EIN%206336%20%284%29

EIN%206336%20%284%29 - EIN6336 Noteson(s,Q)Policies

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Click to edit Master subtitle style  2/16/11 EIN 6336 Notes on (s, Q) Policies
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 2/16/11 We focus on continuous review (s, Q) policies We first summarize results from Chapter 7 Recall that Chapter 7 assumes Q is given We only optimize with respect to safety stock in Chapter  7 SS =  k L The result in Section 7.7.4 simply says that if  k  is  specified, then we set the reorder point as (s, Q) Policy L L k x ˆ s σ + =
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 2/16/11 Expected total cost written as B1 cost per stockout occasion ( 29 ( 29 k p Q DB vr k Q Q AD k ETRC u L + + + = 1 2 σ ( 29 ( 29 k f Q DB vr k ' ETRC L 1 - ( 29 1 DB vr Q k f L = 1 2 2 2 1 DB vr Q k exp L π = -
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 2/16/11 B1 cost per stockout occasion 1 2 2 2 DB vr Q k exp L σ π = - = vr Q DB ln k L 2 2 1 2 = vr Q DB ln k L 2 2 1
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 2/16/11 Expected total cost written as B2 fractional cost per unit short ( 29 ( 29 Q D k G v B vr k Q Q AD k ETRC u L L σ 2 2 + + + = ( 29 ( 29 ( 29 1 2 - Φ + = k Q D v B vr k ' ETRC L L ( 29 ( 29 vr Q D v B k L L = Φ - 2 1 ( 29 ( 29 ( 29 D B Qr k k p u 2 1 = Φ - =
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 2/16/11 Prob{no stockout}   P1 Same as pu„ (k)   1 - P1 Since holding cost is increasing in k and pu (k) is  decreasing in k, an optimal solution sets pu (k) = 1 - P1 or  (k) = P1 Probability of no stockout (P1)
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 2/16/11 1 – ESPRC/Q   P2 Recall that ESPRC/Q is the ratio of expected shortages  to expected demand in a cycle ESPRC = —LGu(k) Solving the first equation gives  LGu(k)/Q /  1 – P2 Since holding cost is increasing in k and Gu(k) is  decreasing in k, an optimal solution sets Gu(k) = (1 – P2)(Q/ L) Minimum fill rate P2
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 2/16/11 Optimizing Q and k We know that if we set the derivative of the above  wrt k equal to zero, we will obtain B1 cost per stockout occasion ( 29 ( 29 k p Q DB vr k Q Q AD k , Q ETRC u L + + + = 1 2 σ = vr Q DB ln k L π 2 2 1
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 2/16/11 We also need the derivative wrt Q: B1 cost per stockout occasion ( 29 k p Q DB vr Q AD Q ETRC u - + - = 2 1 2 2 ( 29 ( 29 ( 29 ( 29 ( 29 k p A B EOQ vr k p B A D Q vr Q k p B A D u u u + = + = = + 1 1 2 1 1 2 2 ( 29 = + = vr Q DB ln k k p A B EOQ Q L u σ π 2 2 1 1 1
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 2/16/11 Optimizing Q and k
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EIN%206336%20%284%29 - EIN6336 Noteson(s,Q)Policies

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