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
Additional Discussion of Periodic-review Systems (s,S)Policies It is difficult to implement a continuous-review solution in a periodic-review environment because the inventory level is likely to overshoot the reorder point R during a period,which makes it impossible to place an order the instant the inventory reaches R. Define two numbers,s and S,to be used as follows:When the level of on-hand inventory is less than or equal to s,an order for the difference between the inventory and S is placed
(s, S) Policies • It is difficult to implement a continuous-review solution in a periodic-review environment because the inventory level is likely to overshoot the reorder point R during a period, which makes it impossible to place an order the instant the inventory reaches R. • Define two numbers, s and S, to be used as follows: When the level of on-hand inventory is less than or equal to s, an order for the difference between the inventory and S is placed. Additional Discussion of Periodic-review Systems
Additional Discussion of Periodic-review Systems (s,S)Policies If u is the starting inventory in any period,then the(s,S)policy is √If u<s,order S-w, √Else,do not order. Approximation:to compute a(O,R)policy using the methods described earlier,and set s=R and S=R+O. This approximation will give reasonable results in many cases,and is probably the most commonly used
(s, S) Policies • If u is the starting inventory in any period, then the (s, S) policy is If u≤s, order S-u; Else, do not order. • Approximation: to compute a (Q,R) policy using the methods described earlier, and set s=R and S=R+Q. This approximation will give reasonable results in many cases, and is probably the most commonly used. Additional Discussion of Periodic-review Systems
Additional Discussion of Periodic-review Systems Service Level in Periodic-Review Systems ·Type 1 service 。 Objective-find the order-up-to point Q so that all of the demand is satisfied in a given percentage of the periods ·F(Q)=0, where F(Q)is the probability that the demand during the period does not exceed Q
Service Level in Periodic-Review Systems • Type 1 service • Objective-find the order-up-to point Q so that all of the demand is satisfied in a given percentage of the periods • F(Q)= , where F(Q) is the probability that the demand during the period does not exceed Q. Additional Discussion of Periodic-review Systems
Additional Discussion of Periodic-review Systems Service Level in Periodic-Review Systems ·Type2 service ■To find the Q to satisfy the Type2 service objectiveβ，itis necessary to obtain an expression for the fraction of demand that stock out each period. Define n(Q),the expected number of demands that stock out at the end of period.n()=(x-)f(x)dx -Since the demand per period is u,then the proportion of demand that stock out each period is n(Q)/u=1-B,giving n(Q) =(1-B)μ
Service Level in Periodic-Review Systems • Type 2 service To find the Q to satisfy the Type 2 service objective , it is necessary to obtain an expression for the fraction of demand that stock out each period. Define n(Q), the expected number of demands that stock out at the end of period. Additional Discussion of Periodic-review Systems ( ) ( ) () Q nQ x Q f x dx Since the demand per period is , then the proportion of demand that stock out each period is n(Q)/ =1- , giving n(Q) =(1- )
Additional Discussion of Periodic-review Systems Example 5.9:Mac,the owner of the newsstand described in Example 5.1,wishes to use a Type 1 service level of 90 percent to control his replenishment of the Computer Journal. √F(R)=0=0.9 Check in Table A-4,z=1.28 √Q*=0z+u=(4.74)1.28)+11.73=17.8≈18
Example 5.9: Mac, the owner of the newsstand described in Example 5.1, wishes to use a Type 1 service level of 90 percent to control his replenishment of the Computer Journal. F(R)= =0.9 Check in Table A-4, z=1.28 Q*= σz+ μ=(4.74)(1.28)+11.73=17.8 ≈18 Additional Discussion of Periodic-review Systems
Additional Discussion of Periodic-review Systems Using a Type 2 service of 90 percent,we obtain n(Q)=(1-β)u=(0.1)11.73)=1.173 It follows that L(z)=n(Q)/o=1.173/4.74=0.2475; From Table A-4,we find z~0.35 Then Q*=oz+u=(4.74)(0.35)+11.73=13.4≈13
Using a Type 2 service of 90 percent, we obtain n(Q)=(1- β) μ=(0.1)(11.73)=1.173 It follows that L(z)=n(Q)/ σ=1.173/4.74=0.2475; From Table A-4, we find z ≈0.35 Then Q*= σz+ μ =(4.74)(0.35)+11.73=13.4 ≈13 Additional Discussion of Periodic-review 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
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
Multiproduct Systems Example 5.10,Performance of 20 Stock Items Selected at Random Part Number Price($) Yearly Demand Dollar Volume($) 5497J 2.25 260 585.00 3K62 2.85 43 122.55 88450 1.50 21 31.50 P001 0.77 388 298.76 2M993 4.45 612 2723.40 4040 6.10 220 1342.00 W76 3.10 110 341.00 J335 1.32 786 1037.52 R077 12.80 14 179.20 70779 24.99 334 8346.66 4J65E 7.75 24 186.00 334Y 0.68 77 52.36 8ST4 0.25 56 14.00 16113 3.89 89 346.21 45000 7.70 675 5197.50 7878 6.22 66 410.52 6193L 0.85 148 125.80 TTR77 0.77 690 531.30 39SS5 1.23 52 63.96 93939 4.05 12 48.60
Multiproduct Systems Example 5.10, Performance of 20 Stock Items Selected at Random Part Number Price ($) Yearly Demand Dollar Volume ($) 5497J 2.25 260 585.00 3K62 2.85 43 122.55 88450 1.50 21 31.50 P001 0.77 388 298.76 2M993 4.45 612 2723.40 4040 6.10 220 1342.00 W76 3.10 110 341.00 JJ335 1.32 786 1037.52 R077 12.80 14 179.20 70779 24.99 334 8346.66 4J65E 7.75 24 186.00 334Y 0.68 77 52.36 8ST4 0.25 56 14.00 16113 3.89 89 346.21 45000 7.70 675 5197.50 7878 6.22 66 410.52 6193L 0.85 148 125.80 TTR77 0.77 690 531.30 39SS5 1.23 52 63.96 93939 4.05 12 48.60