Order Quantities when Demand is Approximately Level

Order Quantities when Demand is Approximately Level Chapter 5 Inventory Management Dr. Ron Tibben-Lembke

Inventory Costs Costs associated with inventory: • Cost of the products • Cost of ordering • Cost of hanging onto it • Cost of having too much / disposal • Cost of not having enough (shortage)

Shrinkage Costs • How much is stolen? • 2% for discount, dept. stores, hardware, convenience, sporting goods • 3% for toys & hobbies • 1.5% for all else • Where does the missing stuff go? • Employees: 44.5% • Shoplifters: 32.7% • Administrative / paperwork error: 17.5% • Vendor fraud: 5.1%

Inventory Holding Costs Category% of Value Housing (building) cost 6% Material handling 3% Labor cost 3% Opportunity/investment 11% Pilferage/scrap/obsolescence 3% Total Holding Cost 26%

ABC Analysis • Divides on-hand inventory into 3 classes • A class, B class, C class • Basis is usually annual $ volume • $ volume = Annual demand x Unit cost • Policies based on ABC analysis • Develop class A suppliers more • Give tighter physical control of A items • Forecast A items more carefully

Classifying Items as ABC % Annual $ Usage A B C % of Inventory Items

ABC Classification Solution Stock # Vol. Cost $ Vol. % ABC 206 26,000 $ 36 $936,000 105 200 600 120,000 019 2,000 55 110,000 144 20,000 4 80,000 207 7,000 10 70,000 Total 1,316,000

ABC Classification Solution

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

EOQ Inventory Level Q* Optimal Order Quantity Decrease Due to Constant Demand Time

EOQ Inventory Level Instantaneous Receipt of Optimal Order Quantity Q* Optimal Order Quantity Time

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

EOQ Inventory Level Q* Average Inventory Q/2 Reorder Point (ROP) Time Lead Time

Total Costs • Average Inventory = Q/2 • Annual Holding costs = rv * Q/2 • # Orders per year = D / Q • Annual Ordering Costs = A * D/Q • Annual Total Costs = Holding + Ordering

How Much to Order? Annual Cost Holding Cost = H * Q/2 Order Quantity

How Much to Order? Annual Cost Ordering Cost = A * D/Q Holding Cost = H * Q/2 Order Quantity

How Much to Order? Total Cost = Holding + Ordering Annual Cost Order Quantity

How Much to Order? Total Cost = Holding + Ordering Annual Cost Optimal Q Order Quantity

Optimal Quantity Total Costs =

Optimal Quantity Total Costs = Take derivative with respect to Q =

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

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

Sensitivity • Suppose we do not order optimal EOQ, but order Q instead, and Q is p percent larger • Q = (1+p) * EOQ • Percentage Cost Penalty given by: • EOQ = 100, Q = 150, so p = 0.5 50*(0.25/1.5) = 8.33 a 8.33% cost increase

Figure 5.3 Sensitivity

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?

Benefits of EOQ • Profit function is very shallow • Even if conditions don’t hold perfectly, profits are close to optimal • Estimated parameters will not throw you off very far

Tabular Aid 5.1 • For A = $3.20 and r = 0.24% • Calculate Dv =total $ usage (or sales) • Find where Dv fits in the table • Use that number of months of supply • D = 200, v = $16, Dv=$3,200 • From table, buy 1 month’s worth • Q = D/12 = 200/12 = 16.7 = 17

How do you get a table? • Decide which T values you want to consider: 1 month, etc. • Use same v and r values for whole table • For each neighboring set of T’s, put them into

How do you get a table? • For example, A = $3.20, r = 0.24 • To find the breakpoint between 0.25 and 0.5 • Dv = 288 * 3.2 / (0.25 * 0.5 * 0.24) • = 921.6 / 0.03 = 30,720 • So if Dv is less than this, use 0.25, more than that, use 0.5 • Find 0.5 and 0.75 breakpoint: • Dv = 288 * 3.2/(0.5 * 0.75 * 0.24) = 10,2240

Why care about a table? • Some simple calculations to get set up • No thinking to figure out lot sizes • Every product with the same ordering cost and holding cost rate can use it • Real benefit - simplified ordering • Every product ordered every 1 or 2 weeks, or every 1, 2, 3, 4, 6, 12 months • Order multiple products on same schedule: • Get volume discounts from suppliers • Save on shipping costs • Savings outweigh small increase from non-EOQ orders

Time Uncoordinated Orders

Simultaneous Orders Time Same T = number months supply allows firm to order at same time, saving freight and ordering expenses Adjusted some T’s, changed order times

Offset Orders Same T = number months supply allows firm to control maximum inventory level by coordinating replenishments With different T, no consistency

Quantity Discounts • How does this all change if price changes depending on order size? • Explicitly consider price:

Discount Example D = 10,000 A = $20 r = 20% Price Quantity EOQ v = 5.00 Q < 500 633 4.50 501-999 666 3.90 Q >= 1000 716

Discount Pricing Total Cost Price 1 Price 2 Price 3 X 633 X 666 X 716 Order Size 500 1,000

Discount Example Order 666 at a time: Hold 666/2 * 4.50 * 0.2= $299.70 Order 10,000/666 * 20 = $300.00 Mat’l 10,000*4.50 = $45,000.00 45,599.70 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

Discount Model 1. Compute EOQ for each price 2. Is EOQ ‘realizeable’? (is Q in range?) If EOQ is too large, use lowest possible value. If too small, ignore. 3. Compute total cost for this quantity 4. Select quantity/price with lowest total cost.

Adding Lead Time • Use same order size • Order before inventory depleted • R = DL where: • D = annual demand rate • L = lead time in years

Order Quantities when Demand is Approximately Level