# EOQ with Nonlinear Cost Fred inventory scenario is a variant of the classic Economic Order Quantity (EOQ) problem, but **the holding cost is nonlinear**: it is proportional to the square of the average inventory level. ## Given Data - Weekly demand: \( 10 \) snakes/week - Weeks per year: \( 50 \) - **Annual demand:** \( D = 10 \times 50 = 500 \) snakes/year - **Cost per snake:** \( c = \$100 \) - **Order cost:** \( S = \$500 \) per order - **Holding cost rate:** \( 20\% \) of purchase price per year - **Holding cost function:** \( H \times \left(\frac{Q}{2}\right)^2 \) per year, where \( Q \) is the order quantity and \( H = \$20 \) ## Total Cost Function The **total annual cost** (\( TC \)) is the sum of: 1. **Ordering cost:** \( \frac{D}{Q}S \) 2. **Purchase cost:** \( Dc \) (irrelevant for optimization) 3. **Holding cost:** \( H\left(\frac{Q}{2}\right)^2 \) So, \[ TC(Q) = \frac{D}{Q}S + H\left(\frac{Q}{2}\right)^2 \] Substitute the given values: \[ TC(Q) = \frac{500}{Q} \times 500 + 20 \left( \frac{Q}{2} \right)^2 \] \[ TC(Q) = \frac{250{,}000}{Q} + 5Q^2 \] ## Finding the Optimal Order Quantity Set the derivative of \( TC(Q) \) with respect to \( Q \) to zero to find the minimum: \[ \frac{dTC}{dQ} = -\frac{250{,}000}{Q^2} + 10Q = \] Solve for \( Q \): \[ 10Q = \frac{250{,}000}{Q^2} \] \[ 10Q^3 = 250{,}000 \] \[ Q^3 = 25{,}000 \] \[ Q^* = (25{,}000)^{1/3} \] ## Final Summary - **The equation to find Fred's optimal order quantity is:** \[ TC(Q) = \frac{250{,}000}{Q} + 5Q^2 \] - **To find the optimal \( Q^* \), solve:** \[ 10Q^3 = 250{,}000 \] \[ Q^* = (25{,}000)^{1/3} \] - **Fred should use this cubic root equation to compute his optimal order size.** --- **Note:** If you want the numerical value: \[ Q^* \approx 29.4 \] snakes per order (rounded to one decimal place).

# EOQ Calculation for Fred's Snake Store Fred's Snake Store requires finding the optimal order quantity for rattlesnakes, given specific demand and cost parameters. ## Given Data - **Weekly demand:** \( 10 \) snakes/week - **Weeks per year:** \( 50 \) - **Annual demand (\(D\))**: \[ D = 10 \times 50 = 500 \text{ snakes/year} \] - **Cost per snake (\(c\))**: \[ c = \$100 \] - **Order cost (\(S\))**: \[ S = \$500 \text{ per order} \] - **Holding cost rate:** \( 20\% \) of purchase price - **Holding cost (\(H\))**: \[ H = 0.2 \times 100 = \$20 \text{ per snake per year} \] ## Total Cost Function The **total annual cost** (\(TC\)) consists of the following components: 1. **Ordering cost:** \[ \frac{D}{Q} S \] 2. **Holding cost:** \[ H \left(\frac{Q}{2}\right)^2 \] Thus, the total cost function is: \[ TC(Q) = \frac{D}{Q} S + H \left(\frac{Q}{2}\right)^2 \] Substituting the known values: \[ TC(Q) = \frac{500}{Q} \times 500 + 20 \left(\frac{Q}{2}\right)^2 \] \[ TC(Q) = \frac{250{,}000}{Q} + 20 \cdot \frac{Q^2}{4} \] \[ TC(Q) = \frac{250{,}000}{Q} + 5Q^2 \] ## Finding the Optimal Order Quantity To find the minimum total cost, we set the derivative of \(TC(Q)\) with respect to \(Q\) to zero: 1. Differentiate \(TC(Q)\): \[ \frac{dTC}{dQ} = -\frac{250{,}000}{Q^2} + 10Q \] 2. Set the derivative to zero: \[ -\frac{250{,}000}{Q^2} + 10Q = 0 \] 3. Rearranging gives: \[ 10Q = \frac{250{,}000}{Q^2} \] \[ 10Q^3 = 250{,}000 \] \[ Q^3 = 25{,}000 \] \[ Q^* = (25{,}000)^{1/3} \] Calculating the cubic root: \[ Q^* \approx 29.7 \text{ snakes (rounded to one decimal place)} \] ## Final Summary - **The total cost function for Fred's Snake Store is:** \[ TC(Q) = \frac{250{,}000}{Q} + 5Q^2 \] - **The optimal order quantity (\(Q^*\)) is approximately:** \[ Q^* \approx 29.7 \text{ snakes per order} \] This quantity minimizes Fred's total costs associated with ordering and holding rattlesnakes.

# EOQ Calculation for Fred's Snake Store Fred's Snake Store needs to determine the optimal order quantity for rattlesnakes based on specific demand and cost parameters. This calculation involves a total cost function that incorporates ordering and holding costs. ## Given Data Fred sells \(10\) rattlesnakes per week for \(50\) weeks, resulting in an annual demand of \(500\) snakes. The cost per snake is \( \$100 \), with an order cost of \( \$500 \) and a holding cost of \( \$20 \) per snake per year. ## Total Cost Function The total annual cost (\(TC\)) consists of ordering and holding costs. The total cost function is represented as: \[ TC(Q) = \frac{250{,}000}{Q} + 5Q^2 \] ## Finding the Optimal Order Quantity To minimize the total cost, the derivative of the total cost function is set to zero. This leads to the equation: \[ 10Q^3 = 250{,}000 \] ## Final Summary The total cost function for the store is: \[ TC(Q) = \frac{250{,}000}{Q} + 5Q^2 \] The optimal order quantity (\(Q^*\)) is approximately \(29.7\) snakes per order, which minimizes Fred's total costs related to ordering and holding snakes.