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To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Sixth Edition 2002 Prentice Hall, Inc. All rights reserved.
Chap ter 13 -
Inventory
Management
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Inven tory Costs
In terest o r
Oppo rtuni ty Cos ts
Storage andHand l ing Costs
Taxes , Insurance, and
Shr inkage Cos ts Order ing and Setup Costs
Transpo rtat ion Costs
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Types o f Inven tory
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Types o f Inven toryCyc le Inven tory
Safety Stoc k Invento ry
Average cycle inventory =Q+ 0
2
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Types o f Inven toryCyc le Inven tory
Safety Stoc k Invento ry
Ant ic ipat ion Inventory
Average cycle inventory =Q+ 0
2
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Types o f Inven toryCyc le Inven tory
Safety Stoc k Invento ry
Ant ic ipat ion Inventory
Pipel ine Invento ry
Average cycle inventory =
Pipeline inventory = DL = dL
Q+ 02
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Types o f Inven tory
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Types o f Inven tory
Cycle inventory = Q/2
Example 13.1
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Types o f Inven tory
Cycle inventory = Q/2= 280/2= 140 drills
Example 13.1
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Types o f Inven tory
Cycle inventory = Q/2= 280/2= 140 drills
Pipeline inventory = DL = dL
Example 13.1
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Types o f Inven tory
Cycle inventory = Q/2= 280/2= 140 drills
Pipeline inventory = DL = dL
= (70 drills/week)(3 weeks)= 210 drills
Example 13.1
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Types o f Inven tory
Figure 13.1
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ABC Analysis
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ABC Analysis
10 20 30 40 50 60 70 80 90 100
Percentage of items
Pe
rcentageofdo
llarvalue
100
90
80
70
60
50
40
30
20
10
0
Figure 13.2
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ABC Analysis
10 20 30 40 50 60 70 80 90 100
Percentage of items
Pe
rcentageofdo
llarvalue
100
90
80
70
60
50
40
30
20
10
0
Figure 13.2
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ABC Analysis
10 20 30 40 50 60 70 80 90 100
Percentage of items
Pe
rcentageofdo
llarvalue
100
90
80
70
60
50
40
30
20
10
0
Class C
Class A
Class B
Figure 13.2
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ABC Analysis
10 20 30 40 50 60 70 80 90 100
Percentage of items
Pe
rcentageofdo
llarvalue
100
90
80
70
60
50
40
30
20
10
0
Class C
Class A
Class B
Figure 13.2
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ABC Analysis
10 20 30 40 50 60 70 80 90 100
Percentage of items
Pe
rcentageofdo
llarvalue
100
90
80
70
60
50
40
30
20
10
0
Class C
Class A
Class B
Figure 13.2
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ABC Analysis
10 20 30 40 50 60 70 80 90 100
Percentage of items
Pe
rcentageofdo
llarvalue
100
90
80
70
60
50
40
30
20
10
0
Class C
Class A
Class B
Figure 13.2
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HowMuch?
When!
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EconomicOrderQuantity
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Economic Order Quantity
1. Demand rate is constant
2. No const rain ts on lot s ize3. Only re levant costs are hold ing and
order ing/setup
4. Decis ions fo r items are independentfrom other i tems
5. No uncer tain ty in lead t ime or supply
Assumpt ions
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Economic Order Quantity
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Economic Order Quantity
On-h
andinventory
(units)
TimeFigure 13.3
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Economic Order Quantity
On-h
andinventory
(units)
TimeFigure 13.3
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Economic Order Quantity
On-h
andinventory
(units)
Time
Receiveorder
Q
Figure 13.3
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Economic Order Quantity
On-h
andinventory
(units)
Time
Receiveorder
Q
1 cycleFigure 13.3
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Economic Order Quantity
On-h
andinventory
(units)
Time
Receiveorder
Inventory depletion(demand rate)
Q
1 cycleFigure 13.3
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Economic Order Quantity
On-h
andinventory
(units)
Time
Receiveorder
Inventory depletion(demand rate)
Q
1 cycleFigure 13.3
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Economic Order Quantity
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Economic Order Quantity
Annualcost(dollars)
Lot Size (Q)Figure 13.4
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Economic Order Quantity
Annualcost(dollars)
Lot Size (Q)
Holding cost (HC)
Figure 13.4
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Economic Order Quantity
Annualcost(dollars)
Lot Size (Q)
Holding cost (HC)
Ordering cost (OC)
Figure 13.4
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Economic Order Quantity
Annualcost(dollars)
Lot Size (Q)
Ordering cost (OC)
Holding cost (HC)
Total cost = HC+ OC
Figure 13.4
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Economic Order Quantity
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Economic Order Quantity
| | | | | | | |50 100 150 200 250 300 350 400
Lot Size (Q)
3000
2000
1000
0
Example 13.2
Annualcost(d
ollars)
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Economic Order Quantity
| | | | | | | |50 100 150 200 250 300 350 400
Lot Size (Q)
3000
2000
1000
0
Total cost = (H) + (S)D
Q
Q2
Holding cost = (H)Q2
Ordering cost = (S)DQ
Example 13.2
Annualcost(d
ollars)
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Economic Order Quantity
Annualcost(d
ollars)
| | | | | | | |50 100 150 200 250 300 350 400
3000
2000
1000
0
Total cost = (H) + (S)D
Q
Q2
Holding cost = (H)Q2
Ordering cost = (S)DQ
Lot Size (Q)
Birdfeeder costs
C= (H) + (S)Q
2
DQ
D= (18 /week)(52 weeks) = 936 unitsH= 0.25 ($60/unit) = $15S= $45 Q= 390 units
Example 13.2
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Economic Order Quantity
Annualcost(d
ollars)
| | | | | | | |50 100 150 200 250 300 350 400
Lot Size (Q)
3000
2000
1000
0
Total cost = (H) + (S)D
Q
Q2
Holding cost = (H)Q2
Ordering cost = (S)DQ
Bird feeder costs
C= (H) + (S)Q
2
DQ
D= (18 /week)(52 weeks) = 936 unitsH= 0.25 ($60/unit) = $15S= $45 Q= 390 units
C = $2925 + $108 = $3033
Example 13.2
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Economic Order Quantity
Annualcost(d
ollars)
| | | | | | | |50 100 150 200 250 300 350 400
Lot Size (Q)
3000
2000
1000
0
Currentcost
Current
Q
Total cost = (H) + (S)D
Q
Q2
Holding cost = (H)Q2
Ordering cost = (S)DQ
Bird feeder costs
C= (H) + (S)Q
2
DQ
D= (18 /week)(52 weeks) = 936 unitsH= 0.25 ($60/unit) = $15S= $45 Q= 390 units
C = $2925 + $108 = $3033
Example 13.2
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Economic Order Quantity
Annualcost(d
ollars)
| | | | | | | |50 100 150 200 250 300 350 400
Lot Size (Q)
3000
2000
1000
0
Currentcost
Current
Q
Total cost = (H) + (S)D
Q
Q2
Holding cost = (H)Q2
Ordering cost = (S)DQ
Bird feeder costs
C= (H) + (S)Q
2
DQ
D= (18 /week)(52 weeks) = 936 unitsH= 0.25 ($60/unit) = $15S= $45 Q= 390 units
C = $2925 + $108 = $3033
Example 13.2
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Economic Order Quantity
Annualcost(d
ollars)
| | | | | | | |50 100 150 200 250 300 350 400
Lot Size (Q)
3000
2000
1000
0
Currentcost
Current
Q
Total cost = (H) + (S)D
Q
Q2
Holding cost = (H)Q2
Ordering cost = (S)DQ
Bird feeder costs
C= (H) + (S)Q
2
DQ
D= (18 /week)(52 weeks) = 936 unitsH= 0.25 ($60/unit) = $15S= $45 Q= 468 units
Example 13.2
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Economic Order Quantity
Annualcost(d
ollars)
| | | | | | | |50 100 150 200 250 300 350 400
Lot Size (Q)
3000
2000
1000
0
Currentcost
Current
Q
Total cost = (H) + (S)D
Q
Q2
Holding cost = (H)Q2
Ordering cost = (S)DQ
Bird feeder costs
C= (H) + (S)Q
2
DQ
D= (18 /week)(52 weeks) = 936 unitsH= 0.25 ($60/unit) = $15S= $45 Q= 468 units
C = $3510 + $90 = $3600
Example 13.2
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Economic Order Quantity
| | | | | | | |50 100 150 200 250 300 350 400
Lot Size (Q)
3000
2000
1000
0
Currentcost
Current
Q
Total cost = (H) + (S)D
Q
Q2
Holding cost = (H)Q2
Ordering cost = (S)DQ
Figure 13.4
Annualcost(d
ollars)
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Economic Order Quantity
| | | | | | | |50 100 150 200 250 300 350 400
Lot Size (Q
)
3000
2000
1000
0
Currentcost
Current
Q
Total cost = (H) + (S)D
Q
Q2
Holding cost = (H)Q2
Ordering cost = (S)DQ
Bird feeder costs
D= (18 /week)(52 weeks) = 936 unitsH= 0.25 ($60/unit) = $15
S= $45 Q= EOQ
C= (H) + (S)Q
2
DQ
EOQ =2DSH
Example 13.3
Annualcost(d
ollars)
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Economic Order Quantity
| | | | | | | |50 100 150 200 250 300 350 400
Lot Size (Q
)
3000
2000
1000
0
Currentcost
Current
Q
Total cost = (H) + (S)D
Q
Q2
Holding cost = (H)Q2
Ordering cost = (S)DQ
Bird feeder costs
D= (18 /week)(52 weeks) = 936 unitsH= 0.25 ($60/unit) = $15
S= $45 Q= 75 units
C= (H) + (S)Q
2
DQ
EOQ =2DSH
Example 13.3
Annualcost(d
ollars)
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Economic Order Quantity
| | | | | | | |50 100 150 200 250 300 350 400
Lot Size (Q)
3000
2000
1000
0
Currentcost
Current
Q
Total cost = (H) + (S)D
Q
Q2
Holding cost = (H)Q2
Ordering cost = (S)DQ
Bird feeder costs
D= (18 /week)(52 weeks) = 936 unitsH= 0.25 ($60/unit) = $15
S= $45 Q= 75 units
C = $562 + $562 = $1124
C= (H) + (S)Q
2
DQ
EOQ =2DSH
Example 13.3
Annualcost(d
ollars)
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Economic Order Quantity
| | | | | | | |50 100 150 200 250 300 350 400
Lot Size (Q)
3000
2000
1000
0
Currentcost
Current
Q
Total cost = (H) + (S)D
Q
Q2
Holding cost = (H)Q2
Ordering cost = (S)DQ
Bird feeder costs
D= (18 /week)(52 weeks) = 936 unitsH= 0.25 ($60/unit) = $15
S= $45 Q= 75 units
C = $562 + $562 = $1124
C= (H) + (S)Q
2
DQ
EOQ =2DSH
Example 13.3
Annualcost(d
ollars)
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Economic Order Quantity
| | | | | | | |50 100 150 200 250 300 350 400
Lot Size (Q)
3000
2000
1000
0
Currentcost
Current
Q
Total cost = (H) + (S)D
Q
Q2
Holding cost = (H)Q2
Ordering cost = (S)DQ
Bird feeder costs
D= (18 /week)(52 weeks) = 936 unitsH= 0.25 ($60/unit) = $15
S= $45 Q= 75 units
C = $562 + $562 = $1124
C= (H) + (S)Q
2
DQ
EOQ =2DSH
Lowestcost
Best Q
(EOQ)Example 13.3
Annualcost(d
ollars)
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Economic Order Quantity
| | | | | | | |50 100 150 200 250 300 350 400
Lot Size (Q)
3000
2000
1000
0
Currentcost
Current
Q
Total cost = (H) + (S)D
Q
Q2
Lowestcost
Best Q
(EOQ)Example 13.3
Annualcost(dollars)
Bird feeder costs
D= (18 /week)(52 weeks) = 936 unitsH= 0.25 ($60/unit) = $15
S= $45 Q= 75 units
C = $562 + $562 = $1124
C= (H) + (S)D
Q
Q2
EOQ =2DSH
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Economic Order Quantity
| | | | | | | |50 100 150 200 250 300 350 400
Lot Size (Q)
3000
2000
1000
0
Currentcost
Current
Q
Total cost = (H) + (S)D
Q
Q2
Birdfeeder costs
D= (18 /week)(52 weeks) = 936 unitsH= 0.25 ($60/unit) = $15
S= $45 Q= 75 units
C = $562 + $562 = $1124
C= (H) + (S)D
Q
Q2
EOQ =2DSH
Lowestcost
Best Q
(EOQ)Example 13.3
D
Q
Time between orders
TBOEOQ = = 75/936 = 0.080 yearEOQ
D
Annualcost(d
ollars)
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Economic Order Quantity
| | | | | | | |50 100 150 200 250 300 350 400
Lot Size (Q)
3000
2000
1000
0
Currentcost
Current
Q
Total cost = (H) + (S)D
Q
Q2
Birdfeeder costs
D= (18 /week)(52 weeks) = 936 unitsH= 0.25 ($60/unit) = $15
S= $45 Q= 75 units
Lowestcost
Best Q
(EOQ)
Time between orders
TBOEOQ = = 75/936 = 0.080 year
TBOEOQ = (75/936)(12) = 0.96 months
TBOEOQ = (75/936)(52) = 4.17 weeks
TBOEOQ = (75/936)(365) = 29.25 days
EOQ
D
Example 13.3
Annualcost(d
ollars)
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HowMuch?
When!
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Continuous Review
Time
Figure 13.7
On-handinve
ntory
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Continuous Review
Time
R
Orderreceived
Q
OH
Figure 13.7
On-handinve
ntory
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Continuous Review
Orderreceived
Q
OH
Orderplaced
IP
TBO
L
R
Figure 13.7
On-handinve
ntory
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Continuous Review
Time
Orderreceived
Orderreceived
Q Q
OH OH
Orderplaced
Orderplaced
IP IP
TBO
L
TBO
L
TBO
L
R
Orderreceived
Q
OH
Orderplaced
IP
Orderreceived
Figure 13.7
On-handinve
ntory
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Continuous Review
Time
Orderreceived
Orderreceived
Q Q
OH OH
Orderplaced
Orderplaced
IP IP
TBO
L
TBO
L
TBO
L
R
Orderreceived
Q
OH
Orderplaced
IP
Orderreceived
Chicken Soup
On-handinve
ntory
Example 13.4
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Continuous Review
Time
Orderreceived
Orderreceived
Q Q
OH OH
Orderplaced
Orderplaced
IP IP
TBO
L
TBO
L
TBO
L
R
Orderreceived
Q
OH
Orderplaced
IP
Orderreceived
Example 13.4
Chicken Soup
R= Average demand during lead time= (25)(4) = 100 cases
On-handinve
ntory
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Continuous Review
On-handinve
ntory
Time
Orderreceived
Orderreceived
Q Q
OH OH
Orderplaced
Orderplaced
IP IP
TBO
L
TBO
L
TBO
L
R
Orderreceived
Q
OH
Orderplaced
IP
Orderreceived
Example 13.4
Chicken Soup
R= Average demand during lead time= (25)(4) = 100 cases
IP = OH+ SR BO
= 10 + 200
0 = 210 cases
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Uncertain Demand
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Uncertain Demand
Time
R
Figure 13.8
On-handinve
ntory
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Uncertain Demand
Time
On-handinve
ntory
Orderreceived
QOH
Orderplaced
Orderplaced
Orderreceived
IP IP
R
TBO1 TBO2 TBO3
L1 L2 L3
Q
Orderplaced
Q
Orderreceived
Orderreceived
Figure 13.6
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Reorder Point / Safety Stock
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Reorder Point / Safety Stock
Averagedemand
duringlead time
Figure 13.9
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Reorder Point / Safety Stock
Probability of stockout(1.0 0.85 = 0.15)
Cycle-service level = 85%
Averagedemand
duringlead time
zL
R
Figure 13.9
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Probability of stockout(1.0 - 0.85 = 0.15)
Cycle-service level = 85%
Averagedemand
duringlead time
zL
R
Reorder Point / Safety Stock
Probability of stockout(1.0 0.85 = 0.15)
Example 13.5
Safety Stock/R
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Probability of stockout(1.0 - 0.85 = 0.15)
Cycle-service level = 85%
Averagedemand
duringlead time
zL
R
Reorder Point / Safety Stock
Probability of stockout(1.0 0.85 = 0.15)
Example 13.5
Safety Stock/R
Safety stock = zL= 2.33(22) = 51.3= 51 boxes
Reorder point = ADDLT + SS= 250 + 51= 301 boxes
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Lead Time Distributions
Figure 13.10
t= 15
+75
Demand for week 1
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Lead Time Distributions
Figure 13.10
t= 15
+
+75
Demand for week 1
75Demand for week 2
t
= 15
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Lead Time Distributions
Figure 13.10
t= 15
=
+
+75
Demand for week 1
75Demand for week 2
75
Demand for week 3
t
= 15
t= 15
L d Ti Di t ib ti
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Lead Time Distributions
Figure 13.10
t= 15
=
+
+75
Demand for week 1
75Demand for week 2
75
Demand for week 3
t= 26
225Demand for
three-week lead time
t
= 15
t= 15
L d Ti Di t ib ti
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Lead Time Distributions
Example 13.6
t= 15
=
+
+75
Demand for week 1
75Demand for week 2
75
Demand for week 3
t= 26
225Demand for
three-week lead time
t= 15
t= 15
Bird feeder Lead Time Distribution
L d Ti Di t ib ti
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Lead Time Distributions
Example 13.6
t= 15
=
+
+75
Demand for week 1
75Demand for week 2
75
Demand for week 3
t= 26
225Demand for
three-week lead time
t= 15
t= 15
Bird feeder Lead Time Distribution
t= 1 week d= 18 L = 2
L d Ti Di t ib ti
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Lead Time Distributions
Example 13.6
t= 15
=
+
+75
Demand for week 1
75Demand for week 2
75
Demand for week 3
t= 26
225Demand for
three-week lead time
t= 15
t= 15
Bird feeder Lead Time Distribution
t= 1 week d= 18 L = 2
L = t L = 5 2 = 7.1
L d Ti Di t ib ti
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Lead Time Distributions
Example 13.6
t= 15
=
+
+75
Demand for week 1
75Demand for week 2
75
Demand for week 3
t= 26
225Demand for
three-week lead time
t= 15
t= 15
Bird feeder Lead Time Distribution
t= 1 week d= 18 L = 2
L = t L = 5 2 = 7.1
Safety stock = zL = 1.28(7.1) = 9.1 or 9 units
Reorder point = dL + Safety stock= 2(18) + 9 = 45 units
L d Ti Di t ib ti
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Lead Time Distributions
Example 13.6
t= 15
=
+
+75
Demand for week 1
75Demand for week 2
75
Demand for week 3
t= 26
225Demand for
three-week lead time
t= 15
t= 15
Bird feeder Lead Time Distribution
t= 1 week d= 18 L = 2
Reorder point = 2(18) + 9 = 45 units
L d Ti Di t ib ti
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Lead Time Distributions
Example 13.6
t= 15
=
+
+75
Demand for week 1
75Demand for week 2
75
Demand for week 3
t= 26
225Demand for
three-week lead time
t= 15
t= 15
Bird feeder Lead Time Distribution
t= 1 week d= 18 L = 2
Reorder point = 2(18) + 9 = 45 units
C = ($15) + ($45) + 9($15)75
2
936
75
C= $562.50 + $561.60 + $135 = $1259.10
P i di R i S t
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Periodic Review Systems
P i di R i S t
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Periodic Review Systems
TimeP P
T
Figure 13.11
On-handin
ventory
P i di R i S t
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Periodic Review Systems
Time
On-handin
ventory
P P
T
Q1
Orderplaced
Figure 13.11
P i di R i S t
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Periodic Review Systems
Time
On-handin
ventory
P P
T
Q1
L
Orderplaced
Figure 13.11
P i di R i S t
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Periodic Review Systems
Time
On-handin
ventory
P P
T
Q1
L
Orderplaced
Orderreceived
Figure 13.11
Periodic Review Systems
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Periodic Review Systems
Time
On-handin
ventory
Orderreceived
Orderreceived
Orderplaced
Orderplaced
Q1Q2
Q3
L L L
P P
T
Orderreceived
Figure 13.11
Periodic Review Systems
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Periodic Review Systems
Time
On-handinventory
IP1
IP3
IP2
Orderreceived
Orderreceived
IP IP
OH OH
Orderplaced
Orderplaced
Q1Q2
Q3
L L L
P P
Protection interval
T
Orderreceived
IP
Figure 13.11
Periodic Review Systems
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Time
On-handinventory
IP1
IP3
IP2
Orderreceived
Orderreceived
IP IP
OH OH
Orderplaced
Orderplaced
Q1Q2
Q3
L L L
P P
Protection interval
TOrderreceived
IP
Periodic Review Systems
Orderreceived
IPTV Set - PSystem
IP= OH+ SRBO
Qt= T- IPt
T= 400 BO= 5OH= 0 SR= 0
IP= 0 + 0 5 =5 sets
Q= 400
(
5) = 405 sets
Periodic Review Systems
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Periodic Review Systems
Example 13.7
Time
On-handinventory
IP1
IP3
IP2
Orderreceived
Orderreceived
IP IP
OH OH
Orderplaced
Orderplaced
Q1Q2
Q3
L L L
P P
Protection interval
TOrderreceived
IP
Periodic Review Systems
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Time
On-handinventory
IP1
IP3
IP2
Orderreceived
Orderreceived
IP IP
OH OH
Orderplaced
Orderplaced
Q1Q2
Q3
L L L
P P
Protection interval
TOrderreceived
IP
Periodic Review Systems
Example 13.8
Bird feeder Calculating Pand T
Periodic Review Systems
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Time
On-handinventory
IP1
IP3
IP2
Orderreceived
Orderreceived
IP IP
OH OH
Orderplaced
Orderplaced
Q1Q2
Q3
L L L
P P
Protection interval
TOrderreceived
IP
Periodic Review Systems
Example 13.8
EOQ = 75 units D= (18 units/week)(52 weeks) = 936 units
t= 18 units L = 2 weeks cycle/service level = 90%
Bird feederCalculating Pand T
Periodic Review Systems
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Time
On-handinventory
IP1
IP3
IP2
Orderreceived
Orderreceived
IP IP
OH OH
Orderplaced
Orderplaced
Q1Q2
Q3
L L L
P P
Protection interval
TOrderreceived
IP
Periodic Review Systems
Example 13.8
EOQ = 75 units D= (18 units/week)(52 weeks) = 936 units
t= 18 units L = 2 weeks cycle/service level = 90%
Bird feederCalculating Pand T
P= (52) = (52) = 4.2 or 4 weeksEOQD
75936
P+L = t P+ L = 5 6 = 12 units
Periodic Review Systems
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Time
On-handinventory
IP1
IP3
IP2
Orderreceived
Orderreceived
IP IP
OH OH
Orderplaced
Orderplaced
Q1Q2
Q3
L L L
P P
Protection interval
TOrderreceived
IP
Periodic Review Systems
Example 13.8
T = Average demand during the protection interval + Safety stock= d(P+ L ) + zP + L= (18 units/week)(16 weeks) + 1.28(12 units) = 123 units
EOQ = 75 units D= (18 units/week)(52 weeks) = 936 units
t= 18 units L = 2 weeks cycle/service level = 90%
Bird feederCalculating Pand T
P= (52) = (52) = 4.2 or 4 weeksEOQD
75936
P+L = t P+ L = 5 6 = 12 units
Periodic Review Systems
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Time
On-handinventory
IP1
IP3
IP2
Orderreceived
Orderreceived
IP IP
OH OH
Orderplaced
Orderplaced
Q1Q2
Q3
L L L
P P
Protection interval
TOrderreceived
IP
Periodic Review Systems
Example 13.8
EOQ = 75 units D= (18 units/week)(52 weeks) = 936 units
t= 18 units L = 2 weeks cycle/service level = 90%
Bird feederCalculating Pand T
P= 4 weeks T= 123 units
Periodic Review Systems
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Time
On-handinventory
IP1
IP3
IP2
Orderreceived
Orderreceived
IP IP
OH OH
Orderplaced
Orderplaced
Q1Q2
Q3
L L L
P P
Protection interval
TOrderreceived
IP
Periodic Review Systems
Example 13.8
EOQ = 75 units D= (18 units/week)(52 weeks) = 936 units
t= 18 units L = 2 weeks cycle/service level = 90%
Bird feederCalculating Pand T
C = ($15) + ($45) + 15($15)4(18)
2
936
4(18)
C= $540 + $585 + $225 = $1350
P= 4 weeks T= 123 units
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Comparison ofQand PSystems
P Systems
Q Systems
Convenient to adm inister
Orders may be combined
IP on ly required at review
Ind iv idual review frequenc ies Possib le quant i ty discounts
Lower, less -expens ive safety sto cks
ABC Analysis Solved Problem 2
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ABC Analysis Solved Problem 2
Comparison of P and Q Systems
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Comparison of P and Q Systems Solved Problem 6
