Business Applications for MATH 122
Exam Relevance for MATH 122
Marginal cost/revenue problems appear regularly. Applied optimization in business contexts.
Introduction: What You Need to Know for the Exam
Important: Economic Lot Size and Economic Order Quantity are essentially the same mathematical problem with the same formula, just applied to different scenarios:
-
Economic Lot Size (Manufacturing): Company makes products. Question: How many units should we produce in each batch? Fixed cost = setup cost (retooling machines, cleaning, calibration).
-
Economic Order Quantity (Purchasing/Retail): Company buys products from suppliers. Question: How many units should we order each time? Fixed cost = ordering cost (shipping, paperwork, processing).
Both use the same core formula: $q = \sqrt{\frac{2fM}{k}}$
Exam Strategy: You will most likely get an exam question on Elasticity of Demand. You probably won't see a full calculation question on lot size/order quantity, but you could get short answer or multiple choice questions testing your understanding of the concepts. You MUST absolutely know Elasticity - it's very likely you will get a question on elasticity on the exam.
Economic Lot Size (Inventory Problem)
What is it? The word "lot" here means a batch or group of items, NOT a plot of land! Companies that manufacture products need to decide how many units to produce in each batch (each "lot"). Making too many at once means high storage costs. Making too few means you're constantly setting up production, which wastes money.
The Goal: Find the perfect batch size ($q$) that minimizes total costs.
Key Variables:
- $q$ = batch size (number of units produced each time - this is what we're solving for!)
- $M$ = total units needed per year
- $f$ = fixed setup cost per batch (doesn't change with batch size)
- $g$ = variable cost per unit (cost to make one item)
- $k$ = storage cost per unit per year
The Math: We create a total cost function $T(q)$ that adds up:
- Manufacturing costs: $gM$ (making all the units)
- Setup costs: $\frac{fM}{q}$ (number of batches × cost per batch)
- Storage costs: $\frac{kq}{2}$ (average inventory × storage cost)
Then we use derivatives to find the minimum.
A manufacturer has a steady annual demand for 15,000 boxes of candy bars. The costs are:
- Storage cost: 12 dollars to store 1 box for 1 year
- Setup cost: 40 dollars to produce each batch (machinery setup, cleaning, etc.)
- Production cost: 18 dollars to produce each box (ingredients, packaging, labor)
Find the number of boxes per batch that should be produced.
Step 1: Identify our variables
- $M = 15{,}000$ boxes per year (total annual demand)
- $k = 12$ dollars per box per year (storage cost)
- $f = 40$ dollars per batch (setup cost)
- $g = 18$ dollars per box (production cost)
- $q = ?$ boxes per batch (what we're solving for)
Step 2: Write the total cost function
$$T(q) = \frac{fM}{q} + gM + \frac{kq}{2}$$
Substituting our values:
$$T(q) = \frac{(40)(15{,}000)}{q} + (18)(15{,}000) + \frac{12q}{2}$$
$$T(q) = \frac{600{,}000}{q} + 270{,}000 + 6q$$
Step 3: Find the derivative
$$T'(q) = -\frac{600{,}000}{q^2} + 6$$
Step 4: Set the derivative equal to zero and solve
$$-\frac{600{,}000}{q^2} + 6 = 0$$
$$6 = \frac{600{,}000}{q^2}$$
$$6q^2 = 600{,}000$$
$$q^2 = 100{,}000$$
$$q = \sqrt{100{,}000} = 316.23$$
Step 5: Use the formula (shortcut method)
We can also use the economic lot size formula directly:
$$q = \sqrt{\frac{2fM}{k}} = \sqrt{\frac{2(40)(15{,}000)}{12}} = \sqrt{\frac{1{,}200{,}000}{12}} = \sqrt{100{,}000} \approx 316.23$$
Answer: The factory should produce approximately 316 boxes per batch to minimize costs.
What does this mean in practice?
- The factory needs 15,000 boxes per year
- If they make 316 per batch, they'll run about $\frac{15{,}000}{316} \approx 47$ batches per year
- This balances the cost of frequent setups against the cost of storing too many boxes
Economic Order Quantity (Reorder Problem)
What is it? This is similar to the lot size problem, but for companies that buy products instead of making them. How often should you reorder, and how many units should you order each time?
The Goal: Find the perfect order size ($q$) that minimizes total ordering and storage costs.
Key Variables:
- $q$ = order quantity (units per order)
- $M$ = total units needed per year
- $f$ = fixed cost per order (shipping, paperwork, etc.)
- $k$ = storage cost per unit per year
The Math: Almost identical to the lot size problem! The formula is the same: $q = \sqrt{\frac{2fM}{k}}$. The only difference is we don't include the $gM$ term because the per-unit purchase cost doesn't change with order size.
A bookstore has an annual demand for 120,000 copies of a best-selling novel. The costs are:
- Storage cost: 0.40 dollars to store 1 copy for 1 year
- Ordering cost: 75 dollars to place an order (shipping, paperwork, processing fees)
Find the optimum number of copies per order.
Step 1: Identify our variables
- $M = 120{,}000$ copies per year (total annual demand)
- $k = 0.40$ dollars per copy per year (storage cost)
- $f = 75$ dollars per order (ordering cost)
- $q = ?$ copies per order (what we're solving for)
Step 2: Write the total cost function
For ordering problems, we don't include the purchase cost since it doesn't change with order size:
$$T(q) = \frac{fM}{q} + \frac{kq}{2}$$
Substituting our values:
$$T(q) = \frac{(75)(120{,}000)}{q} + \frac{0.40q}{2}$$
$$T(q) = \frac{9{,}000{,}000}{q} + 0.20q$$
Step 3: Find the derivative
$$T'(q) = -\frac{9{,}000{,}000}{q^2} + 0.20$$
Step 4: Set the derivative equal to zero and solve
$$-\frac{9{,}000{,}000}{q^2} + 0.20 = 0$$
$$0.20 = \frac{9{,}000{,}000}{q^2}$$
$$0.20q^2 = 9{,}000{,}000$$
$$q^2 = 45{,}000{,}000$$
$$q = \sqrt{45{,}000{,}000} \approx 6{,}708$$
Step 5: Use the formula (shortcut method)
We can also use the economic order quantity formula directly:
$$q = \sqrt{\frac{2fM}{k}} = \sqrt{\frac{2(75)(120{,}000)}{0.40}} = \sqrt{\frac{18{,}000{,}000}{0.40}} = \sqrt{45{,}000{,}000} \approx 6{,}708$$
Answer: The bookstore should order approximately 6,708 copies per order to minimize costs.
What does this mean in practice?
- The bookstore needs 120,000 copies per year
- If they order 6,708 copies at a time, they'll place about $\frac{120{,}000}{6{,}708} \approx 18$ orders per year
- This balances the cost of frequent ordering against the cost of storing too many books
Elasticity of Demand
What is it? Elasticity measures how sensitive customers are to price changes. If you raise the price by 10%, will demand drop by 5%? By 20%? This ratio tells you.
The Goal: Understand whether demand is elastic (customers are very price-sensitive) or inelastic (customers will buy anyway).
Key Variables:
- $E$ = elasticity of demand
- $p$ = price
- $q$ = quantity demanded (or $q = f(p)$ as a function)
- $\frac{dq}{dp}$ = rate of change of demand with respect to price
The Formula:
$$E = -\frac{p}{q} \cdot \frac{dq}{dp}$$
The Three Cases:
-
Inelastic Demand ($E < 1$): Customers aren't very price-sensitive. Even if there's a big change in price, customers will still buy these products because they need them. Examples: gasoline, medicine, electricity.
-
Elastic Demand ($E > 1$): Customers are very price-sensitive. If there's a big change in price, customers will definitely change their purchasing habits and stop buying this product (or buy much less). Examples: luxury goods, restaurant meals, concert tickets.
-
Unit Elastic ($E = 1$): The percent change in price exactly equals the percent change in demand. This is a special "balanced" case.
Why the negative sign? Price and demand move in opposite directions (higher price → lower demand), so we use a negative sign to make $E$ positive.
The short-term demand for crude oil in the United States in 2008 can be approximated by:
$$q = f(p) = 2{,}431{,}129p^{-0.06}$$
where $p$ represents the price of crude oil in dollars per barrel and $q$ represents the per capita consumption of crude oil. Calculate and interpret the elasticity of demand when the price is 40 dollars per barrel.
Step 1: Identify what we know
- Demand function: $q = 2{,}431{,}129p^{-0.06}$
- Price we're evaluating at: $p = 40$ dollars per barrel
- We need to find: $E$ (elasticity) at this price
Step 2: Find the derivative $\frac{dq}{dp}$
Using the power rule:
$$\frac{dq}{dp} = 2{,}431{,}129 \cdot (-0.06)p^{-0.06-1}$$
$$\frac{dq}{dp} = -145{,}867.74p^{-1.06}$$
Step 3: Find $q$ when $p = 40$
$$q = 2{,}431{,}129(40)^{-0.06}$$
$$q = 2{,}431{,}129 \cdot 0.7237$$
$$q \approx 1{,}759{,}458 \text{ (per capita consumption)}$$
Step 4: Find $\frac{dq}{dp}$ when $p = 40$
$$\frac{dq}{dp} = -145{,}867.74(40)^{-1.06}$$
$$\frac{dq}{dp} = -145{,}867.74 \cdot 0.01809$$
$$\frac{dq}{dp} \approx -2{,}639$$
Step 5: Calculate elasticity using the formula
$$E = -\frac{p}{q} \cdot \frac{dq}{dp}$$
$$E = -\frac{40}{1{,}759{,}458} \cdot (-2{,}639)$$
$$E = \frac{40 \cdot 2{,}639}{1{,}759{,}458}$$
$$E = \frac{105{,}560}{1{,}759{,}458}$$
$$E \approx 0.06$$
Answer: The elasticity of demand is approximately 0.06 when crude oil is 40 dollars per barrel.
Interpretation: Since $E = 0.06 < 1$, crude oil has highly inelastic demand. This makes sense! Even if the price of gas changes significantly, people still need to drive to work, heat their homes, and transport goods. A 10% increase in price would only decrease demand by about 0.6% - people will pay more rather than stop using oil. This is why oil-producing countries can raise prices without losing much business.
Economic Lot Size / Economic Order Quantity
Minimizes total cost by balancing setup/ordering costs and storage costs. Same formula works for both manufacturing (lot size) and purchasing (order quantity) problems!
Variables:
- $q$:
- optimal batch/order size (units per production run or order)
- $f$:
- fixed cost per batch or order
- $M$:
- total annual demand (units per year)
- $k$:
- storage cost per unit per year
Total Cost Function (Manufacturing/Lot Size)
Total annual cost for manufacturing: setup costs + production costs + storage costs
Variables:
- $T(q)$:
- total annual cost
- $\frac{fM}{q}$:
- setup costs (number of batches × cost per batch)
- $gM$:
- total manufacturing cost (only for production, not ordering)
- $\frac{kq}{2}$:
- storage costs (average inventory × storage cost)
- $q$:
- batch size
Total Cost Function (Ordering/Reordering)
Total annual cost for ordering: ordering costs + storage costs (no gM term since purchase cost is constant)
Variables:
- $T(q)$:
- total annual cost
- $\frac{fM}{q}$:
- ordering costs (number of orders × cost per order)
- $\frac{kq}{2}$:
- storage costs (average inventory × storage cost)
- $q$:
- order quantity
Elasticity of Demand
Measures how responsive quantity demanded is to price changes. E < 1 is inelastic (customers keep buying), E > 1 is elastic (customers stop buying), E = 1 is unit elastic
Variables:
- $E$:
- elasticity of demand
- $p$:
- price
- $q$:
- quantity demanded
- $\frac{dq}{dp}$:
- rate of change of demand with respect to price (always negative)
This skill is taught in the following courses. Create an account to access practice exercises and full course materials.