CDAE 266 - Class 26Dec. 5

Last class:

5. Inventory analysis and applications

Today: Problem set 5 5. Inventory analysis and applications Quiz 7 Class evaluation

Next class: 5. Inventory analysis and applications Review for the final exam

Reading: Inventory decisions with certain factors

CDAE 266 - Class 26Dec. 5

Important dates: Problem set 5, due Thursday, Dec. 7

Problems 6-1, 6-2, 6-3, 6-4, and 6-13 from the reading package Final exam: 8:00-11:00am, Thursday, Dec. 14

5. Inventory analysis and applications

5.1. Basic concepts

5.2. Inventory cost components

5.3. Economic order quantity (EOQ) model

5.4. Inventory policy with backordering

5.5. Inventory policy and service level

5.6. Production and inventory model

A graphical presentation of the EOQ model:– The constant environment described by the EOQ

assumptions leads to the following observation

THE OPTIMAL EOQ POLICY ORDERS THE SAME AMOUNT EACH TIME.

This observation results in the inventory profile below:Q QQ

5.3. The economic order quantity (EOQ) model

5.3.3. Mathematical model

-- Total annual inventory cost = annual order costs + annual holding cost + annual item costs

-- Total annual relevant (variable) cost:

-- Examples

AcQ

hckQ

A

2

2

Qhck

Q

ATC

5.3. The economic order quantity (EOQ) model

5.3.3. Mathematical model

-- Optimal solution:

hc

AkQ

2*

5.3. The economic order quantity (EOQ) model

5.3.4. Reorder point if lead time > 0 If L > 0, R= L x A

Note that L and A must have consistent units

5.3.5. Examples

(1) Liquor store (pp. 209-211)

Additional question: If the sale price is $3 per case, what will be the total “gross” profit per

year?

5.3. The economic order quantity (EOQ) model

5.3.4. Examples

(1) Liquor store (pp. 209-211)Available information: A = 5200 cases/yr k = $10/orderc = $2 per case h = $0.20 per $ per yr.

(a) Current policy: Q = 100 cases/order R = (5200/365) * 1 = 15 cases T = Q/A = 100/5200 (year) = 7 days

TC = $540 per year (see page 210)

(b) Optimal policy: Q* = 510 cases/order R = (5200/365) * 1 = 15 cases T = Q*/A = 510/5200 (year) =36

days TC = $204 per year If the retail price is $3 per case, Gross profit = 5200*3 – 5200*2 – 204 = $4996

Class exercise

1. Draw a graph to show the following inventory policy for

a business with no backordering: the annual demand is 3650 units and the business opens 365 days a year, the order quantity is 305 units and the lead time is 4 days.

2. If some customers of the above business are willing to take backorders and the maximum backorders are 50

units, draw another graph to show the inventory policy (there is no change in order quantity and lead time)

3. Take-home exercise: Example on pp. 215-216 with the annual demand (A) increased to 1200 units and the lead time to be 3 days.

Problem set 56.3. A = 20x5x52 = 5200 batteries per year

c = 10 + 2 = $12 per batteryL = 1 dayInterest rate = 1.5% per month = 18% per yearK = $50 per orderhc = per unit holding cost = 0.5 + 0.18*12 = $2.66 per battery per yrh = 0.5/12 + 0.18 = $0.2217 per $ value per yrhc = h x C = 0.2217 x 12 = $2.66 per battery per yr

Q* = T =

5.3. The economic order quantity (EOQ) model

5.3.5. Lead time (L), reorder point (R) and safety stock (SS) and their impacts

(1) Inventory policy: Q: order quantityR: reorder point (Note that R is related to T

but they are two different variables)

(2) In the basic EOQ model:Q* =L = 0 ==> R* = L x A = 0

(3) If L > 0, R= L x A (the units of L & A must be consistent)

5.3. The economic order quantity (EOQ) model

5.3.5. Lead time (L), reorder point (R) and safety stock (SS) and their impacts

(4) If the lead time is zero (L=0) and the co. wants to keep a safety stock (SS),

R = L x A + SS = SS

(5) If the lead time is greater than zero (L>0) and the co. wants to keep a safety stock,

R = L x A + SS

(6) Impacts of L & SS on R*, Q* and TC:No impact on Q*L ==> no impact on TCSS ==> increase TC by (hc * SS)

Take-home exercise

For the liquor store example (pp. 209-211), what will be the optimal inventory policy (Q* and R) and what will be the TC if the interest rate is reduced from 10% to 6%, the net cost of each case to the store increased from $2 to $3, and the lead time increased from 1 day to 3 days?

Additional question: what is the TC if the store keeps a safety stock (SS) of 20 units?

5.4. Inventory policy with backordering

5.4.1. A graphical presentation (page 214) A = Annual demand (e.g., 7300 kg per year) Q = order quantity [e.g., 200 kg per order (delivery)] S = Maximum on-hand inventory (e.g, 150 kg)

Q - S = Maximum backorders (e.g., 50 kg) T = Q/A = time for each inventory cycle

(e.g., T = 200/7300 = 0.0274 yr = 10 days) T1 = S/A = the time with on-hand inventory

(e.g., T1 = 150/7300 = 0.0206 yr = 7.5 days) T2 = (Q-S)/A = T - T1

(e.g., T2= 50/7300 = 0.00685 yr = 2.5 days) T1/T = Proportion of time with on-hand inventory

T2/T = Proportion of time without on-hand inventory Lead time and reorder point (e.g., L = one day)

5.4. Inventory policy with backordering

5.4.2. Total relevant (variable) inventory cost

TC = annual ordering cost + annual holding cost + annual shortage (goodwill cost)

= …… (see page 214)

p = per unit goodwill (shortage) cost per year (e.g., p=$2 per unit per year)

5.4.3. Optimal inventory policy (page 215) Q* = S* = R =

5.4. Inventory policy with backordering

5.4.4. Example (pp. 215-216) A = 1000 cases of wine per year K = $100 per order (delivery) C = $20 per case h = $0.20 per dollar value per year p = $3.65 per unit of shortage per year L = 0

Q* = S* = R =

5.5. Production lot size model

5.5.1. A graphical presentation (page 220) Production phase Inventory-only phase

5.5.2. Available information: A = annual demand K = fixed cost per production run B = annual production rate C = production cost per unit h = holding cost per dollar value per year

5.5. Production lot size model

5.5.3. Annual total variable production cost = start-up cost + holding cost =

5.5.4. Optimal solutions (page 221)

Q* =

5.5.6. An example (pp. 221-222)

5.5.7. Other useful results Maximum inventory = T1 = T2 =

T1/T = T2/T =

Summary and comparison of the three models EOQ model Backordering model Production

modelGraphGiven info A, c, h, K, L, SS A, c, h, K, L, P A, c, h, K, BSolutions Q*, R, TC, T Q*, S*, R, TC, T, T1, T2 Q*, TC, T, T1, T2Key words backorder, shortage, Production rate

goodwill costVariable definitions:A = Annual demand Q* = Optimal order (production) quantityL = lead time S* = maximum on-hand inventoryc = Per unit value or price R = reorder pointh = holding cost per $ value per year TC = total relevant inventory costhc = holding cost per unit per year T = Length of each inventory cycleP = goodwill (shortage) cost per unit per year T1 =B = production rate per year T2 =

K = fixed cost per order or production run