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
Lot Size-Reorder Point System The procedure of computing Q and R:Qo=EOQ 1-F(R)=2h/pa check z,and L(z,value in Table A-4 R=u+2,0 n(R)=oL(3) =K+pn(R,)/h No,i=i+1 Qi,Qit close? ↓Yes Stop
The procedure of computing Q and R: Q 0=EOQ Lot Size-Reorder Point System 1 () / F R Qh p i i check z and L(z ) value in Table A-4 i i R z i i ( ) i i n R Lz 1 2 [ ( )] / Q K pn R h i i Qi, Qi+1 close? Yes Stop No, i=i+1
Lot Size-Reorder Point System Example 5.4 Harvey's Specialty Shop sells a popular mustard that purchased from English company.The mustard costs $10 a jar and requires a six-month lead time for replenishment stock.The holding cost is computed on basis 20%annual interest rate;the lost-of-goodwill cost is $25 a jar;and bookkeeping expenses for placing an order amount to about $50.During the six-month lead time,average 100 jars are sold,but with substantial variation from one six-month period to the next.The demand follows normal distribution and the standard deviation of demand during each six- month period is 25.How should Harvey control the replenishment of the mustard?
Lot Size-Reorder Point System Example 5.4 Harvey’s Specialty Shop sells a popular mustard that purchased from English company. The mustard costs $10 a jar and requires a six-month lead time for replenishment stock. The holding cost is computed on basis 20% annual interest rate; the lost-of-goodwill cost is $25 a jar; and bookkeeping expenses for placing an order amount to about $50. During the six-month lead time, average 100 jars are sold, but with substantial variation from one six-month period to the next. The demand follows normal distribution and the standard deviation of demand during each sixmonth period is 25. How should Harvey control the replenishment of the mustard?
Lot Size-Reorder Point System Solution to Example 5.4 To find the optimal values of R and Q The mean lead time demand in six-month lead time is 100,the mean yearly demand is 200,giving A=200; ·h=10×0.20=2,K=50;P=25; Initialization:Q,=E0Q=V2K/h=√2×50×200/2=100， Step1:1-F(R)=Qh/p2=100*2/(25*200)=0.04; Step 2:check in Table A-4Zo=1.75 and L(zo)=0.0162; Step3:R=4+2oo=100+1.75*25=144; Step4:n(R)=oL(zo)=25*0.0162=0.405, 22K+pnR】=,2×20050+25×0.405] Step =110, h 2 Compare:Qo and Q are not close,so return to step 1 and continue
Lot Size-Reorder Point System Solution to Example 5.4 To find the optimal values of R and Q • The mean lead time demand in six-month lead time is 100, the mean yearly demand is 200, giving =200; • h=10 0.20=2; K=50; P=25; 0 Initialization: =EOQ= 2 / 2 50 200 Q Kh / 2 1 0 0; Step 1: 1 ( ) / 100*2 / 25*200 0.04; F R Qh p 0 0 Step 2: check in Table A-4 z =1.75 and L(z ) =0.0162; 0 0 Step 3: 100 1.75*25 144; R z 0 0 Step 4: ( ) 25*0.0162 0.405; n R Lz 0 0 0 1 2 [ ( )] 2 200[50 25 0.40 11 5] Step 5: 2 0; K pn R Q h Compare: Q and Q are not close, so return to step 1 and continue. 0 1
Lot Size-Reorder Point System 9=110; Step1:1-F(R)=2h/p2=110*2/(25*200)=0.044; Step 2:check in Table A-4z=1.70 and L(z)=0.0183; Step3:R=4+z,o=100+1.70*25=143, Step4:n(R)=oL(z)=25*0.0183=0.4575, 22[K+pn(R】_ 2×200[50+25×0.4575] ≈111 2 Compare:Q and Q,are close,stop. Substitue Q,=111 into 1-F(R)=Oh/pA=0.044, z2=Z,=1.70,R2=R1=143
Lot Size-Reorder Point System 1 1 1 1 1 1 1 1 1 2 =110; Step 1: 1 ( ) / 110*2 / 25*200 0.044; Step 2: check in Table A-4 z =1.70 and L(z ) =0.0183; Step 3: 100 1.70*25 143; Step 4: ( ) 25*0.0183 0.4575; 2 [ Step 5: Q F R Qh p R z n R Lz K Q 1 1 2 ( )] 2 200[50 25 0.4575] ; 2 Compare: Q and Q are close, sto 1 . 1 1 p pn R h 2 22 21 2 1 Substitue Q =111 into 1 ( ) / 0.044, z =z =1.70, R =R =143 F R Qh p
Lot Size-Reorder Point System Results for Example 5.4:The optimal values of(Q,R)=(111, 143),that is,when Harvey's inventory of this type mustard hits 143 jars,he should place an order for 111 jars. Example 5.4 (Cont.):determine the following (1)Safety stock; (2)The average annual holding,setup,and penalty costs associated with the inventory control of the mustard: (3)The average time between placement of orders; (4)The proportion of order cycles in which no stock-outs occur>Among given number of order cycles,how many order cycles do not have stock-outs? (5)The proportion of demands that are not met
Lot Size-Reorder Point System • Results for Example 5.4: The optimal values of (Q, R)=(111, 143), that is, when Harvey’s inventory of this type mustard hits 143 jars, he should place an order for 111 jars. • Example 5.4 (Cont.): determine the following (1) Safety stock; (2) The average annual holding, setup, and penalty costs associated with the inventory control of the mustard; (3) The average time between placement of orders; (4) The proportion of order cycles in which no stock-outs occur>Among given number of order cycles, how many order cycles do not have stock-outs? (5) The proportion of demands that are not met
Lot Size-Reorder Point System Solution to Example 5.4(Cont.) 1)The safety stock is s=R-u=143-100=43 jars; 2)Three costs: The holding cost is h(Q/2+s)=2(111/2+43)=$197/jar; The setup cost is KA/Q=50x200/111=$90.09/jar; The penalty cost is pA n(R)/Q=25 x 200x0.4575/111=$20.61/jar Hence,the total average cost under optimal inventory control policy is $307.70/iar. 3)The average time between placement of orders: T=Q/=111/200=0.556yr=6.7 months; 3)Compute the probability that no stock-out occurs in the lead time,which is the same as that the probability that the lead time demand does not exceeds the reorder point:P(D<R)=F(R)=1-Qh/p=1-0.044=0.956: 4)The proportion of demand that stock out is n(R)/Q=0.4575/111=0.004
Lot Size-Reorder Point System Solution to Example 5.4 (Cont.) 1) The safety stock is s=R- =143-100=43 jars; 2) Three costs: The holding cost is h(Q/2+s)=2(111/2+43)=$197/jar; The setup cost is K /Q=50 200/111=$90.09/jar; The penalty cost is p n(R)/Q=25 200 0.4575/111=$20.61/jar Hence, the total average cost under optimal inventory control policy is $307.70/jar. 3) The average time between placement of orders: T=Q/ =111/200=0.556 yr=6.7months; 3) Compute the probability that no stock-out occurs in the lead time, which is the same as that the probability that the lead time demand does not exceeds the reorder point: P(D R)=F(R)=1-Qh/p =1-0.044=0.956; 4 ) The pro portion of demand that stock out is n ( R ) / Q=0.4575/111=0.004
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
Service Levels in (Q,R)System In reality,it is difficult to determine the exact value of stock-out cost p.A common substitute for a stock-out cost is a service level. Service level generally refers to the probability that a demand or a collection of demand is met. Service level can be applied both to periodic review and continuous review systems,that is,(Q,R)system Two types of service levels for continuous review system:Type 1 and Type 2
Service Levels in (Q, R) System • In reality, it is difficult to determine the exact value of stock-out cost p. A common substitute for a stock-out cost is a service level. • Service level generally refers to the probability that a demand or a collection of demand is met. • Service level can be applied both to periodic review and continuous review systems, that is, (Q, R) system. • Two types of service levels for continuous review system: Type 1 and Type 2