Production and Operation Managements Inventory Control Subject to Unknown Demand Dr.Na GENG Department of Industrial Engineering Management Shanghai Jiao Tong University
Production and Operation Managements Dr. Na GENG Department of Industrial Engineering & Management Shanghai Jiao Tong University Inventory Control Subject to Unknown Demand
Inventory Control Subject to Unknown Demand Contents 。Introduction ·The newsboy model Lot Size-Reorder Point System; Service Level in (Q,R)System; Additional Discussion of Periodic-review Systems Multiproduct Systems
Inventory Control Subject to Unknown Demand Contents • Introduction • The newsboy model • Lot Size-Reorder Point System; • Service Level in (Q, R) System; • Additional Discussion of Periodic-review Systems • Multiproduct Systems
The newsboy model-Optimal Policy for Discrete Demand .In some cases,accurate representation of the observed pattern of demand in term of continuous distribution is difficult or impossible. .In the discrete case,the critical ratio will generally fall between two values of F(Q) The optimal solution procedure is to locate the critical ratio between two values of F(Q)and choose the Q corresponding to the higher value
The newsboy model- Optimal Policy for Discrete Demand •In some cases, accurate representation of the observed pattern of demand in term of continuous distribution is difficult or impossible. •In the discrete case, the critical ratio will generally fall between two values of F(Q). • The optimal solution procedure is to locate the critical ratio between two values of F(Q) and choose the Q corresponding to the higher value
The newsboy model-Optimal Policy for Discrete Demand Example 5.2-Mac's newsstand f(4)=3/52 is obtained by dividing frequencies 4(the numbers of times 3 that a given weekly demand 4 occur during a year,i.e. 52 weeks)by 52; The critical ratio is 0.77,which corresponds to a value of F(Q) between Q=14 and Q=15. Q fQ) F(Q) Q fQ) F(Q) Q fQ) F(Q) 0 1/52 0.02 8 4/52 0.25 16 1/52 0.81 1 0 0.02 9 6/52 0.37 17 3/52 0.87 2 0 0.02 10 2/52 0.40 18 3/52 0.92 3 0 0.02 11 5/52 0.50 19 3/52 0.98 4 3/52 0.08 12 4/52 0.58 20 0 0.98 5 1/52 0.10 13 1/52 0.60 21 0 0.98 6 2/52 0.13 14 5/52 0.69 22 1/52 1.00 7 2/52 0.17 15 5/52 0.79
The newsboy model- Optimal Policy for Discrete Demand Example 5.2- Mac’s newsstand • f(4) =3/52 is obtained by dividing frequencies 4 (the numbers of times 3 that a given weekly demand 4 occur during a year, i.e. 52 weeks) by 52; • The critical ratio is 0.77, which corresponds to a value of F(Q) between Q=14 and Q=15. Q f(Q) F(Q) Q f(Q) F(Q) Q f(Q) F(Q) 0 1/52 0.02 8 4/52 0.25 16 1/52 0.81 1 0 0.02 9 6/52 0.37 17 3/52 0.87 2 0 0.02 10 2/52 0.40 18 3/52 0.92 3 0 0.02 11 5/52 0.50 19 3/52 0.98 4 3/52 0.08 12 4/52 0.58 20 0 0.98 5 1/52 0.10 13 1/52 0.60 21 0 0.98 6 2/52 0.13 14 5/52 0.69 22 1/52 1.00 7 2/52 0.17 15 5/52 0.79
The newsboy model-Extension to Include Starting Inventory Suppose that the starting inventory is some value u and u>0 The optimal policy is simply to modify that for u-=0. The same ideal is that we still want to be at Q*after ordering If uQ*,do not order. Note that Q*should be understood as order-up-to point rather than the order quantity when u>0. Example 5.2 (Cont.)-Suppose that Mac has received 6 copies of the Journal at the beginning of the week from other supplier.The optimal policy still calls for having 15 copies on hand after ordering,thus he would order the difference 15-6=9 copies
The newsboy model- Extension to Include Starting Inventory Suppose that the starting inventory is some value u and u>0. The optimal policy is simply to modify that for u=0. The same ideal is that we still want to be at Q* after ordering. If uQ*, do not order. Note that Q* should be understood as order-up-to point rather than the order quantity when u>0. Example 5.2 (Cont.)-Suppose that Mac has received 6 copies of the Journal at the beginning of the week from other supplier. The optimal policy still calls for having 15 copies on hand after ordering, thus he would order the difference 15-6=9 copies
The newsboy model-Extension to Multiple Planning Periods The ending inventory in any period becomes the starting inventory in the next period. Consider the case in which there are infinitely many periods remaining: If excess demand is back-ordered,interpret c,as the loss- of-goodwill cost and c,as the holding cost. If excess demand is lost,interpret c,as the loss-of- goodwill cost plus the lost profit and c.as the holding cost
The newsboy model- Extension to Multiple Planning Periods The ending inventory in any period becomes the starting inventory in the next period. Consider the case in which there are infinitely many periods remaining: If excess demand is back-ordered, interpret c u as the lossof-goodwill cost and c o as the holding cost. If excess demand is lost, interpret c u as the loss-ofgoodwill cost plus the lost profit and c o as the holding cost
The newsboy model-Extension to Multiple Planning Periods Example 5.3:Suppose that Mac is considering how to replenish the inventory of a very popular paperback thesaurus that is ordered monthly.Copies of the thesaurus unsold at the end of a month are still kept on the shelves for future sales. Assume that customers who request copies of the thesaurus when they are out of stock will wait until the following month.Mac buys the thesaurus for $1.25 and sells it for 3.75.Mac estimates a loss-of-goodwill cost of 80 cents each time a demand for a thesaurus must be back-ordered.Monthly demand for the book is fairly closely approximated by a normal distribution with mean 20 and standard deviation 10. Mac uses a 20 percent annual interest rate to determine his holding cost.How many copies of the thesaurus should be purchased at the beginning of each month?
Example 5.3: Suppose that Mac is considering how to replenish the inventory of a very popular paperback thesaurus that is ordered monthly. Copies of the thesaurus unsold at the end of a month are still kept on the shelves for future sales. Assume that customers who request copies of the thesaurus when they are out of stock will wait until the following month. Mac buys the thesaurus for $1.25 and sells it for $ 3.75. Mac estimates a loss-of-goodwill cost of 80 cents each time a demand for a thesaurus must be back-ordered. Monthly demand for the book is fairly closely approximated by a normal distribution with mean 20 and standard deviation 10. Mac uses a 20 percent annual interest rate to determine his holding cost. How many copies of the thesaurus should be purchased at the beginning of each month? The newsboy model- Extension to Multiple Planning Periods
The newsboy model-Extension to Multiple Planning Periods Answer for Example 5.3: u=20 and standard deviation o=10 ©c。=1.25*0.2/12=0.208 holding cost ©c=0.80. The critical ratio is c/(c+c)-0.8/(0.208+0.8)-0.794 Hence,he ought to purchase enough copies to satisfy all of the monthly demand with probability 0.794.The optimal Q* is the 79.4th percentile of the demand distribution. ©Q*=0z+μ=10×0.82+20=28.2≈28
Answer for Example 5.3: =20 and standard deviation =10 c o=1.25*0.2/12=0.208 holding cost c u=0.80. The critical ratio is c u/(c o+c u)=0.8/(0.208 +0.8)=0.794 Hence, he ought to purchase enough copies to satisfy all of the monthly demand with probability 0.794. The optimal Q* is the 79.4th percentile of the demand distribution. Q*= z+ =10 0.82+20=28.2 28 The newsboy model- Extension to Multiple Planning Periods
The newsboy model-Extension to Multiple Planning Periods Assume the excess demands are lost,and the cost for lost-of- sale is 0.8: u=20 and standard deviation o=10 co=1.25*0.2/12=0.208 holding cost cu=0.80+2.5=3.3. The critical ratio is cu/(co+c)=3.3/(0.208+3.3)=0.94 Hence,he ought to purchase enough copies to satisfy all of the monthly demand with probability 0.94.The optimal Q*is the 94th percentile of the demand distribution. Q*=oz+μ=10×1.56+20=35.6≈36
Assume the excess demands are lost, and the cost for lost-ofsale is 0.8: =20 and standard deviation =10 c o=1.25*0.2/12=0.208 holding cost c u=0.80+2.5=3.3. The critical ratio is c u/(c o+c u)=3.3/(0.208 +3.3)=0.94 Hence, he ought to purchase enough copies to satisfy all of the monthly demand with probability 0.94. The optimal Q* is the 94th percentile of the demand distribution. Q*= z+ =10 1.56+20=35.6 36 The newsboy model- Extension to Multiple Planning Periods
The newsboy model-Extension to Multiple Planning Periods One serious limitation of the multi-period newsboy model it did not include a setup cost for placing an order. In most real systems,there are fixed costs associated with ordering,and it is not optimal to place orders each period. The problem of random demand is approached when a fixed charge for ordering is present in a different way
The newsboy model- Extension to Multiple Planning Periods One serious limitation of the multi-period newsboy model : • it did not include a setup cost for placing an order. In most real systems, there are fixed costs associated with ordering, and it is not optimal to place orders each period. The problem of random demand is approached when a fixed charge for ordering is present in a different way
