A foundry receives an order for four custom-designed castings. The customer will pay $30,000 for each of four good castings. The customer will accept neither fewer nor more than four castings. It will cost $15,000 to produce each casting. Each casting is produced independently; the probability of a casting being good is estimated to be 0.90. a. How many castings should be produced? b. What is the probability of losing money on the transaction? c. It has been argued that the estimate of cost to produce each casting is not very accurate. Analyze the sensitivity of the optimum batch production quantity to errors in estimating the values of the cost parameters; determine how much the cost of producing each casting has to decrease in order to increase by one the number of castings to produce. d. It has been argued that the estimate of the probability of a casting being good is inaccurate. Analyze the sensitivity of the optimum batch production quantity to changes in the probability. n = number of castings = (4x30,000) -(15000n) Per the Excel table below the optimum is when n=5 castings would be produced with a profit of $35,224.5 n P(X >=4 ) Profit Optimal 4 0.6561 60000 39366.00 5 0.59049 45000 26572.05 6 0.531441 30000 15943.23 7 0.478297 15000 7174.45 8 0.430467 0 0.00 Good Cast # of Scheduled Cast Good Cast # of Scheduled Cast 4 5 6 7 4 5 6 7 0 0.0001 0 0 0 0 -60000 -75000 -90000 -105000 1 0.0036 0.0005 0.0001 0 1 -60000 -75000 -90000 -105000 2 0.0486 0.0081 0.0012 0.0002 2 -60000 -75000 -90000 -105000 3 0.2916 0.0729 0.0146 0.0026 3 -60000 -75000 -90000 -105000 4 0.6561 0.3281 0.0984 0.023 4 60000 45000 30000 15000 5 0 0.5905 0.3543 0.124 5 0 45000 30000 15000 6 0 0 0.5314 0.372 6 0 0 30000 15000 7 0 0 0 0.4783 7 0 0 0 15000 Good Cast # of Scheduled Cast 4 5 6 7 0 -6 0 0 0 1 -216 -37.5 -9 0 2 -2916 -607.5 -108 -21 3 -17496 -5467.5 -1314 -273 4 39366 14764.5 2952 345 5 0 26572.5 10629 1860 6 0 0 15942 5580 7 0 0 0 7174.5 Expected Profit 18732 35224.5 28092 14665.5 The probability of losing money when 5 cast are scheduled is when only 4 good castings are produced .The probability of less than 4 good castings would be 0005+.0081+.0729 or .0815 Good Cast # of Scheduled Cast 4 5 6 7 0 0.0001 0 0 0 1 0.0036 0.0005 0.0001 0 2 0.0486 0.0081 0.0012 0.0002 3 0.2916 0.0729 0.0146 0.0026 4 0.6561 0.3281 0.0984 0.023 5 0 0.5905 0.3543 0.124 6 0 0 0.5314 0.372 7 0 0 0 0.4783 Good Cast # of Scheduled Cast 4 5 6 7 0 -6 0 0 0 1 -216 -37.5 -9 0 2 -2916 -607.5 -108 -21 3 -17496 -5467.5 -1314 -273 4 39366 14764.5 2952 345 5 0 26572.5 10629 1860 6 0 0 15942 5580 7 0 0 0 7174.5 Expected Profit 18732 35224.5 28092 14665.5 Prob of Losing -6112.5

A foundry receives an order for four custom-designed castings. The customer will pay $30,000 for each of four good castings. The customer will accept neither fewer nor more than four castings. It will cost $15,000 to produce each casting. Each casting is produced independently; the probability of a casting being good is estimated to be 0.90. a. How many castings should be produced? b. What is the probability of losing money on the transaction? c. It has been argued that the estimate of cost to produce each casting is not very accurate. Analyze the sensitivity of the optimum batch production quantity to errors in estimating the values of the cost parameters; determine how much the cost of producing each casting has to decrease in order to increase by one the number of castings to produce. d. It has been argued that the estimate of the probability of a casting being good is inaccurate. Analyze the sensitivity of the optimum batch production quantity to changes in the probability. n = number of castings = (4x30,000) -(15000n) Per the Excel table below the optimum is when n=5 castings would be produced with a profit of $35,224.5 n P(X >=4 ) Profit Optimal 4 0.6561 60000 39366.00 5 0.59049 45000 26572.05 6 0.531441 30000 15943.23 7 0.478297 15000 7174.45 8 0.430467 0 0.00 Good Cast # of Scheduled Cast Good Cast # of Scheduled Cast 4 5 6 7 4 5 6 7 0 0.0001 0 0 0 0 -60000 -75000 -90000 -105000 1 0.0036 0.0005 0.0001 0 1 -60000 -75000 -90000 -105000 2 0.0486 0.0081 0.0012 0.0002 2 -60000 -75000 -90000 -105000 3 0.2916 0.0729 0.0146 0.0026 3 -60000 -75000 -90000 -105000 4 0.6561 0.3281 0.0984 0.023 4 60000 45000 30000 15000 5 0 0.5905 0.3543 0.124 5 0 45000 30000 15000 6 0 0 0.5314 0.372 6 0 0 30000 15000 7 0 0 0 0.4783 7 0 0 0 15000 Good Cast # of Scheduled Cast 4 5 6 7 0 -6 0 0 0 1 -216 -37.5 -9 0 2 -2916 -607.5 -108 -21 3 -17496 -5467.5 -1314 -273 4 39366 14764.5 2952 345 5 0 26572.5 10629 1860 6 0 0 15942 5580 7 0 0 0 7174.5 Expected Profit 18732 35224.5 28092 14665.5 The probability of losing money when 5 cast are scheduled is when only 4 good castings are produced .The probability of less than 4 good castings would be 0005+.0081+.0729 or .0815 Good Cast # of Scheduled Cast 4 5 6 7 0 0.0001 0 0 0 1 0.0036 0.0005 0.0001 0 2 0.0486 0.0081 0.0012 0.0002 3 0.2916 0.0729 0.0146 0.0026 4 0.6561 0.3281 0.0984 0.023 5 0 0.5905 0.3543 0.124 6 0 0 0.5314 0.372 7 0 0 0 0.4783 Good Cast # of Scheduled Cast 4 5 6 7 0 -6 0 0 0 1 -216 -37.5 -9 0 2 -2916 -607.5 -108 -21 3 -17496 -5467.5 -1314 -273 4 39366 14764.5 2952 345 5 0 26572.5 10629 1860 6 0 0 15942 5580 7 0 0 0 7174.5 Expected Profit 18732 35224.5 28092 14665.5 Prob of Losing -6112.5

Practical Management Science

6th Edition

ISBN:9781337406659

Author:WINSTON, Wayne L.

Publisher:WINSTON, Wayne L.

Chapter2: Introduction To Spreadsheet Modeling

Section: Chapter Questions

Problem 20P: Julie James is opening a lemonade stand. She believes the fixed cost per week of running the stand...

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Question

*Hello I have the below problem and i already have solved questions a & b i need help solving c & d ONLY***

2.30 A foundry receives an order for four custom-designed castings. The customer will pay
$30,000 for each of four good castings. The customer will accept neither fewer nor
more than four castings. It will cost $15,000 to produce each casting. Each casting is
produced independently; the probability of a casting being good is estimated to be 0.90.
a. How many castings should be produced?
b. What is the probability of losing money on the transaction?
c. It has been argued that the estimate of cost to produce each casting is not very accurate.
Analyze the sensitivity of the optimum batch production quantity to errors in estimating
the values of the cost parameters; determine how much the cost of producing each
casting has to decrease in order to increase by one the number of castings to produce.
d. It has been argued that the estimate of the probability of a casting being good is inaccurate.
Analyze the sensitivity of the optimum batch production quantity to
changes in the probability.

n = number of castings

= (4x30,000) -(15000n)

Per the Excel table below the optimum is when n=5 castings would be produced with a profit of $35,224.5

n	P(X >=4 )	Profit	Optimal
4	0.6561	60000	39366.00
5	0.59049	45000	26572.05
6	0.531441	30000	15943.23
7	0.478297	15000	7174.45
8	0.430467	0	0.00


Good Cast	# of Scheduled Cast		Good Cast	# of Scheduled Cast
4	5	6	7		4	5	6	7
0	0.0001	0	0	0		0	-60000	-75000	-90000	-105000
1	0.0036	0.0005	0.0001	0		1	-60000	-75000	-90000	-105000
2	0.0486	0.0081	0.0012	0.0002		2	-60000	-75000	-90000	-105000
3	0.2916	0.0729	0.0146	0.0026		3	-60000	-75000	-90000	-105000
4	0.6561	0.3281	0.0984	0.023		4	60000	45000	30000	15000
5	0	0.5905	0.3543	0.124		5	0	45000	30000	15000
6	0	0	0.5314	0.372		6	0	0	30000	15000
7	0	0	0	0.4783		7	0	0	0	15000

Good Cast	# of Scheduled Cast
4	5	6	7
0	-6	0	0	0
1	-216	-37.5	-9	0
2	-2916	-607.5	-108	-21
3	-17496	-5467.5	-1314	-273
4	39366	14764.5	2952	345
5	0	26572.5	10629	1860
6	0	0	15942	5580
7	0	0	0	7174.5
Expected Profit	18732	35224.5	28092	14665.5

The probability of losing money when 5 cast are scheduled is when only 4 good castings are produced .The probability of less than 4 good castings would be 0005+.0081+.0729 or .0815

Good Cast	# of Scheduled Cast
4	5	6	7
0	0.0001	0	0	0
1	0.0036	0.0005	0.0001	0
2	0.0486	0.0081	0.0012	0.0002
3	0.2916	0.0729	0.0146	0.0026
4	0.6561	0.3281	0.0984	0.023
5	0	0.5905	0.3543	0.124
6	0	0	0.5314	0.372
7	0	0	0	0.4783

Good Cast	# of Scheduled Cast
4	5	6	7
0	-6	0	0	0
1	-216	-37.5	-9	0
2	-2916	-607.5	-108	-21
3	-17496	-5467.5	-1314	-273
4	39366	14764.5	2952	345
5	0	26572.5	10629	1860
6	0	0	15942	5580
7	0	0	0	7174.5
Expected Profit	18732	35224.5	28092	14665.5
Prob of Losing		-6112.5

1
1
2
4
3 5
4 6
5 7
A
n
7
8
9
10 Good Cast 4
11 0
12 1
13 2
14 3
15 4
16 5
17 6
18 7
19
20
0.0001
0.0036
0.0486
0.2916
0.6561
B
P(X>-4)
=0.9*0.9*0.9*0.9
(4*30000)-(15000*4)=B2*C2
=(4*30000)-(15000*5) =B3*C3
=0.9*0.9*0.9*0.9*0.9
=0.9*0.9*0.9*0.9*0.9*0.9
(4*30000)-(15000*6)=B4*C4
-0.9*0.9*0.9*0.9*0.9*0.9*0.9
(4*30000)-(15000*7) =B5*C5
|=0.9*0.9*0.9*0.9*0.9*0.9*0.9*0.9 (4*30000)-(15000*8) |=B6*C6
0
0
0
# of Scheduled Cast
5
0
C
Profit
0.0005
0.0081
0.0729
0.3281
0.5905
0
0
D
Optimal
6
0
7
0
0
E
0.0001
0.0012 0.0002
0.0146 0.0026
0.023
0.0984
0.3543 0.124
0.5314
0.372
0
0.4783
F
Good Cast 4
0
1
|2
3
4
G
15
16
7
-60000
-60000
-60000
-60000
5
- 75000
-75000
- 75000
-75000
(4*30000)-(15000*4) 45000
=B16*C2
H
0
0
# of Scheduled Cast
6
-90000
-90000
-90000
-90000
30000
(4*30000)-(15000*5) 30000
0
0
|=(4*30000)-(15000*6)
0
7
-105000
-105000
-105000
-105000
15000
15000
15000
K
(4*30000)-(15000*7)
L
M
0
1
2
3
4
5
6
7
Good Cast
4
N
=B11*H11
=B12*H12
=B13*H13
=B14H14
=B15*H15
=B16*H16
=817*H17
=B18*H18
Expected Profit |=SUM(N11:N18)
Prob of Losing
5
=C11*111
=C12*112
=C13*113
|=C14*114
=C15*115
=C16*116
=C17*117
=C18*118
O
# of Scheduled Cast
6
P
=D11*J11
=D12*J12
=D13*J13
=D14*J14
=D15*J15
=D16*J16
=D17"J17
=D18*J18
=SUM(011:018) =SUM(P11:P18)
=SUM(012:014)
7
Q
|=E11*K11
=E12*K12
=E13*K13
=E14*K14
=E15*K15
=E16*K16
=E17*K17
=E18*K18
|=SUM(Q11:Q18)

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