Re-Order Point Problems
Set 3: Advanced
Problem 1: Problem 7.2 – Average Inventory With Isafety
Weekly demand for DVD-Rs at a retailer is normally distributed
with a mean of 1,000 boxes and a standard deviation of 150.
Currently, the store places orders via paper that is faxed to the
supplier. Assume 50 working weeks in a year. Lead time for
replenishment of an order is 4 weeks. Fixed cost (ordering and
transportation) per order is $100. Each box of DVD-Rs costs $1.
Annual holding cost is 25% of average inventory value. The
retailer currently orders 20,000 DVD-Rs when stock on hand
reaches 4,200.
R/week = N(1000,150)
a.1) How long, on average, does a box of
DVD-R spend in the store?
Icycle =Q/2 = 20000/2
Isafety = ROP – LTD
LTD = L×R = 4×1000 = 4000
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
L = 2 week
Q = 20,000
ROP = 4,200
50 weeks /yr
Ave. R /year = 50,000
S =100
C=1
H=0.25C = $0.25
Sep-2012
2
Problem 1: Problem 7.2 – Average Inventory With Isafety
Isafety = 4200 - 4000 = 200
Average inventory = Icycle + Isafety
I = Average Inventory = 20000/2 + 200 = 10200
RT = I =  1000T = 10200  T = 10.2 weeks
a.2) What is the annual ordering and holding
cost.
Number of orders R/Q
R/Q = 50,000/20,000 = 2.5
Ordering cost = 100(2.5) = 250
R/week = N(1000,150)
L = 2 week
Q = 20,000
ROP = 4,200
50 weeks /yr
Ave. R /year = 50,000
S =100
C=1
H=0.25C = $0.25
Annual holding cost = 0.25% ×1 ×10200 =2550
Total inventory system cost excluding purchasing = 250+2550 =
2800. OC ≠ CC for two reason, (i) Not EOQ, (ii) Isafety
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
3
Problem 1: Problem 7.2 – Average Inventory With Isafety
b.1) Assuming that the retailer wants the probability of stocking
out in a cycle to be no more than 5%, recommend an optimal
inventory policy: Q and R policy.
2(50000)(100)
EOQ 
=6325
0.25
 LTD : L R  LTD : 4 (150)  300 LTD : N (4000,300) Z(95%) = 1.65
Z(95%) = 1.65
Isafety = zσLTD =1.65(300) = 495
ROP = 4000+195 = 4495
Optimal Q and R Policy: Order 6325 whenever inventory on hand
is 4495
b.2) Under your recommended policy, how long, on average,
would a box of DVD-Rs spend in the store?
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
4
Problem 1: Problem 7.2 – Average Inventory With Isafety
Average inventory = Icycle+Isafety = Q/2 + Isafety
Average inventory = I = 6325/2 + 495 = 3658
RT = I  1000T = 3658  T = 3.66 weeks
c) Reduce lead time form 4 to 1. What is the impact on cost and
flow time?
 LTD : 1 R  LTD  150 LTD : N (1000,150) I s  1.65(150)  247.5
2
2
2
Safety stock reduces from 495 to 247.5 247.5 units reduction
That is 0.25(247.5) = $62 saving
Average inventory = Icycle + Isafey = Q/2 + Isafety
Average Inventory = I = (6,325/2) + 247.5 = 3,410.
Average time in store  I = RT
3410 = 1000T  T = 3.41 weeks
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
5
Problem 2: Problem 7.8 – Centralization
R/week in each warehouse follows Normal distribution with
mean of 10,000, and Standard deviation of 2,000. Compute mean
and standard deviation of weekly demand in all the warehouse
together – considered as a single central warehouse.
Mean (central) = 4(10000) = 40,000 per week
Variance (central) = SUM(variance at each warehouse)
Variance (central) = 4(variance at each warehouse)
Variance at each warehouse = (2000)2 = 4,000,000
Variance (central) = 4(4,000,000) = 16,000,000
Standard deviation (central) =
16,000,000  4,000
2 RS
2(4000)(50)(1000)
Q

H
2.5
R (central) = Normal(40,000, 4,000)
Flow Variability; Safety Inventory
=40000
Ardavan Asef-Vaziri
Sep-2012
6
Problem 2: Problem 7.8 – Centralization
The replenishment lead time (L) = 1 week.
Standard deviation of demand during lead time in the centralized
system
 LTD ( of all)  N  LTD ( of each)
 LTD ( of all)  4 2000  4000
Safety stock at each store for 95% level of service
Isafety (central) = 1.65 x 4,000 = 6,600.
Average inventory in the centralized system
I = Icycle +Isafety
I = Q/2 +Isafety
40000/2 +6600
=26600
Average Centralized Inventory
= 26600
Average Decentralized Inventory = 53200
Average time spend in inventory (Centralized): RT =I
4000T = 26600  T = 0.67 weeks
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
7
Problem 2: Problem 7.8 – Centralization
S(R/Q) = 1000(4)(500,000/40,000) = 50,000
H(Isafety + Icycle) = 2.5(26,000) = 66,500
Purchasing cost = CR = 10(500,000) = 5,000,000
We do not consider RC because it does not depend on the
inventory policy. But you can always add it.
Inventory system cost for four warehouse in centralized system
116,500
Inventory system cost for four warehouses in decentralized
system
233,000
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
8
Problem 3: Problem 7.3- Lead Time vs Purchase Price
Home and Garden (HG) chain of superstores imports decorative planters from
Italy. Weekly demand for planters averages 1,500 with a standard deviation of
800. Each planter costs $10. HG incurs a holding cost of 25% per year to carry
inventory. HG has an opportunity to set up a superstore in the Phoenix region.
Each order shipped from Italy incurs a fixed transportation and delivery cost
of $10,000. Consider 52 weeks in the year.
a) Determine the optimal order quantity (EOQ).
EOQ 
2(78000)(10000)
=24980
2.5
b) If the delivery lead time is 4 weeks and HG
wants to provide a cycle service level of 90%,
how much safety stock should it carry?
 LTD : L R
 LTD : 4 (800)  1600
z  1.28  LTD  1600
Flow Variability; Safety Inventory
I s  1.28(1600)
R/week = N(1500,800)
C = 10
h = 0.25
H =0.25(10) = 2.5
52 weeks /yr
R = 78000/yr
S =10000
L = 4 weeks
SL = 90%
LTD : N (6000,1600)
I s  2048
Ardavan Asef-Vaziri
Sep-2012
9
Problem 3: Problem 7.3- Lead Time vs Purchase Price
c) Reduce L from 4 to 1, Increase C by 0.2 per unit. Yes or no?
c.1) Rough computations.
Purchasing cost increase  0.2(78000) = 15600
 LTD : 1(800)  800
z  1.28
 LTD  800
LTD : N (6000,800)
I s  1.28(800)
I s  1024
Safety stock decrease  from 2048 to 1024  1024 units reduction
Safety stock cost saving = 2.5(1024) = 2560
15600-2560 = 13040 increase in cost
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
10
Problem 3: Problem 7.3- Lead Time vs Purchase Price
c.1) Detailed computations.
For the original case of C = 10 and H =2.5
TC *(every thing)  TC * +CR +HIs
TC *  2 RSH  62450
R/week = N(1500,800)
C = 10 + 0.2
H = .25(10+.2) = 2.55
52 weeks /yr
R = 78000
S =10000
L = 4 weeks
SL = 90%
Change in C  Change in H. Not only purchasing cost changes
But also cost of EOQ and Is Changes
Reduce L from 4 to 1
Purchasing cost increase = 0.2(78000) = 15600
Safety stock reduces from 2048@2.5 to 1024@2.55
Safety stock cost saving = 2.5(2048)-2.55(1024) = 2509
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
11
Problem 3: Problem 7.3- Lead Time vs Purchase Price
Total inventory cost (ordering + carrying) will
also increase
TC  2RSH 1
*
1
*
2
*
1
TC

TC
TC2*  2RSH 2
2 RSH 2
2 RSH 1
*
2
*
1
TC

TC
H2
H1
TC2*

*
TC1
R/week = N(1500,800)
C = 10 + 0.2
H = .25(10+.2) = 2.55
52 weeks /yr
R = 78000
S =10000
L = 4 weeks
SL = 90%
2.55
2. 5
TC2*  1.00995 TC1*
Total inventory cost (ordering + carrying) increase =
0.00995*62450 = 621
Total impact = +621+15600-2509 = 13712
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
12
Problem 3: Problem 7.3- Lead Time vs Purchase Price
2(500000)(1000)
2 RS

Q
2.5
H
z(0.95) = 1.65, σLTD = 2,000
# of warehouses = 4
R/week at each warehouse
N(10,000, 2,000)
50 weeks per year
C = 10
H=0.25(10) = 2.5 /year
S = 1000
L = 1 week
SL = 0.95
=20000
Is = 1.65 x 2,000 = 3,300.
ROP = LTD + Is = 1×10,000 + 3,300 = 13,300.
Average inventory at each warehouse :
Each time we order Q=20000  Icycle = Q/2 = 20000/2=10000
Average Inventory = Ic + Is =10000+ 3300
=13300
Average Decentralized inventory in 4 warehouses =
I = 4(13300) = 53200
Demand in in 4 warehouses = 4(10000) =40000/w
RT = I  40000T= 53200
Flow Variability; Safety Inventory
T = 1.33 weeks
Ardavan Asef-Vaziri
Sep-2012
13
Problem 3: Problem 7.3- Lead Time vs Purchase Price
OC = S(R/Q) = 1000[(50×10,000)]/Q = 25,000.
CC = H (Icycle + Isafety) = H(Q/2+Isafety) = 2.5 (13300) =32,250.
We do not consider RC because it does not depend on the
inventory policy. But you can always add it.
RC = 10 [(50×10,000)]= 5,000,0000
Inventory system cost for one warehouse excluding purchasing
TC = 25,000 + 32,000 = 58,200
Inventory system cost for four warehouses = 4(58,250)
Inventory system cost for four warehouses = 233,000
Inventory system cost for four warehouses including purchasing
= 4(5,000,000) + 233,000 = 20,233,000
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
14