1. $10,000 Invested at 10% Compounded Annually for 5 Years Results in ______

Review Final ENM 505Spring2013

1. $10,000 invested at 10% compounded annually for 5 years results in ______

(FGP 10E3 10 5)  $16,105.10 = 10000(F/P, 10%, 5)

2. The value of $1322.50 at year 2 from today is equivalent to ______now at i = 15% per year.

(PGF 1322.50 15 2)  $1000 = 1322.5(P/F, 15%, 2)

3. How long for money to quadruple at 5% compounded annually?

(NGPFI 1 4 5) 28.41 years

4 = (1.05)n => n = Ln 4 / Ln 1.05 => n = 28.41

4.You get $5K 3 years from now and $10K 5 years from now. What is the annual worth over the 5 years at i = 7%?

(AGP (+ (PGF 5E3 7 3) (PGF 10E3 7 5)) 7 5) $2734.34

5. Which of the two mutually exclusivesis better?

MARR = 20% A BB - A340(F/P, 16.5/12 $, 11) = $395.11.

First cost$4200$7000$2800

Annual revenues 6000 8000 2000

Annual costs 4000 5100 1100

Salvage value 420 600 180

Life (years) 10 10 10

PWA(20%) = (+ -4200 (PGA 2000 20 10) (PGF 420 20 10)) 4252.78

PWB(20%) = (+ -7000 (PGA (- 8000 5100) 20 10) (PGF 600 20 10)) $5255.07

PWB-A(20%) = (+ -2800 (PGA 900 20 10) (PGF 180 20 10))1002.30

PWA=B(20%) = -x + 2000(P/A, 20%, 10) + 420(P/F, 20, 10)

=> x = $8452.78

AFC = (+ (PGA 2E3 20 10) (PGF 420 20 10) -5255.07)  $3197.71

= 4200 – 1002.3 (Beth's algorithm)

6. 10 shares of stock at $20 each are bought and 6 years later are worth $47.50 each. Compute rate of return. 10K(/F, 2%, N) + 2500(F/A, 2%, N quarters)

(P/A, 2%, N quarters) = 22K/2500 = 8.8 => 10 quarters

(IGPFN 20 47.50 6) 15.51%

7. Policy Level1 Level2 Level 3 EV Laplace Maxmin Regret Hurwicz(0.2)
M1 10 20 30 21.5 20 103014
M2 22 26 26 24.8 24.67 22*1822.8*
M3 40 30 15 26.25 28.33* 1515*20
Probability 0.3 0.25 0.45

The regret matrix is 30 10 0 30 PI = 0.3 * 40 + 0.25 * 30 + 0.45 * 30 = 33
18 4 4 18

0 0 15 15*
EVPI = 26.25 => EOL = 33 – 26.25 = 6.75

Probability 0.3 0.25 0.45 

9 2.5 0 11.5
5.4 1 1.8 8.2
0 0 6.75 6.75 *
(EV '(0.3 0.25 0.45) '(18 4 4))  8.2

7b. Profit
AlternativesS1 S2 Laplace Maximin Regret Hurwicz 0.2
A18 7 7.5 7 12 7.2
A214 5 9.5 5 6 6.8
A320 -9 5.5 -9 16 -3.2

Probabilities 0.8 0.2

Sensitivity Analysis: Change the probabilities slightly and re-analyze.

E(A1) = 0.8*8 + 0.2*7 = 7.8; E(A2) = 12.2; E(A3) = 14.2 *** => E(PI)

PI = 0.8 * 20 + 0.2 * 7 = 17.4

PI – EVPI = 17.4 – 14.2 = 3.2 = EOL

Regret Matrix12 0 EOL(A1) = 0.8 * 12 + 0.2 * 0 = 9.6 6 2 EOL(A2) = 0.8 * 6 + 0.2 * 2 = 5.2 0 16 EOL(A3) = 0.8 * 0 + 0.2 * 16 = 3.2  Best

7c. Repeat 7a for a cost matrix

8. Find the expected value RV X with density table shown below

X 0 1 2
P(X) 0.1 0.5 0.4

V(X) = E(X2) – [E(X)]2

(EV '(0.1 0.5 0.4) '(0 1 2)) 1.3

9. Given RVX with density function f(x) = 3x2 on [0, 1], find the variance.

E(X) = ¾; E(X2) = 3/5; V(X) = 3/5 – (¾)2 = 3/80.

V(X) =

(+ 3/5 -9/8 9/16) 3/80

E(X - )2 = V(X)

10. Find and identify all critical points of f(x) = 3x4 - 4x3 - 6x2 + 12x

f' = 12x3 -12x2 -12x +12 = 0 or x3 -x2 -x +1 = 0

(cubic 1 -1 -1 1) (-1 1 1); (1, 5) is and (-1, -11)

f'' = 36x2 -24x -12 = 0 or 3x2 – 2x -1 = 0

Points of inflection: (1, 5) and (-1/3, - 4.4815)

f''(1) = 0; f''(-1) = > 0 => rel min

(polyval '(36 -24 -12) '(1 -1))[0 48] => (-1 -11) is relative minimum
(1, 5) & (-1/3, -4.4815) are pts of inflection. (-1,-11)

(polyval '(3 -4 -6 12 0) '(-1 1)) (-11 5); Roots([3 -4 -6 12 0])  0, -1.5683

10b. Find and identify all critical points of f(x) = x4 - 4x3 - 2x2 + 12x

roots([1 -4 -2 12 0])  0, -1.6458, 3.6458, 2
polyder([1 -4 -2 12 0])  4 -12 -4 12
roots([4 -12 -4 12])  1, -1, 1 (critical x-values)
polyval(polyder([1 -4 -2 12]), [3 -1 1]) 1, 9, -7 => rel min, rel min, rel max
polyval([1 - 4 -2 12 0], [3 -1 1])  -63, -11, 7 => (3,-63); (-1,-11), (1,7)
roots([12 -24 -4])  2.1547, -0.1547 x-pts of inflection

x x x x

11. Find the inverse of #2A((1 5 -7)(3 6 1)(0 0 3)) using the Genie. Hand calculate to
find the determinant.

(inverse #2A((1 5 -7)(3 6 1)(0 0 3)))#2A((-2/3 5/9 -47/27)(1/3 -1/9 22/27)(0 0 1/3))

(det #2A((1 5 -7)(3 6 1)(0 0 3))) -27

inv([1 5 -7;3 6 1; 0 0 3])  -0.6667 0.555 -1.7407
0.3333 -0.1111 0.8148
0 0 0.3333
det([1 5 -7;3 6 1; 0 0 3])  -27

12. Find the system reliability under CFR for 2 components in series with hazards of 1 and 1/2 and in parallel with component with hazard rate of 2.

(CFR11/221000)0.3283 = e-1 * e-0.5 + e-2 - e-1 * e-0.5 * e-2

13. Find the probability of RV X ~N(30, 4) exceeding 34.5. Find x for P(X < x) = 0.378.

(U-normal 30 4 34.5) 0.012224

(inv-normal 30 4 0.378)  29.3794;
To check, (normal 30 4 29.3794)0.378166

14. Find P(X = 2 in 2 hours) for a Poisson RV with mean rate of 4 per hour.

(Poisson 8 2)  0.010735

15. Write the dual of Max 5X1 + 7X2

s.t. 2X1 + 3X2 <= 12
7X1 + 4X2 <= 28.

Xi >= 0

How does one write the dual of constraint X1 + X2 = 2?

X1 + X2 = 2

-X1 -X2 >= -2

Write X1 + X2 <= 2 AND X1 + X2 >=2 and then convert the latter into

-X1 – X2 <= -2 as the constraint so that the single constraint X1 +X2 = 2 is replaced by

the two constraints: X1 + X2 <= 2 and –X1 – X2 <= -2.

Solve the original using the Genie command (LP '((-5 -6 0)(2 3 12)(7 4 28))) and the dual by hand. Compare the dual variables.

16. f(x, y) = -2x2 – y2 + xy + 8x + 3y subject to 3x + y = 10.

L(x, y, ) = -2x2 – y2 + xy + 8x + 3y + (3x + y – 10)

Lx = -4x + 1y + 8 + 3 = 0(1)

Ly = -2y + 1x + 3 -  = 0(2)

L= 3x + 1y -10 = 0(3)

(solve '((-4 1 3 -8)(1 -2 -1 -3)(3 1 0 10)))(33/14 41/14 -1/2)= (x y )

The Hessian or matrix determiner for classifying the point is 8 indicating a max.

= 7 with Lxx and Lyy negative => relative maximum

17. Given the demand is 500 per year and the ordering cost is $5 and each item costs 40 cents with a holding cost of 8 cents each, find the EOQ. How many orders per year areneeded?

TC = (500 * 0.40) + 5* 500/Q + 0.08Q/2

dTC/dQ = -2500/Q2 + 0.08/2 = 0 when Q* = (2 * CoD/Ch)1/2 = (5000/0.08)1/2 = 250

500/250 = 2 orders per year.

(EOQ 500 5 0.08) Q-star = 250 units; Cycle time = 182.50 working days

18. a) If cars arrive at 10 per hour at a fast food window and service time is 4 minutes a car, how manycars on average are expected in the waiting line?

b)What is the probability if being idle? 1/3 How many on average are awaiting service? 4/3

c) What is average time in minutes spent in system? 0.2 hours or 12 minutes.

How many customers per hour will be servedby the servers?

If always busy, 15 customers, but since only working 2/3 time, imply 2/3 of 15 or 10 customers, since in steady state, 10 arrive and 10 depart.

 = 10/15

(MM1 10 15) 

P0 = 0.333

WS = 0.200 or 12 minutes

WQ = 0.133

LS = 2.000

LQ = 1.333 UTILIZATION = 0.667

19. Graphically solve Max Z = 2X1 – X2

Subject to 1X1 – X2 <= 1

2X1 + X2 >= 6

Beware unbounded.

6

1 3

(LP '((-2 1 0)(1 -1 1)(-2 -1 -6)))

20. max z = 3X1 + 2X2 Solve graphically and verify using final tableau.

s.t. 2X1 + 1X2 <= 100

1X1 + 1X2 <= 80

1X1 + 0X2 <= 40

0 0 1 1 0 180

0 1 -1 2 0 60

0 0 -1 1 1 20

1 0 1 -1 0 20

21. Find the median of (38 13 2 85 79 14 83 91 38 62) and of f(x) = 3x2 on [0, 1].

22. Find the expected value of RV X with density f(x) = e-xdx.

23. Given X ~ N(43, 12), find P(X > 42) and P(40 < X < 43)

(- 1 (normal 43 12 42))

24. Solve for x and y given that 1x + 2y = 17

5x – 2y = 1

25. Repeat #24 using matrices. -2 -2 [17
-5 1 1] x=3; y = 7

-12

26. Find the mean of [3 -4 -6 12 0].

27. Find the determinant of matrix A = [4 5 6;3 7 9;9 2 5]. 2 2

28. Find the trace of A. Trace is the sum of the diagonal elements

29. Solve Ax = [7 9 11] for matrix A defined in #27.

(Solve '((4 5 6 7)(3 7 9 9)(9 2 5 11)))(5/8 -3/4 11/8) = (0.6250, -0.7500, 1.3750]

(Linsolve #2A((4 5 6 )(3 7 9 )(9 2 5 )) #2A((7)(9)(11))) ((5/8) (-3/4) (11/8))

30. The daily random variable flaw ratewas Poisson and resulted in:
5 4 4 5 1 1 1 4 5 2 forthe last 10 days. 3.2 = 3*3.21/2

Create the x-bar chart for L = 3.

31. If C = A + B is valid, show that c = ab is also valid with truth tables.

ABCabcA+Bab
000111 0 11
001110 0 1
010101 1 0
011100 1 0
100011 1 0
101010 1 0
110001 1 0
111000 1 0

32. Compute probability of the third flip of a fair coin showing the first head.

P(X = x) = qx-1p (½)2(½)

33. Compute your monthly mortgage payment on a loan of $250,000 from a bank charging 6% interest compounded monthly for 60 months.

A = A/P(250K 0.005 60)  $4833.20

34. Compute the present worth of the following cash flow.

$1000

5% 7% 8%

0 3 5 7

35. There are 4 arrivals every 12 minutes which can be serviced at a rate of 30 per hour. How long isthe queue idle on average?

36. If the maximum LP formulation is unbounded, will the dual also be?

37. Find all solutions to: 1x1 + 2x2 = 3
2x1 + 4x2 = 5

38. Find the determinant of matrix [1 2 3; 4 5 6; 7 8 9]. 2 1 2
5 1

39.Solve min z = 50x1 + 100x2 and the dual graphically.14

Subject to 7x1 + 2x2 >= 28
2x1 + 12x2 > = 24

4

40. Given exponential distribution f(t) = e-t, t  0, write the survivor function and its mean and variance.

S(t) = e-t;  = E(T) =

41. Give a quick test to determine if a random sample came from an exponential or a Poisson. (mu-svar data)

42. How long will a bicyclist take to cover 60 laps at a constant rate of one lap per 2 minutes and 23 seconds? Answer within 10 seconds

43. A brick and a half weigh 2½ pounds. How much does the brick weigh?

44. 3 6 18 108 1944 209952 ?

45. The daily flaw rate was Poisson and resulted in: 5 4 4 5 1 11 4 5 2 for last 10 days. Create the x-bar chart for L = 3.

46. If C = A + B, show that c = ab with truth tables.

ABCabcA+Bab
000111 0 11
001110 0 1
010101 1 0
011100 1 0
100011 1 0
101010 1 0
110001 1 0
111000 1 0

47. Given f(x) = x3 - ax2 + bx + c, find a, b and c given that the critical x-values occur at x = -1 and x = 3 and that the maximum value is -29,

f'(x) = 3x2 – 2ax + b

3 + 2a + b = 0 or 2a + b = -3

27 – 6a + b = 0 or -6a + b = -27

8a = 24 => a = 3

b = -9

f(-1) = -1 -3 + 9 + c = -29 => c = -34

48. A particle is shot upward from a point 112 ft above the ground with initial velocity of 96ft/sec.

a) How fast is it moving when it is 240 ft above the ground?

a = -g => v = -gt+ 96 => s = -gt2/2 + 96t + 112

240 = -gt2/2+ 96t +112

(quadratic -16 96 -128)(4, 2)

v(2) = -32(2) + 96 = 32 ft/sec

b) When will it reach its highest point?

-gt = -96 or when t = 3 sec

c) At what speed will it strike the ground?

s(t) = 0 = -16t2 + 96t + 112

(quadratic -16 96 112)(7 -1)

v(7) = -32 (7) + 96

=-128 ft/sec or speed = 128 ft/sec

49. What is the minimum number of identical components, each with reliability 0.2 that must be put in parallel to get a system probability of 0.95?

1 – 0.95 = 0.8n => n = 13.43 or 14

1