Inventory: Stable Demand Dr. Ron Lembke

Economic Order Quantity Assumptions Demand rate is known and constant No order lead time Shortages are not allowed Costs: S - setup cost per order H - holding cost per unit time

EOQ Decrease Due to Inventory Constant Demand Level Q* Instantaneous Optimal Order Quantity Instantaneous Receipt of Optimal Order Quantity Average Inventory Q/2 Time

Total Costs Average Inventory = Q/2 Annual Holding costs = H * Q/2 # Orders per year = D / Q Annual Ordering Costs = S * D/Q Annual Total Costs = Holding + Ordering

How Much to Order? Total Cost = Holding + Ordering Annual Cost Ordering Cost = S * D/Q Holding Cost = H * Q/2 Optimal Q Order Quantity

Optimal Quantity Total Costs = Take derivative with respect to Q = Set equal to zero Solve for Q:

EOQ Inventory Level Q* Reorder Point (R) Time Lead Time

Adding Lead Time Use same order size Order before inventory depleted Where: = demand rate (per day) L = lead time (in days) both in same time period (wks, months, etc.)

A Question: If the EOQ is based on so many horrible assumptions that are never really true, why is it the most commonly used ordering policy? Cost curve very flat around optimal Q, so a small change in Q means small increase in Total Costs If overestimate D by 10%, and S by 10%, and H by 20%, they pretty much cancel each other out Have to overestimate all in the wrong direction before Q affected

Sensitivity Suppose we do not order optimal Q*, but order Q instead. Percentage profit loss given by: Should order 100, order 150 (50% over): 0.5*(0.66 + 1.5) =1.08 an 8%cost increase

Quantity Discounts

Quantity Discounts- Price Break How does this all change if price changes depending on order size? Holding cost as function of cost: H = i * C Explicitly consider price:

Discount Example D = 10,000 S = $20 i = 20% Price Quantity EOQ C = 5.00 Q < 500 633 4.50 500-999 667 3.90 Q >= 1000 716 Must Include Cost of Goods:

Discount Pricing X 633 X 667 X 716 Total Cost C=$5 Price 1 Price 2 500 1,000 Order Size

Discount Example Order 667 at a time: Hold 667/2 * 4.50 * 0.2= $300.15 Order 10,000/667 * 20 = $299.85 Mat’l 10,000*4.50 = $45,000.00 45,600.00 Order 1,000 at a time: Hold 1,000/2 * 3.90 * 0.2= $390.00 Order 10,000/1,000 * 20 = $200.00 Mat’l 10,000*3.90 = $39,000.00 39,590.00

Excel graph of costs

Discount Model 1. Compute EOQ for next cheapest price 2. Is EOQ feasible? (is EOQ in range?) If EOQ is too small, use lowest possible Q to get price. 3. Compute total cost for this quantity Repeat until EOQ is feasible or too big. Select quantity/price with lowest total cost.

Laundry soap prices - 2012 $9.99 129 oz. $0.077 $4.99 51 oz. $0.0978 $8.99 103 oz. $0.087 $6.99 77 oz. $/oz savings $0.091 7% 4.4% 11.5%

Value Pack - 2015 Regular: Value Pack $1.88 for 10.56 oz = $0.178 per ounce Value Pack $2.98 for 15.85 oz = $0.188 per ounce 5.6% higher price!

Triscuits - 2015 $1.98 8 oz. $0.2475 $3.88 12 oz. $0.323 $/oz 30% higher price!

Goldfish - 2015 $2.97 11 oz. $0.270 $1.97 6.6 oz. $0.298 $6.48 30 oz. $0.216 $/oz savings 9.4% 20%

Summary Economic Order Quantity Discount Model Perfectly balances ordering and holding costs Very robust, errors in input quantities have small impact on correctness of results Discount Model Start with EOQ calculations, using H= iC Compute EOQ for each price, Determine feasible quantity Compute Total Costs: Holding, Ordering, and Cost of Goods.