Maximizing Profit: Evaluating an Equation for Selling Cookies at a Fundraiser

[Mk]

This question should be pretty easy compared to others. Can someone help me evaluate an equation for finding the profit after selling cookies for 75¢. I will make as many as I can make, with a budget of $100, and sell them all.

Each batch needs:

• 2 pounds of flour
• 4 eggs
• 1 pound of sugar
• 4 cups of milk

Wholesale costs:

• $8 per 2 pounds of flour
• $2 per 12 eggs
• $8 per 1 pound of sugar
• $3 per 16 cups of milk

I tried to find a relationship between the costs over time, but mit was inconclusive. Costs $21 for the first batch, $8 for the second, $8 for the 3rd, $10 for the 4th, $8 for the 5th. Theres a lot more different ingredients than amount of batches.
Remember that I can use surplus materials from the last batch that weren't used.

Cost per cookie times number of cookies equals income. Profit equals the income minus the cost of ingredients. But, I don't know how many batches, so I don't know how many cookies.

I give much thanks for your future contributions.

 

[honestrosewater]

Each batch uses 1 unit of flour @ $8/unit, 1/3 unit of eggs @ $2/unit, 1 unit sugar @ $8/unit, and 1/4 unit of milk @ $3/unit. Total budget is $100. If you make 4 batches, you'll need
4 u of f, 4/3 u of e, 4 u of s, 4/4 u of m.
You can't buy a fraction of unit, so you need to round up to the next integer:
4(8) + 2(2) + 4(8) + 1(3) = $71.
Keep doing this until you reach or pass $100. (I would just pay the extra $6.)

 

[owl3951]

Although honestrosewater's iterative solution will tell you how many batches you can make for $100, nothing can tell you the profit you will make unless you specify how many cookies are in a batch. This is because your income seems to be based upon a per-cookie, and not a per-batch, price. If 75 cents is your per-batch price, you need a bigger truck, if you are familiar with that joke.

"Cost per cookie times number of cookies" does not equal "income," as you have said. It equals gross cost. Gross income is 75 cents per cookie times the total number of cookies sold. Profit is gross income less gross cost.

Until we know the number of cookies in a batch, we cannot multiply the 75 cents times anything, to get gross income.

The problem you have is a typical problem in the field of operations research, in which you try to maximize profit (or minimize cost, travel time, etc.) subject to a series of constraints. Typically the constraints are a series of equations suitable for a matrix, and include both equalities (e.g., gross income = 75 cents times number of cookies sold) and inequalities (e.g., gross cost <= $100). The method of finding the optimal solution is called linear programming.

You can google "operations research" or "quantitative decision making" or "linear programming" and probably come up with websites that will demonstrate for you how these problems are solved. One such site is listed below.

http://commerce.concordia.ca/bourjolly/lp.html#Begin

 

[honestrosewater]

Figuring the profit would be easy if your variable costs were, eh, constant. You just find the cost per unit (cookie). The production cost C is a function of volume (units produced). C(x) = (fixed costs) + (variable costs)x. Total revenue R(x) = (unit price)x. If you sell all units produced, profit P(x) = R(x) - C(x).
Assuming this is a real problem (though $8 for a pound of sugar?! Sugar is about $.40 per pound in the US!), here's a quick run-through. You presumably have no fixed costs to speak of, and you've determined the unit price, so you just need to find the variable cost, which is just your ingredients. Recipes with similar ratios of ingredients yield 5 dozen cookies for 5 cups of flour. 1 cup = 1/2 pound, so let's say your batches yield 4 dozen cookies each. 5 batches = 20 dozen units = 240 units @ $90 is $.375 per unit. This may seem like a step backwards, but it isn't- you need volume to find revenue. Anywho, x = 240, C(240) = .375(240) = 90, R(240) = .75(240) = 180, P(240) = 180 - 90 = 90. So your profit at 240 units is $90 (using the figures I picked, of course). Keep in mind that your variable costs will vary, so your cost per unit will vary as well- you'll need to find the new variable cost for a new number of batches. Have fun rolling out those 200+ cookies!

 

[Mk]

To clear things up, there are 36 cookies per batch, and sorry, sugar is 8 dollars per 5 pounds.

 

[honestrosewater]

Well, the problem has been worked out for other values. How far can you get trying to solve for the new values?

 

[Mk]

C(240) = .375(240) = 90, R(240) = .75(240) = 180, P(240) = 180 - 90 = 90. So your profit at 240 units is $90 (using the figures I picked, of course).

Sorry, where did the $90 come from before you knew it was the profit? And what about 180?

 

[honestrosewater]

Sorry, where did the $90 come from before you knew it was the profit? And what about 180?

90 was the cost of producing 5 batches. 180 was total revenue.

Here's a summary of what you're doing:
Let f(x) = smallest integer > x. Find the greatest b such that
1) f(b)*8 + f(b/5)*8 + f(b/4)*3 + f(b/3)*2 < 100
2) x = 36*b
3) C(x) = f(b)*8 + f(b/5)*8 + f(b/4)*3 + f(b/3)*2
4) R(x) = .75*x
5) P(x) = R(x) - C(x)
(1) tells you how many batches to make
(2) tells you how many cookies you're making. Of course, you could just figure everything for batches- just use price per batch instead of per cookie. Anyway...
(3) your production costs
(4) your total revenue (income)
(5) profit IF you sell every cookie you produce