INVENTORY MANAGEMENT Part 2 Professor Robert Saltzman Operations Analysis

Robert Saltzman © 2005

Financial EOQ Example • Murray has most of his savings invested in a money market (MM) account that yields about 7% annually. • Murray maintains a checking account in a bank that pays only 2% interest annually on the balance, so he tries not to keep too much money in this account. • Occasionally, Murray must transfer money from his MM account to his checking account. In a year, he needs to transfer about $6,400 per year into his checking account. • The MM institution charges Murray $10 each time he transfers funds from his MM fund to the bank where Murray has his checking account. • How much & how often should Murray transfer funds from his MM account to his checking account? Robert Saltzman © 2005

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EOQ with Quantity Discounts • Back to film (Example 1): P = $3/package S = $30/order H = $0.25/package/month D = 200 packages/month • Optimal policy: Order Q* = 219 packages when inventory falls to R* = dL packages Robert Saltzman © 2005

All-Units Discount if Q is Big • Now suppose an “all-units discount” is offered: • P1 = $3.00/package, if Q ≤ 99 • P2 = $2.75/package, if 100 ≤ Q ≤ 399 • P3 = $2.50/package, if Q ≥ 400 • It’s no longer clear what the best order quantity Q is, since the unit price P depends on Q • Temporarily, think of 3 total cost curves: TCj = PjD + DS/Q + QH/2, for j = 1, 2, 3 Robert Saltzman © 2005

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Which Q minimizes total costs? Total cost per month

0

100

219

400

Q

Robert Saltzman © 2005

Finding Best Overall Quantity • Q*EOQ corresponds to only 1 particular price • Calculate TC(Q*EOQ) using corresponding price • Calculate TC(Q’) for any price-break quantity Q’ that’s larger than Q*EOQ • Overall best quantity to order has lowest TC EX: Here, compare – TC2(219) vs. – TC3(400) Robert Saltzman © 2005

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EOQ with Probabilistic Demand • • • •

View demand as a random variable, not a constant During the lead time (L > 0), you may run out while waiting for the order to arrive Businesses often carry safety stock, just in case How much safety stock (SS) to carry depends on: 1. LTD = Lead time demand distribution (you can observe); 2. SL = Service level: the % of cycles you’d like to meet all demand (you can set this as desired) SL = P(LTD ≤ R*) = 1 – P(stockout during this cycle) Robert Saltzman © 2005

Probabilistic Demand Example 1 •

Suppose a particular product has: – –

•

Desired Service Level SL = 95%, i.e., –

• • •

Lead time L = 3 days Observed LTD = Normal(µ = 100, σ = 20) P(LTD ≤ R*) = 0.95 or P(LTD > R*) = 0.05

R* = 95th percentile of the Normal distribution R* = µ + z0.95σ = 100 + 1.645(20) = 133 Here, safety stock SS = z0.95σ = 33 Robert Saltzman © 2005

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Probabilistic Demand Example 2 Observed LTD is:

If desired SL = 85%, then R* = ____ If desired SL = 95%, then R* = ____

30%

30%

P r o b a b ility

If desired SL = 70%, then R* = ____

35%

25%

25% 20% 15%

15%

15% 10%

10%

5%

5% 0% 5

15

(µ = 24; rest is SS)

25

35

45

55

Lead Time Demand Robert Saltzman © 2005

Probabilistic Demand Summary • • • • •

With or without SS, the inventory policy works same way: Order Q* when inventory falls to R* Calculate Q* as before Reorder point R* is higher than before by the amount of safety stock SS On average, you carry SS more inventory than before (which has a cost) Time between orders varies from cycle to cycle (since demand isn’t really constant) Robert Saltzman © 2005

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An Example with Production • Consider a company that makes speakers: S = $1,000/batch (Setup cost) H = $1.20/speaker/month (Holding cost) D = 600 speakers/month (Demand rate) speakers/batch • Q*EOQ = • But this is unrealistic: Producers need time to manufacture large batches of goods • Need to account for gradual production Robert Saltzman © 2005

Production Order Quantity Model • Suppose only p = 50 speakers/day can be made – Note: daily production rate p ≠ unit price or cost P

• Assuming 30 days per month: – Daily demand rate d = 20 speakers/day

• Inventory builds up at rate p – d = 30 speakers/day: – During production, Inventory(t) = (p – d)t

• How long does it take to make Q speakers? • What’s the peak level of inventory? • How long does it take to demand all Q speakers? Robert Saltzman © 2005

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Production Order Quantity Model Inv.

0

Q/p

Q/d

time (days)

Robert Saltzman © 2005

POQ: How to find best Q? • • • • •

Find Q* again by minimizing total costs/unit time Monthly Production Costs = DP Monthly Setup Costs = DS/Q Monthly Holding Costs = ½((p – d)/p)QH Total Costs = Production + Setup + Holding: TC(Q) = DP + (DS)Q-1 + ((p – d)/p)QH/2 • Q*POQ = Q*EOQ p /( p − d ) Robert Saltzman © 2005

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