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Unformatted text preview: 8.2 All oustanding orders at a time t must arrive by time HI. and no order placed
after I can arrive by t+L. Therefore,_ the net stock at time t+L must be equal to
the inventory position at time rminus any demand x0 in t to r+L. Now the
inventory position at time ris s+j,j=1,2,..,,Q with probability IIQ (i) If pm(nolj) is the probability that the net stock at r+L takes on the value no given
an inventory position of s+j at rI then (n I j: ppo(s+j—nofL),s+j—n020
pm 0 J Ootherwise The expected onhand inventory, given the inventory position s+j is 181 fj = inoPMG‘o I "0‘0 = g(5+j'xo)Ppo(xo '37:.) In=0 Thus, Q11"? i=2— (s+j—no)ppo(xo IJ‘cL) using (1).
1:1 quD Furthermore, the probability that a particular demand required backordering is
equal to the probability that the net stock is zero or less. That is, a demand is not satisﬁed
prob given an inventory position = z pmmo l j)
s + j at time t ""‘m = :Ppobe I fit.) xo=s+j
Over all initial inVentory positions,
prob{a demand is not satisﬁed] = 2 lamb:a IEL)
i=1 xu=s+j The expected shortage costs per unit time are CI = (cost per shortage) * (expected demand per unit time) * prob {a demand is not satisﬁed} = ENDié :Ppob‘o 13;) i=1 xn=s+1 Expected total relevant costs per unit time are EIFC(S) = fvri C, 132 1 a; i=1 + Bny‘ i=1 ETRC(S +1) — ETRCU) = O a . = wﬁ i=1 => rippos(s+jta)=321) 1—] ) i
Q xo=s+j 1 :+' I
a;(+l)ppo(xo 13L)+ 0} E(S+j—xo)Ppo(xo Iii.) 19:0 :Ppooco IfI.) 1
+ semi—{— ppo(s + j I m}: 0 1=1 :Ppucs'i'le—EL) F1 3.8 ESPRC = 0.2(0) + (O.8)(8)Gu[ ESPR C
Q 0.02 = 6.4G“ [5 "840 ) /100 Gu["'340]=.3125 0—102): 5—40
8 194 J From Table GR(0.19) = .3125 "40 =0.19 s = 41.52 say 42 units b. SS = s — it, = 42 — 0.8(40) = 10 units 8.15 8.16 a. The total cost per year of the current item with standard deviation of lead time
demand = 100 is $871.78,_ from Problem 8.3. By reducing O'L to 70, there is no
change to the EOQ or to k. However, the reorder point 5 = 500 + 2.1300) = 649,
and the total cost using the formula given in the solution to Problem 8.3 reduces
to $685 .24, a saving of $186.54. This saving would continue until the underlying
parameters change again. Other factors include ease of forecasting, higher service
level, the possibility of using the new system for other items. So the minimum
the company would be willing to pay is $186, but they would probably pay
signiﬁcantly more because of the other factors. ZAD
vr EOQ = = 120 units. Air Alberta is using a B, shortage cost, so the safety factor is determined as in the
appendix to Chapter '7. Check that A = 5.32 >1 w/ZJIQVO'LI'
Thus set k = «Din 5.32 = 1.83 Hence 3 = J’EL +k0'L = 150+ 1.83(60) = 260 units. 13. The expected total relevant costs per year are Emc=%+{9+k0L }r+%mz(kl 2 Q (1) 193 c. He is right—there is a small drop in ETRC. The EOQ minimizes the sum of
replenishing and carrying costs. Increasing Q above the EOQ raises the total of 8.17 8.18 = 240+(60+110)4+ = $1027.527yr. these costs but also reduces Kl pﬂUc), and this reduction outweighs the increase (at ﬁrst). Check: Q = 1.05 EOQ = 126 units DB 1920 * 200 k = 1.80 (proceedingas in part a.) (0.0336) and ETRC = $1021.98/yr., a small reduction. a. Equation 8.3 shows that vn’D = 20 and D = 8, so use an order quantity of 1. Eq.
8.4’s right hand side is 0.0521. The left hand side is 0.2353 when s = 0 and is 0.0349 when r = 1. Therefore the optimal s = 1. b. Chapter '1' methods yield an EOQ of 1 (rounded up). Qii’DB2 = 0.0521, as in part
a. So I: = 1.62 and s = 8(2 weeks)7'52 weeks per year + 1.62(0.55) = 1. The result is the same in this case. Beginning with the EOQ = 165, we iterate using the two equations in the problem.
Note that there is a slight error in the equation that should be pointed out to students: The numerator of the equation for Q should read 2(AD + DBzvolijD. This solution uses the correct formula. Iteration
1 2 3 G (k) Q
165.380
0.0971 172.901
0.1027 173.327
0.103 173.352
0.103 173.353
0.103 173.353 199 k 0.9186 0.88119 0.8861 0.8860 0.8860 0.8860 0.1792 0.1873 0.1878 0.1873 0.1878 0.1878 Therefore, Q = 173 and s = 32 + 0.886042) = 45. 8.19 a. With order quantity Q = 1, the inventory position is always equal to the value 5 +
1. If p;i (ho) is the probability that a backorder of size be is outstanding at a random point in time, then
px(s+1+bo),bo >0 = 3+1
p3(b0) sz(xo)rbo=0
15:0 where p, (x0) is the probability that the iead time demand takes on the value x0.
Hence the expected backorder of a random point in time is E(bo)= Zbap3(ba)
bad) i (x — (s +1))px(xo) xo=s+l 'b. Recall that the average stock on hand equals
Inventory position  Emuorder) + E(bo) =(s+1)+JT:L+E(bD) since Q = 1. Thus the expected total relevant costs as a function of s are 5mm) = [(s + 1) — 52L + 5(3)o )]vr + B3vE(bD) =(s+1)vr— am 3941+ r) 2(x—(s+1))px(xo) x°=s+l One is indifferent between reorder levels s and s+1 when 200 01' ETRC(s) _= ETRC(S +1) (5 +1)vr— iLvr + Bzv(1+ r) :(x — (s +1))px(xo) x°=s+l = (s + 2)vr  iLvr + Bzv(1+ r) i (x — (S + 2)),0; (x0) xn=c+2 which implies that ‘Bzv(1+r) i px(x0)=vr—232v(l+r) 2px(x0) xn=5+1 Xo=$+2
" vr
=> x x = ——
10:2,}: ( a) Bzv(1+ r)
=> P» (5 +1) r = Bz(l+r) ...
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 Spring '08
 Wilson
 inventory position, Total relevant costs, ETRC

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