isye3232_hw04_sol
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Date
Dec 6, 2023
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ISyE 3232
Stochastic Manufacturing and Service Systems
Spring 2023
L.N. Steimle
Homework 4
Due: 11:59PM ET, February 7th, using Gradescope
Students should access Gradescope through Canvas, navigate to ISYE 3232, and find this week’s home-
work assignment. Students should submit a scanned pdf of their legible work and clearly-marked answers
to the Gradescope submission box, and they should assign each question to the page it appears on, when
prompted. The grading team will not consider missing pages, cut off images, illegible work, or corrupted
files when grading. For each question,
show your work and explain your reasoning.
Please use a PDF
scanner instead of a photo to ensure your upload is easily read.
1. Ford Motor Company will use a certain chemical solution for part of its production process for next
month’s production. Assume that there is an ordering cost of $5,000 incurred whenever an order for
the chemical solution is placed and the chemical solution costs $40 per liter.
Due to short product
life cycle, unused chemical solution cannot be used in following months. There will be a $10 disposal
charge for each liter of chemical solution left over at the end of the month.
If there is a shortage
of chemical solution, the production process is seriously disrupted at a cost of $100 per liter short.
Assume that demand is discrete with Pr
{
D
= 500
}
= Pr
{
D
= 800
}
= 1
/
8, Pr
{
D
= 600
}
= 1
/
2 and
Pr
{
D
= 700
}
= 1
/
4.
(a) What is the expected demand?
(b) What is the optimal order quantity when initial inventory is
m
for each case below?
•
m
= 0,
•
m
= 300,
•
m
= 400,
•
m
= 500,
•
m
= 600
(c) What is the optimal ordering policy for arbitrary initial inventory level
m
? (You need to specify
the critical value
m
*
in addition to the optimal order-up-to-quantity
y
*
.
When
m
≤
m
*
, you
make an order. Otherwise, do not order.)
(d) Assume Ford will order the optimal quantity. What is Ford’s total expected cost when the initial
inventory
m
= 0? What is Ford’s total expected cost when the initial inventory
m
= 500?
2. Redo Problem 1 for the case where the demand is governed by the continuous uniform distribution
varying between 500 and 900 liters.
3. Delta operates a plane between ATL and DTW (Detroit). The plane has 150 seats for economy class
with two price tiers: low fare at $300 and high fare at $450. There is an unlimited demand for the
low fare seats among travelers.
So seats offered at $300 will always be sold out so long as they are
offered the day before the flight. The high fare seats are aimed at business travelers because business
travels plan their trips at the last minute and are willing to pay the higher price. The airline decides
to reserve some seats for their sales to business travelers. Any seat that is reserved so that it could be
sold to a business traveler cannot be sold to other travelers. If a reserved seat doesn’t get bought by a
business traveler, it will remain empty on the flight. Suppose that Delta determines that the number
of business travelers that follows this route is Poisson with mean 25. Note that the PMF of a Poisson
distribution with mean
λ
,
P
(
X
=
x
) can be calculated in Excel using the
=POISSON.DIST(x,
λ
,FALSE)
function and the CDF
F
(
x
) can be calculated using
=POISSON.DIST(x,
λ
,TRUE)
. You may also want
to look up the
=SUMPRODUCT()
in Excel.
(a) Suppose that Delta currently reserves 30 seats for business travelers. What is the expected number
of seats sold to business travelers? What is the expected number of seats that sit empty? (You
may leave Σ in the expression.)
(b) Derive an expected profit formula which is a function of the number of seats reserved for business
travelers,
y
.
•
First, write this expression leaving
E
and
D
in the formula (where
D
is the distribution of
business travelers who follow this route).
•
Then, simplify further but you may leave this formula with Σ in the expression.
(c) What is the optimal number of seats that the company should reserve for business travelers and
what is the expected revenue associated with this policy? (The answers to these questions should
be reported as numbers)
Hint: Here are two different ways you might approach this: 1) You might try to infer what the
overstock and understock costs are in this setting. What does it mean to be over/understocked
in this setting? 2) Alternatively, you may want to use your expected profit formula from the
question above (still written in terms of
E
) to infer what the corresponding values of
b, h, c
v
and
p
are in this setting. Choose whichever approach makes the most sense to you.
(d) Now suppose that any seats that were reserved for business travelers that go unsold can now
instead be sold on a discounted fare website the night before the flight to bargain-hunting cus-
tomers. There will be an unlimited demand for these tickets because they are sold for a low price
of $80 per seat (these tickets can no longer be sold for $300 per seat because it is too late of notice
for most customers).
Does this change the optimal reservation policy?
If so, what is the new
policy and the new expected revenue? If not, why not and what is the new expected revenue?
2
ISyE 3232
Stochastic Manufacturing and Service Systems
Spring 2022
Solutions to Homework 4
1
a)
E
[
Demand
] = 500(1
/
8) + 600(1
/
2) + 700(1
/
4) + 800(1
/
8) = 637
.
6
In addition, we should find y*.
critical ratio=
c
u
-
c
v
c
u
+
c
o
=
100
-
40
100+10
=
6
11
and
F
(500) = 1
/
8
<
6
/
11
, F
(600) = 1
/
8 + 1
/
2 = 5
/
8
>
6
/
11. So y*=600.
b) When we order, we will use the equation
E
[
CostOrder
] = 5000 + 40(600
-
m
) + 100
E
[(
D
-
600)
+
] + 10
E
[(600
-
D
)
+
]
and when we do not order, we will use the equation
E
[
CostNoOrder
] = 100
E
[(
D
-
m
)
+
] + 10
E
[(
m
-
D
)
+
]
.
The terms that lead with 100 represent the expected understocking cost, and the terms that lead with 10
represent the expected overstocking cost.
Let’s start with the highest m values and work down until we no longer should place an order.
For
m
= 600, we know that we should
not order
since we are already at y*.
For
m
= 500, let’s calculate and compare the costs of ordering versus not ordering:
E
[
CostOrder
] = 5000 + 40(600
-
500) + 100
E
[(
D
-
600)
+
] + 10
E
[(600
-
D
)
+
]
E
[
CostOrder
] = 5000 + 40(600
-
500) + 100(
700
-
600
4
+
800
-
600
8
) + 10(
600
-
500
8
)
E
[
CostOrder
] = 14125
E
[
CostNoOrder
] = 100
E
[(
D
-
500)
+
] + 10
E
[(500
-
D
)
+
]
E
[
CostNoOrder
] = 100(
600
-
500
2
+
700
-
500
4
+
800
-
500
8
) + 10(0)
E
[
CostNoOrder
] = 13750
We should still
not order
at m=500
For
m
= 400, let’s calculate and compare the costs of ordering versus not ordering:
E
[
CostOrder
] = 5000 + 40(600
-
400) + 100
E
[(
D
-
600)
+
] + 10
E
[(600
-
D
)
+
]
E
[
CostOrder
] = 5000 + 40(600
-
400) + 100(
700
-
600
4
+
800
-
600
8
) + 10(
600
-
500
8
)
3
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E
[
CostOrder
] = 16125
E
[
CostNoOrder
] = 100
E
[(
D
-
400)
+
] + 10
E
[(400
-
D
)
+
]
E
[
CostNoOrder
] = 100(
500
-
400
8
+
600
-
400
2
+
700
-
400
4
+
800
-
400
8
) + 10(0)
E
[
CostNoOrder
] = 23750
We should now
order 200 items, from m=400 to y*=600
We know that any lower m value will still result in us
ordering up to y*
. Thus, at m=300 we should
order 300
, and at m=0, we should
order 600
.
c) We have deduced that m*, the point of initial inventory where the cost of ordering and the cost of not
ordering are equal, is between m=400 and m=500 since we ordered at 400 but not at 500. We can use this
to our advantage, and now solve for m*:
E
[
CostOrder
] =
E
[
CostNoOrder
]
5000 + 40(600
-
m
*
) + 100
E
[(
D
-
600)
+
] + 10
E
[(600
-
D
)
+
] = 100
E
[(
D
-
m
*
)
+
] + 10
E
[(
m
*
-
D
)
+
]
I will use results from earlier to simplify the left side down to:
34125
-
40
m
*
= 100
E
[(
D
-
m
*
)
+
] + 10
E
[(
m
*
-
D
)
+
]
We know that, if m* is below 500, we will never be overstocked and always be understocked at that inventory
level, so we can ignore the overstocking term and continue our work:
34125
-
40
m
*
= 100(
500
-
m
*
8
+
600
-
m
*
2
+
700
-
m
*
4
+
800
-
m
*
8
)
34125
-
40m
*
=
63750
-
100m
*
m
*
=
493
.
75
d) We have derived equations in terms of m for the cost of ordering and the cost of not ordering, bolded
above.
For m=0, we know that we should order, so
34125
-
40(0) = 34125
For m=500, we know not to order, so
63750
-
100(500) = 13750
as calculated earlier.
4
2
a) E[Demand]=
500+900
2
= 700, and y*=(900
-
500)(
6
11
) + 500 = 718
.
18.
c) We are going to answer part c, solving for m*, first, and use it to answer part b. Let’s do this problem in
the exact same way as question 1.
E
[
CostOrder
] =
E
[
CostNoOrder
]
5000 + 40(718
.
18
-
m
*
) + 100
E
[(
D
-
718
.
18)
+
] + 10
E
[(718
.
18
-
D
)
+
] = 100
E
[(
D
-
m
*
)
+
] + 10
E
[(
m
*
-
D
)
+
]
5000 + 40(718
.
18
-
m
*
) + 100
Z
900
718
.
18
(1
/
400)(
x
-
718
.
18)
dx
+ 10
Z
718
.
18
500
(1
/
400)(718
.
18
-
x
)
dx
=
100
Z
900
m
*
(1
/
400)(
x
-
m
*
)
dx
+ 10
Z
m
*
500
(1
/
400)(
m
*
-
x
)
dx
Using an integral calculator:
38454
.
54
-
40
m
*
=
m
*
2
-
1800
m
*
+ 810000
8
+
m
*
2
-
1000
m
*
+ 250000
80
38454
.
54
-
40m
*
=
11
m
*
2
-
19000
m
*
+ 8350000
80
0 = 11
m
*
2
-
15800
m
*
+ 5273636
.
8
m
*
= 527
.
489
b) So, at m=0, 300, 400, and 500 we should
order
and at m=600 we should
not order
.
d) We derived a formula for cost of ordering, bolded above. At both m=0 and m=500, we should order.
The cost of each of these are:
m=0:
38454
.
54
-
40(0) = 38454
.
54
m=500:
38454
.
54
-
40(500) = 18454
.
54
3
a)
E
[
SeatsSold
] =
E
[
min
(30
, D
)]
=
∞
X
k
=0
min
(30
, k
)
*
e
-
25
25
k
k
!
=
30
X
k
=0
k
*
e
-
25
25
k
k
!
+ 30
∞
X
k
=31
e
-
25
25
k
k
!
5
=24.55 seats sold, and, as a result, 5.45 seats empty.
b)
E
[
Profit
] = 450
E
[
min
(
D, y
)] + 300(150
-
y
)
(1)
= 450
∞
X
k
=0
min
(
y, k
)
*
e
-
25
25
k
k
!
+ 45000
-
300
y
= 450
y
X
k
=0
k
*
e
-
25
25
k
k
!
+ 450
y
∞
X
k
=
y
+1
e
-
25
25
k
k
!
+ 45000
-
300
y
c) There are two ways to think about this.
You could think about determining the understock and
overstock costs: understocking cost= $150 (you miss out on $450-$300=$150 if you reserve too few seats),
overstocking cost= $300 (you miss out on $300-$0=$300 if you reserve too many seats)
So critical ratio=
c
u
-
c
v
c
u
+
c
o
=
150
-
0
150 + 300
=
1
3
OR
You might look at the Expected profit formula in equation (1) from part (b) and see that we can make some
equivalences here of
p
= 450 since 450 is multiplied by our expected units sold (
E
[min(
y, D
)]), and
c
v
= 300
since there is a
-
300
y
term where
y
is our ‘order quantity’ in the profit formula. In that case: critical ratio=
p
+
b
-
c
v
p
+
b
+
h
=
450
-
300
450
=
1
3
These are two different approaches you could use to think about how to determine the critical ratio. One
may resonate with you more than the other.
Using Excel to investigate the CDF of our Poisson, we obtain
y
*
= 23. And
E
[
Revenue
] = 450
E
[
min
(23
, D
)] + 300(127) = 47945
.
6
using the equation that we already derived.
d) understocking cost= $150, overstocking cost= $300-80=220
So critical ratio=
c
u
-
c
v
c
u
+
c
o
=
150
-
0
150 + 220
=
.
405405
and, using the Excel tool provided, y*=24.
And
E
[
Revenue
] = 450
E
[
min
(24
, D
)] + 80
E
[(24
-
D
)
+
] + 300(126) = 48039
.
53
using the equation that we already derived. (Notice if you use the
p
,
c
v
approach, you now have the salvage
value of
s
= 80 which shows up in the denominator of the criterion value.)
6
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