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Logistical and Transportation Planning Methods
Problem Set #5
Problem 1
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Logistical and Transportation Planning Methods
1.203J / 6.281J / 13.665J / 15.073J / 16.76J / ESD.216J
Logistical and Transportation Planning Methods
1.203J / 6.281J / 13.665J / 15.073J / 16.76J / ESD.216J
Logistical and Transportation Planning Methods
We have :
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Logistical and Transportation Planning Methods
1.203J / 6.281J / 13.665J / 15.073J / 16.76J / ESD.216J
Logistical and Transportation Planning Methods
Problem 2
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Logistical and Transportation Planning Methods
1.203J / 6.281J / 13.665J / 15.073J / 16.76J / ESD.216J
Logistical and Transportation Planning Methods
1.203J / 6.281J / 13.665J / 15.073J / 16.76J / ESD.216J
Logistical and Transportation Planning Methods
Problem 3
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Logistical and Transportation Planning Methods
Problem 4
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Logistical and Transportation Planning Methods
1.203J / 6.281J / 13.665J / 15.073J / 16.76J / ESD.216J
Logistical and Transportation Planning Methods
1.203J / 6.281J / 13.665J / 15.073J / 16.76J / ESD.216J
Logistical and Transportation Planning Methods
1.203J / 6.281J / 13.665J / 15.073J / 16.76J / ESD.216J
Logistical and Transportation Planning Methods
1.203J / 6.281J / 13.665J / 15.073J / 16.76J / ESD.216J
Logistical and Transportation Planning Methods
Problem 5
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Logistical and Transportation Planning Methods
a) We are in the case of congestion pricing.
Type 1 customers: λ1 = 30 /hr and c1 = $2/min = $120/hr
Type 2 customers: λ 2 = 24 /hr and c 2 = $3/min = $180/hr
Thus λ = λ1 + λ 2 = 54/hr.
And for both types of customers: E[S] = 1 min =
1
hr and σ S = 0.
60
The total cost is given by C = c Wq .
54(
1
+ 0)
λ[ E 2 [ S ] + σ S2 ]
60 2
With Wq =
=
= 0.075 hr.
54
2(1 − ρ )
2(1 − )
60
Therefore, the cost is:
C = ( c1λ1 + c 2 λ 2 ) Wq = (120 × 30+180 × 24) × 0.075 = $594
b) The marginal cost MC1 for Type 1 customers is given by
MC1 =
dWq
dC
= c1Wq + c
dλ1
dλ
The first term gives the internal cost, and we have c1Wq = $9/hr.
The second term gives the external cost and we have:
c
dWq
dλ
= ( c1λ1 + c 2 λ 2 )[
E 2 [ S ] 1 λE 2 [ S ]
+
] = $110/hr.
2(1 − ρ ) µ 2(1 − ρ )
For Type 2 customers, we have MC 2 =
dWq
dC
= c 2Wq + c
.
dλ 2
dλ
The internal cost is c 2Wq = $13.5 /hr.
The external cost c
dWq
dλ
is the same as for Type 1 customers: c
dWq
dλ
= $110/hr.
The external cost is the same because the service time for the two types of customers is the same.
1
hr.
µ1
120
1
13
Let S 2 be service time of Type 2 customers: E[ S 2 ] =
= 1.625 min =
hr.
µ2
480
c) Let S1 be service time of Type 1 customers: E[ S1 ] =
1
= 0.5 min =
For the entire set of facility users, we have:
λ = 54/hr
1
µ
= E[S] =
λ1 1 λ2 1
1
+
=
hr
λ µ1 λ µ 2
60
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Logistical and Transportation Planning Methods
ρ=
λ
= 0.9
µ
c=
λ1
λ
440
c1 + 2 c 2 =
$/hr
λ
λ
3
The total cost is given by C = cλ Wq , with Wq =
We have E[S 2 ] =
λ[ E 2 [ S ] + σ S2 ]
.
2(1 − ρ )
λ1 1 λ2 1
7
+
=
hr, thus the waiting time is given by:
2
2
λ µ 2 19200
λ µ1
54
7
λ[ E 2 [ S ] + σ S2 ] λE[ S 2 ]
63
Wq =
=
= 11520 =
,
2(1 − ρ )
2(1 − ρ ) 2(1 − 0.9) 640
and C =
440
63
× 54 ×
≈ $720
3
640
d) The same calculations as in part b) give us:
Type of
customers
1
2
Marginal Cost ($)
Internal
External
11.81
67.72
17.72
240.2
Now, the external costs are different. The external cost is higher for Type 2 customers because
their service time is higher.
e) Type 1 customers should be assigned priority because the ratio f1 =
c1
= 4 $/min is greater
E[ S1 ]
that Type 2 customers’ ration f 2 = 1.85 $/min.
f) Let’s compute the total cost of waiting at the facility per hour, given that Type 1 customers
have the priority over Type 2 customers.
C = c1 λ1Wq1 + c 2 λ 2Wq 2
W0 =
λ1 E[ S12 ] + λ 2 E[ S 22 ]
2
Wq1 =
=
λ1 E 2 [ S1 ] + λ 2 E 2 [ S 2 ]
2
=
63
6400
W0
W0
W0
21
=
=
=
λ
1 − a1 1 − ρ1
1600
1− 1
µ1
Wq 2 =
Therefore, C =
W0
=
(1 − a1 )(1 − a 2 )
W0
(1 −
λ1
λ λ
)(1 − 1 − 2 )
µ1
µ1 µ 2
2457
= $614.25/hr
4
This is a 14.7% improvement compared to the cost in c).
=
21
160
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Logistical and Transportation Planning Methods
g) We have the two rations f1 = 4 $/min and f 2 = 1.85 $/min. Since f1 > f 2 , Type 1 customers
should be assigned priority, in order to minimize the expected cost of the total time that the all
customers spend in the system.
h) The service times are no longer deterministic. Therefore, the expected waiting times will
increase.
W0 =
λ1 E[ S12 ] + λ 2 E[ S 22 ]
2
=
λ1 ( E 2 [ S1 ] + σ S21 ) + λ 2 ( E 2 [ S 2 ] + σ S22 )
2
=
63
(This is twice as
3200
much as previously.)
Wq1 =
W0
W0
W0
21
=
=
=
λ
1 − a1 1 − ρ1
800
1− 1
µ1
Wq 2 =
W0
=
(1 − a1 )(1 − a 2 )
W0
(1 −
λ1
λ λ
)(1 − 1 − 2 )
µ1
µ1 µ 2
=
21
80
The waiting times are twice as high as in part f), therefore, the cost is going to be twice as
important as in f):
C = c1 λ1Wq1 + c 2 λ 2Wq 2 = $1228.5/hr