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
Quantity Discount Model The suppliers may charge less per unit for larger orders to encourage the customer to buy their products in larger batches. Two popular ways of discounts: All-units:discount is applied to all of the units in an order; Incremental:only applied to additional units beyond the breakpoints;
•The suppliers may charge less per unit for larger orders to encourage the customer to buy their products in larger batches. •Two popular ways of discounts: All-units: discount is applied to all of the units in an order; Incremental: only applied to additional units beyond the breakpoints; Quantity Discount Model
Quantity Discount Model Example 4.4 Weighty Trash Bag Company's pricing schedule for its large trash can liners: .For orders of less than 500 bags,charges 30 cents per bag; .for orders of 500 or more but fewer than 1,000 bags,charges 29 cents per bag;and .for orders of 1,000 or more,charges 28 cents per bag
Quantity Discount Model Example 4.4 Weighty Trash Bag Company’s pricing schedule for its large trash can liners: •For orders of less than 500 bags, charges 30 cents per bag; •for orders of 500 or more but fewer than 1,000 bags, charges 29 cents per bag; and •for orders of 1,000 or more, charges 28 cents per bag
Quantity Discount Model-Optimal Policy for All- Units Discount Schedule The breakpoints are 500 and 1,000.The discount schedule is all-units; The order cost function C(Q)is defined as 0.309for0≤9<500， g c2-28 0=.301.29 C(Q)=0.299for500≤9<1,000， 0.28Qfor1,000≤Q 5001.000 Q- Fig 4-9 All-Units Discount Order Cost Function
The breakpoints are 500 and 1,000. The discount schedule is all-units; The order cost function C(Q) is defined as Fig 4-9 All-Units Discount Order Cost Function Quantity Discount Model- Optimal Policy for AllUnits Discount Schedule 0.30 0 500, ( ) 0.29 500 1,000, 0.28 1,000 Q for Q C Q Q for Q Q for Q
Quantity Discount Model-Optimal Policy for All- Units Discount Schedule Example 4.4:If Weighty uses trash bags at a fairly constant rate of 600 per yr,how to place order? Suppose that fixed cost of placing an order is $8,and holding costs are based on 20%annual interest rate. First compute EOQ values corresponding to each of the unit cost. 2KA 2×8×600 2= =400,C,0≤9<500 ICo 0.2×0.3 2K1 2×8×600 9= = 406,C1for500≤9<1000 IC 0.2×0.29 2KA 2×8×600 2= = IC2 414,C2forQ≥1000 V0.2×0.28
Example 4.4: If Weighty uses trash bags at a fairly constant rate of 600 per yr, how to place order? Suppose that fixed cost of placing an order is $8, and holding costs are based on 20% annual interest rate. First compute EOQ values corresponding to each of the unit cost. Quantity Discount Model- Optimal Policy for AllUnits Discount Schedule 0 0 0 2 2 8 600 400, 0 500 0.2 0.3 K Q CQ IC 1 1 1 2 2 8 600 406, 500 1000 0.2 0.29 K Q C for Q IC 2 2 2 2 2 8 600 414, 1000 0.2 0.28 K Q C for Q IC
Quantity Discount Model-Optimal Policy for All- Units Discount Schedule .Each curve is valid only for certain values of Q,thus the average annual cost function is given by discontinuous curves. 240 2 G,(Q)=c,+元K/Q+1cQ/2 220 GolQ) GlQ) for j=0,1,and 2 210 G2lQ) 200 G(Q)for0≤Q<500, 190 G(Q)=G(2)for500≤Q<1,000, 18 G2(Q)for1,000≤Q 1002003004005006007008009001,0001,1001,200 Q- Fig.4-12 All-Units Discount Average Annual Cost Function
Quantity Discount Model- Optimal Policy for AllUnits Discount Schedule •Each curve is valid only for certain values of Q, thus the average annual cost function is given by discontinuous curves. 0 1 2 ( ) / /2 0,1, 2 ( ) 0 500, ( ) ( ) 500 1,000, ( ) 1,000 j GQ c KQ IQ jj C for j and G Q for Q G Q G Q for Q G Q for Q Fig. 4-12 All-Units Discount Average Annual Cost Function
Quantity Discount Model-Optimal Policy for All- Units Discount Schedule 20 230 22 GolQ) 400for0≤Q<500, 210 gi(Q) GIQ) E09={406for500≤Q<1,000, 200 414for1,000≤Q 190 18 1002003004005006007008009001,0001,1001,200 Q- Fig.4-12 All-Units Discount Average Annual Cost Function An EOQ value is realizable,if it falls within the interval of EOQ that corresponds to the unit cost that has been used to compute it. (Qo for the example);
Quantity Discount Model- Optimal Policy for AllUnits Discount Schedule 400 0 500, 406 500 1,000, 414 1,000 for Q EOQ for Q for Q Fig. 4-12 All-Units Discount Average Annual Cost Function • An EOQ value is realizable, if it falls within the interval of EOQ that corresponds to the unit cost that has been used to compute it. (Q0 for the example);
Quantity Discount Model-Optimal Policy for All- Units Discount Schedule The goal is to find the minimum of this discontinuous curve, which corresponds to the EOQ. Generally,the optimal solution will be either the largest realizable EOQ or one of the breakpoints that exceeds it. The three candidates are 400,500,and 1,000. G(400)=G(400)=600×0.3+600×8/400+0.2×0.3×400/2=$204.00 G(500)=G(500)=600×0.29+600×8/500+0.2×0.29×500/2=$198.10 G(1,000)=G2(1,000)=600×0.28+600×8/1,000+0.2×0.28×1,000/2=$200.80 Conclusion:the optimal solution is to place a standing order for 500 units with Weighty at an annual cost of $198.10
The goal is to find the minimum of this discontinuous curve, which corresponds to the EOQ. Generally, the optimal solution will be either the largest realizable EOQ or one of the breakpoints that exceeds it. The three candidates are 400, 500, and 1,000. Quantity Discount Model- Optimal Policy for AllUnits Discount Schedule 0 G G (400) (400) 600 0.3 600 8/ 400 0.2 0.3 400 / 2 $204.00 1 G G (500) (500) 600 0.29 600 8 / 500 0.2 0.29 500 / 2 $198.10 2 G G (1,000) (1,000) 600 0.28 600 8/1,000 0.2 0.28 1,000 / 2 $200.80 Conclusion: the optimal solution is to place a standing order for 500 units with Weighty at an annual cost of $198.10
Quantity Discount Model-Optimal Policy for All-Units Discount Schedule Summary of the Solution Technique for All-Units Discounts Determine the largest realizable EOQ value.The most efficient way to do this is to compute the EOQ for the lowest price first,and continue with the next higher price.Stop when the first EOQ is realizable (that is,within the correct interval) Compare the value of the average annual cost at the largest realizable EOQ and at all the price breakpoints that are greater than the largest realizable EOQ.The optimal Q is the point at which the average annual cost is a minimum
Quantity Discount Model- Optimal Policy for All-Units Discount Schedule Summary of the Solution Technique for All-Units Discounts • Determine the largest realizable EOQ value. The most efficient way to do this is to compute the EOQ for the lowest price first, and continue with the next higher price. Stop when the first EOQ is realizable (that is, within the correct interval) • Compare the value of the average annual cost at the largest realizable EOQ and at all the price breakpoints that are greater than the largest realizable EOQ. The optimal Q is the point at which the average annual cost is a minimum