CDAE Class 26 Dec. 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 Class 26 Dec. 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: QQQ

5.3. The economic order quantity (EOQ) model Mathematical model -- Total annual inventory cost = annual order costs + annual holding cost + annual item costs -- Total annual relevant (variable) cost: -- Examples

5.3. The economic order quantity (EOQ) model Mathematical model -- Optimal solution:

5.3. The economic order quantity (EOQ) model Reorder point if lead time > 0 If L > 0, R= L x A Note that L and A must have consistent units Examples (1) Liquor store (pp ) 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 Examples (1) Liquor store (pp ) Available information: A = 5200 cases/yrk = $10/order c = $2 per caseh = $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 with the annual demand (A) increased to 1200 units and the lead time to be 3 days.

Problem set A = 20x5x52 = 5200 batteries per year c = = $12 per battery L = 1 day Interest rate = 1.5% per month = 18% per year K = $50 per order hc = per unit holding cost = *12 = $2.66 per battery per yr h = 0.5/ = $ per $ value per yr hc = h x C = x 12 = $2.66 per battery per yr Q* =T =

5.3. The economic order quantity (EOQ) model Lead time (L), reorder point (R) and safety stock (SS) and their impacts (1) Inventory policy: Q: order quantity R: 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 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 TC SS ==> increase TC by (hc * SS)

Take-home exercise For the liquor store example (pp ), 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 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 = yr = 10 days) T1 = S/A = the time with on-hand inventory (e.g., T1 = 150/7300 = yr = 7.5 days) T2 = (Q-S)/A = T - T1 (e.g., T2= 50/7300 = 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 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) Optimal inventory policy (page 215) Q* = S* = R =

5.4. Inventory policy with backordering Example (pp ) 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 A graphical presentation (page 220) Production phase Inventory-only phase 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 Annual total variable production cost = start-up cost + holding cost = Optimal solutions (page 221) Q* = An example (pp ) Other useful results Maximum inventory = T1 =T2 = T1/T =T2/T =

Summary and comparison of the three models EOQ model Backordering model Production model Graph Given info A, c, h, K, L, SS A, c, h, K, L, P A, c, h, K, B Solutions Q*, R, TC, T Q*, S*, R, TC, T, T1, T2 Q*, TC, T, T1, T2 Key words backorder, shortage, Production rate goodwill cost Variable definitions: A = Annual demandQ* = Optimal order (production) quantity L = lead timeS* = maximum on-hand inventory c = Per unit value or priceR = reorder point h = holding cost per $ value per yearTC = total relevant inventory cost hc = holding cost per unit per yearT = Length of each inventory cycle P = goodwill (shortage) cost per unit per yearT1 = B = production rate per yearT2 = K = fixed cost per order or production run