Production and Operation Managements Inventory Control Subject to Known Demand Prof.JIANG Zhibin Dr.GENG Na Department of Industrial Engineering Management Shanghai Jiao Tong University
Production and Operation Managements Prof. JIANG Zhibin Dr. GENG Na Department of Industrial Engineering & Management Shanghai Jiao Tong University Inventory Control Subject to Known Demand
Contents .Introduction .Types of Inventories .Motivation for Holding Inventories; .Characteristics of Inventory System; Relevant Costs; .The EOQ Model: EOQ Model with Finite Production Rate Quantity Discount Models .Resource-constrained multiple product system .EOQ models for production planning Power-of-two policies
Contents •Introduction •Types of Inventories •Motivation for Holding Inventories; •Characteristics of Inventory System; •Relevant Costs; •The EOQ Model; •EOQ Model with Finite Production Rate •Quantity Discount Models •Resource-constrained multiple product system •EOQ models for production planning •Power-of-two policies
The EOO Model-Basic Model 国 Express the average annual cost as a function of the lot size Order cost in each order cycle:C(Q)=K+cQ; The average holding cost during one order cycle is hQ/2; The average annual cost(suppose there are n cycles in a year) G(Q)=(Ktcon ho K+cO hO K元 nT 2 O/1 2 Q +2c+一 2 KA h 2KA >0for9>0 G@)=0→0= 2KA h
The EOQ Model-Basic Model Express the average annual cost as a function of the lot size • Order cost in each order cycle: C(Q)=K+cQ; • The average holding cost during one order cycle is hQ/2; • The average annual cost (suppose there are n cycles in a year) : ( ) ( ) 2 /2 2 K cQ n hQ K cQ hQ K hQ GQ c nT Q Q 2 3 2 '( ) ; ''( ) 0 0; 2 Kh K G Q G Q for Q Q Q * 2 '( ) 0 K GQ Q h
The EOO Model-Basic Model Example 4.1 Pencils are sold at a fairly steady rate of 60 per week; Pencils cost 2 cents each and sell for 15 cents each; Cost $12 to initiate an order,and holding costs are based on annual interest rate of 25%. Determine the optimal number of pencils for the book store to purchase each time and the time between placement of orders Solutions √Annual demand rate=60×52=3,120; The holding cost is the product of the variable cost of the pencil and the annual interest-h=0.02 x0.25=0.05 ○* 2KA 2×12×3,120 =3,870 T= Q 3,870 1.24yr = h 0.05 3,120
• Example 4.1 Pencils are sold at a fairly steady rate of 60 per week; Pencils cost 2 cents each and sell for 15 cents each; Cost $12 to initiate an order, and holding costs are based on annual interest rate of 25%. Determine the optimal number of pencils for the book store to purchase each time and the time between placement of orders The EOQ Model-Basic Model • Solutions Annual demand rate =6052=3,120; The holding cost is the product of the variable cost of the pencil and the annual interest-h=0.02 0.25=0.05 * 2 2 12 3,120 3,870 0.05 K Q h 3,870 1.24 3,120 Q T yr
The EOQ Model-Considering Lead Time Since there exits lead time t(4 moths for Example 4.1),order should be placed some time ahead of the end of a cycle; Reorder point R-determines when to place order in terms of inventory on hand,rather than time. R=A=20m)x3,120mis/W）=l,040 4 E Q=3,870 .1.24 years- 4 months R=1,040 Order placed↑ Order arrives Fig.4-6 Reorder Point Calculation for Example 4.1
Since there exits lead time (4 moths for Example 4.1), order should be placed some time ahead of the end of a cycle; Reorder point R -determines when to place order in terms of inventory on hand, rather than time. The EOQ Model-Considering Lead Time Fig. 4-6 Reorder Point Calculation for Example 4.1 4 ( ) 3,120( / ) 1,040 12 R yr units yr
The EOQ Model-Considering Lead Time Determine the reorder point Computing R for placing order when the lead time exceeds a 2.31 cycles ahead is the same as cycle. that 0.31 cycle ahead. Example: E0Q=25; e .31 =500/yr; cycle Q=25 .0155 year ●T=6WkS, T=25/500=2.6wkS; R=8 t/T=2.31--2.31 Order placed Order arrives cycles are included 2.31 cycles =.1154 year in LT. .Action:place every order 2.31 cycles in Fig 4-7 Reorder Point Calculation for advance. Lead Times Exceeding One Cycle
Determine the reorder point when the lead time exceeds a cycle. The EOQ Model-Considering Lead Time Fig 4-7 Reorder Point Calculation for Lead Times Exceeding One Cycle Example: •EOQ=25; • =500/yr; • =6 wks; Computing R for placing order 2.31 cycles ahead is the same as that 0.31 cycle ahead. •T=25/500=2.6 wks; • /T=2.31---2.31 cycles are included in LT. •Action: place every order 2.31 cycles in advance
The EOQ Model-Sensitivity How sensitive is the annual cost function to errors in the calculation of Q? >Considering Example By substituting Q=1,000, 4.1.Suppose that the we can find the average bookstore orders pencils annual cost for this lot size. in batches of 1,000,rather than 3,870 as the optimal G(O)=KA/0+h0/2 solution indicates.What =(12)3,120)/1,000+(0.005)1,000)/2 additional cost is it =$39.94 incurring by using a suboptimal solution? Which is considerably larger than the optimal cost of $19.35
How sensitive is the annual cost function to errors in the calculation of Q? The EOQ Model- Sensitivity Considering Example 4.1. Suppose that the bookstore orders pencils in batches of 1,000, rather than 3,870 as the optimal solution indicates. What additional cost is it incurring by using a suboptimal solution? By substituting Q=1,000, we can find the average annual cost for this lot size. ( ) / /2 (12)(3,120)/1,000 (0.005)(1,000)/ 2 $39.94 G Q K Q hQ Which is considerably larger than the optimal cost of $19.35
The EOQ Model-Sensitivity Let's obtain a universal solution to the sensitivity problem. Let G*be the average annual holding and setup cost at the optimal solution.Then G*=K元/Q*+hQ*/2 It follows that for any Q, KA h 2KA G(O)Kil+he/2 √2K/h2Vh G* 2KAh h =22 2K元 h 2V2K1 =√2Kh Ox 20 2Q* 2 +品
The EOQ Model- Sensitivity Let’s obtain a universal solution to the sensitivity problem. Let G* be the average annual holding and setup cost at the optimal solution. Then * / * */2 2 2 / 2 2 2 2 G K Q hQ K hK K h h K h K h ( ) / /2 * 2 1 2 2 22 * 2 2* 1 * 2 * It follows that for any Q, G Q K Q hQ G K h K Q h Qh K Q Q Q Q Q Q Q Q
The EOQ Model-Sensitivity 国 To see how one would use this result,consider using a suboptimal lot size in Example 4.1. The optimal solution was Q*=3,870,and we wished to evaluate the cost error of using Q=1,000.Forming the ratio Q*/Q gives 3.87.Hence, G(Q)/G*=(0.5)(3.87+1/3.87)=(0.5)(4.128)=2.06. This says that the average annual holding and setup cost with Q=1,000 is 2.06 times the optimal average holding and setup cost
To see how one would use this result, consider using a suboptimal lot size in Example 4.1. The optimal solution was Q*=3,870, and we wished to evaluate the cost error of using Q=1,000. Forming the ratio Q*/Q gives 3.87. Hence, G(Q)/G*=(0.5)(3.87+1/3.87)=(0.5)(4.128)=2.06. This says that the average annual holding and setup cost with Q=1,000 is 2.06 times the optimal average holding and setup cost. The EOQ Model- Sensitivity
The EOQ Model-Sensitivity In general,the cost function G(Q)is relative insensitive to errors in Q.For example, if Q is twice as large as Q*,then G/G(Q*)=1.25, meaning that an error of 100%in Q will generate an error of 25%in annual average cost. 国 And suppose that the order quantity differed from the optimal by△Q units.A value of Q=Q*+△Q would result in a lower average annual cost than a value of Q-Q*-AQ.---Not symmetric
In general, the cost function G(Q) is relative insensitive to errors in Q. For example, if Q is twice as large as Q*, then G/G(Q*) ＝1.25 , meaning that an error of 100% in Q will generate an error of 25% in annual average cost. And suppose that the order quantity differed from the optimal by ∆Q units. A value of Q=Q*+ ∆Q would result in a lower average annual cost than a value of Q=Q*- ∆Q. ---Not symmetric. The EOQ Model- Sensitivity