixed Expenses = $1,750 / 25 days = $70.00/day.
2. What is her profit for this day?
   Revenue from sales: (80 × $4.00) + (20 × $0.25) = $320 + $5 = $325.00.
   Total expenses: $70 + (100 × $0.75) + (20 × $0.50) = $70 + $75 + $10 = $155.00.
   Profit for this day: $325.00 − $155.00 = $170.00.

5. Using information from the above problem, one day Kiana makes 100 donuts, but 120 people come wanting to buy her donuts; she has to send some of them away unhappy.

1. What is her profit for this day?
   Revenue from sales: 100 × $4.00 = $400.00.
   Total expenses: $70 + (100 × $0.75) + (20 × $0) [no leftover clearance expense] = $70 + $75 = $145.00.
   Profit for this day: $400.00 − $145.00 = $255.00.

Point of Indifference: Location Decision

To determine the point of indifference between staying at the current location and moving to the new location, calculate the number of services (or customers) needed each month to make profits at both locations equal.

Current Location

• Rent & Utilities: $5,528.00
• Salaries: $6,297.00
• Insurance: $1,778.00
• Total Fixed Costs: $5,528.00 + $6,297.00 + $1,778.00 = $13,603.00
• Supplies (Variable Cost per Service): $2.00
• Revenue per Service: $61.00
• Profit per Service: $61.00 − $2.00 = $59.00
• Profit Equation: 59x − 13,603

New Location

• Rent & Utilities: $10,184.00
• Salaries: $6,744.00
• Insurance: $2,111.00
• Total Fixed Costs: $10,184.00 + $6,744.00 + $2,111.00 = $19,039.00
• Supplies (Variable Cost per Service): $7.00
• Revenue per Service: $149.00
• Profit per Service: $149.00 − $7.00 = $142.00
• Profit Equation: 142x − 19,039

Indifference Calculation

Set total profit equations equal to each other:
59x − 13,603 = 142x − 19,039
142x − 59x = 19,039 − 13,603
83x = 5,436
x = 5,436 / 83 ≈ 65.49 services.

Probability and Simulation with VLOOKUP

Probability = Number of Favorable Outcomes / Sum of All Possible Outcomes

Simulation Problem 1: 21 Total Donuts

Step 1: Calculate the Total Number of Donuts Purchased

• Chocolate Cream Filled: 4
• Strawberry: 2
• Kale Frosted: 4
• Blueberry: 1
• Cream Filled Surprise: 10
• Total Donuts: 4 + 2 + 4 + 1 + 10 = 21

Step 2 & 3: Individual and Cumulative Probability Table

Step 4: Match Random Number to Cumulative Probability
The random number provided is 0.64:

• Chocolate Cream Filled: 0 to 0.1905 (Does not match)
• Strawberry: 0.1905 to 0.2857 (Does not match)
• Kale Frosted: 0.2857 to 0.4762 (Does not match)
• Blueberry: 0.4762 to 0.5238 (Does not match)
• Cream Filled Surprise: 0.5238 to 1.0000 (Matches, since 0.64 is between 0.5238 and 1.0000)

Simulation Problem 2: 30 Total Donuts

Step 1: Total Donuts Purchased
Total = 5 + 9 + 1 + 7 + 8 = 30.

Step 2 & 3: Probabilities and Cumulative Distribution

• Chocolate Cream Filled: 5/30 ≈ 0.1667; Cumulative: 0.1667
• Strawberry: 9/30 = 0.3000; Cumulative: 0.1667 + 0.3000 = 0.4667
• Kale Frosted: 1/30 ≈ 0.0333; Cumulative: 0.4667 + 0.0333 = 0.5000
• Blueberry: 7/30 ≈ 0.2333; Cumulative: 0.5000 + 0.2333 = 0.7333
• Cream Filled Surprise: 8/30 ≈ 0.2667; Cumulative: 0.7333 + 0.2667 = 1.0000

Step 4: Create the VLOOKUP Table
The first column contains the lower limit of each cumulative probability range:

Step 5: Answer the Question
What value goes in the 1st column for the row containing Kale Frosted?
Final Answer: 0.4667.

Simulation Problem 3: VLOOKUP Range Intervals

Using the 21-donut sample:

Step 5 & 6: Determine Range for Strawberry
The cumulative probability for Strawberry is between 0.1905 and 0.2857. Therefore, a random number between 0.1905 and 0.2857 returns a Strawberry donut.
Matching the provided options, the closest range is c) 0.19 to 0.29.