The critical path is A–D–E because it has the longest time duration. The expected completion time is 19 days.

Where T = 21 days, = 19 days, and the sum of the variances for critical path A–D–E is (1.00 + 0.44 + 1.00) = 2.44.
Assuming the normal distribution applies (which is questionable for a sample of three activities), we use the table for the normal probability distribution. Given z = 1.28, the probability that the project can be completed in 21 days is 0.8997, or about 90%.

Because the normal distribution is symmetrical, the probability the project can be completed in 17 days is (1 – 0. 8997) = 0. 1003, or about 10%.
10.

The AON diagram is:

Critical path is B–D–F. Expected duration of the project is 15 weeks.

Activity slacks for the project are:



Start

Finish


Critical



Activity

Earliest

Latest

Earliest

Latest

Slack

Path?



A

0

4

5

9

4

No



B

0

0

3

3

0

Yes



C

5

9

7

11

4

No



D

3

3

8

8

0

Yes



E

8

11

12

15

3

No



F

8

8

15

15

0

Yes

Ch3

Dr. Gulakowicz
Fixed cost, F = $150,000
Revenue per patient, p = $3,000
Variable cost per unit, c = $1000
Breakeven volume, patients
3. Baker Machine Company

Closeness





Department

Rating

Current Plan

Proposed Plan

Pair

w_{ij}

Distance (d_{ij})

w_{ij}d_{ij}

Distance (d_{ij})

w_{ij}d_{ij}

1–2

8

3

24

3

24

1–3

3

1

3

1

3

1–5

9

1

9

2

18

1–6

5

2

10

1

5

2–4

3

1

3

1

3

3–5

8

2

16

3

24

3–6

9

3

27

2

18

4–6

3

2

6

1

3

5–6

3

1

3

1

3



wd =

101

wd =

101

There is no change in the weighteddistance score. These layouts can be assessed using the Layout solver of OM Explorer, as shown following for the current plan.

Baker Machine Block Plan
A good plan would locate the following department pairs close together: 1–2, 3–5, 1–5, 3–6, 1–6. The following layout satisfies these requirements and leaves department 3 unmoved. It also provides oneunit distances for department pairs 2–4 and 4–6.

The weighteddistance (wd) score is:
[8(l) + 3(2) + 9(l) + 5(l) + 3(l) + 8(l) + 9(l) + 3(l) + 3(2)] = 57, a 43.6% reduction over Problem 3’s solution.
Ch4

Gasoline Stations

The gas station in part (b) has a more efficient flow from the perspective of the customer because traffic moves in only one direction through the system.

The gas station in part (a) creates the possibility for a random direction of flow, thereby causing occasional conflicts at the gas pumps.

At the gas station in part (b) a customer could pay from the car. However, this practice could be a source of congestion at peak periods.

Just Like Home Restaurant

The summary of the process chart should appear as follows:

Each cycle of making a singlescoop ice cream cone takes
1.70 + 0.80 + 0.25 + 0.50 = 3.25 minutes. The total labor cost is
($10/hr)[(3.25 min/cone)/60 min](10 cones/hr)(10 hr/day)(363 day/yr)
= $19,662.50.
To make this operation more efficient, we can eliminate delay and reduce traveling by having precleaned scoops available. The improved process chart follows.

Perrotti’s Pizza Pareto chart

Although the frequency of partly eaten pizza is low, it is a serious quality problem because it is deliberate rather than accidental. It is likely to cause extreme loss of goodwill. A common root cause of many of these problems could be miscommunication between the customer and the order taker, between the order taker and production and between production and distribution. This chart was created using OM Explorer.

Causeandeffect diagram

Smith, Schroeder, and Torn (short moves)

The tally sheet given in the problem is essentially a horizontal bar chart. To create a Pareto diagram, the categories are arranged in order of decreasing frequency. This diagram was created using OM Explorer.

Causeandeffect diagram
25.Grindwell, Inc.

Scatter diagram

Correlation coefficient . There is a negative relationship between permeability and carbon content, although it is not too strong.

Carbon content must be increased to reduce permeability index.
The cycle time is reduced to 1.65 + 0.45 + 0.25, or 2.35 minutes. The total labor cost is ($ 10/hr)[(2.35 min/cone)/60 min](10 cones/hr)(10 hr/day)(363 day/yr)
= $14,217.50.
Therefore, the annual labor saving is $19,662.50 – $14,217.50 = $5,445.00.
Ch5.
3. Garcia’s Garage
, n = 100, z = 2
At 8 of 100, the number of returns for service is below average, but this observation is within the control limits. The repair process is still in control.
4. Canine Gourmet Company
= 45 grams, n = 10, = 6 grams

From Table 5.1,
= 0.308, = 0.223, = 1.777
= 1.777(6 grams) = 10.662 grams
= 0.223(6 grams) = 1.338 grams
= 45 grams + 0.308(6 grams) = 46.848 grams
= 45 grams – 0.308(6 grams) = 43.152 grams

The range is in statistical control; however, the averages of samples 2, 4, and 5 are out of statistical control, therefore, the process is out of control.

Marlin Company

Bottle



Sample

1

2

3

4


R

1

.604

.612

.588

.600

.601

.024

2

.597

.601

.607

.603

.602

.010

3

.581

.570

.585

.592

.582

.022

4

.620

.605

.595

.588

.602

.032

5

.590

.614

.608

.604

.604

.024

6

.585

.583

.617

.579

.591

.038








For a quick overview of the data, we can use the Statistics module of POM for Windows, which shows among other things that and = 0.0128”. The graph tracks the cap diameters over the 6 samples, with four in each sample.

, n = 4,
From Table 5.1,
, ,

If the process passes the process capability index test, the process is capable.
Process capability index:
C_{pk} = 1.21
The process is not capable of foursigma quality. The target is 1.33. Consequently, we test to see if the process variability is too large.
The process variability is below foursigma quality, which has a target of 1.33. Management and employees should look for ways to reduce the variability in the process and then recheck the process capability index.

We initially assume the historical grand average is adequate for the central line of the chart:

Student


Year

1

2

3

4

5

6

7

8

9

10

Average

1

63

57

92

87

70

61

75

58

63

71

69.7

2

90

77

59

88

48

83

63

94

72

70

74.4

3

67

81

93

55

71

71

86

98

60

90

77.2

4

62

67

78

61

89

93

71

59

93

84

75.7

5

85

88

77

69

58

90

97

72

64

60

76.0

6

60

57

79

83

64

94

86

64

92

74

75.3

7

94

85

56

77

89

72

71

61

92

97

79.4

8

97

86

83

88

65

87

76

84

81

71

81.8

9

94

90

76

88

65

93

86

87

94

63

83.6

10

88

91

71

89

97

79

93

87

69

85

84.9













The average for the process, and the standard deviation of the 100 historical data points in Table 5.2 is 13.
Although the process is in control, the last four observations are all above the average and exhibit an everincreasing trend. MegaByte should explore for causes of corruption, such as instructor or performance measures, which give incentives for improved test scores. It is possible that students are getting brighter or are becoming more highly motivated. Perhaps admissions standards have been raised. It is possible that teaching methods have improved. The point shown here is: the process must be stable while data are collected for setting control limits.

Hospital administrator

= Total absent/Total observations
= 49/15(64) = 0.051
= 0.0275
= 0.051 + 2.58(0.0275) = 0.1219
= 0.051 – 2.58(0.0275) = – 0.01995, adjusted to zero.

The data from the last three weeks fall within the control limits. Therefore we accept the estimate of 5.1% absenteeism. You must now assess whether this amount of absenteeism is typical for nurse’s aides.

Textile manufacturer


Because the last two samples with 22 and 21 irregularities plot outside the upper control limit, we conclude that the process is out of control.
15. The Money Pit

Lower Specification Calculation
Upper Specification Calculation

Because and have values less than 1, the process is not capable of meeting specifications. Yes, valid because the process is under statistical control, as can be shown by plotting the last 15 observations on control charts. Ask students to demonstrate that the process is in statistical control.

The variability of the process must be greatly reduced. Also, the process should be better centered between the specification limits.
Ch6.

Capacity requirements in five years
This year’s capacity requirement, allowing instead for just a 5percent capacity cushion, is 52.63 (or 50 / [1.0 – 0.05]) customers per day. Essentially you should divide by the desired utilization rate. Five years from now, if demand is only 75 percent of the current level, the customer requirement will be 39.47 (or 52.63 ´ 0.75) customers per day.

Airline company
This year's capacity requirement, allowing for a 25percent capacity cushion, is 93.3 (or 70 / [1.0  0.25] ) customers per day. Three years from now, if demand increases by 20 percent, the customer requirement will be about 112 (or 93.3 ´ 1.2) customers per day for this flight segment.

Automobile brake supplier

The total machine hour requirements for all three demand forecasts:

Pessimistic Forecast

Expected Forecast

Optimistic Forecast


Process

Setup

Process

Setup

Process

Setup

Component

Time
Dp

Time
(D/Q)s

Time
Dp

Time
(D/Q)s

Time
Dp

Time
(D/Q)s

A

750

250.0

900

300.0

1,250

416.7

B

2,000

562.5

2,600

731.3

3,400

956.3

C

850

1,161.7

1,250

1,708.3

2,000

2,733.3


3,600

+ 1,974.2

4,750

+ 2,739.6

6,650

+ 4,106.3

Demand


5,574.2


7,489.6


10,756.3

The number of hours (N) provided per machine is:
N = (2 shifts/day ´ 8 hours/shift ´ 5 days/week ´ 52 weeks/year)(1.0 – 0.2)
= 3,328 hours/machine
The capacity requirements for three forecasts are:
Pessimistic: M = 5,574.2/3328 = 1.67 or 2 machines
Expected: M = 7,489.6/3328 = 2.25 or 3 machines
Optimistic: M = 10,756.3/3328 = 3.23 or 4 machines

The current capacity is sufficient for the pessimistic and expected forecasts. However, there is a gap of one machine for the optimistic forecast. The gap drops to zero when the 20 percent increase from shortterm options is included.
3 machines ´ 3,328 hours/machine ´ 1.2 = 11,981 hours.
This is greater than 10,756 hours required.
Ch7

CKC
Station X is the bottleneck – 2600 minutes
Work Station

Product A

Product B

Total Load

W

10*90=900

14*85=1190

2090

X

10*90=900

20*85=1700

2600

Y

15*90=1350

11*85=935

2285


CKC

Traditional Method: Product B has the higher contribution margin/unit

Product A

Product B

Price

55.00

65.00

Raw and Purchased Parts

5.00

10.00

Contribution Margin

50.00

55.00

Work Station

Minutes at Start

Mins. Left after Making 85 Bs

Mins. Left after Making 90 As

Can Only Make 70 As

W

2400

1210

310


X

2400

700


700/10 = 70

Y

2400

1465

115


85 units of B and 70 units of A (Product B will use 1700 minutes at station X leaving 700 for Product A.
Product

Overhead

Raw Mat’l

Labor

Purchase Parts

Total Costs

Revenues

A


70 x 2 =140


70 x 3 = 210


70 x $55 = 3850

B


85 x 5 = 425


85 x 5 = 425


85 x $65 = 5525

Totals

3500

565

3 x $6 x 40 hrs = 720

635

5420

9375

Revenue – costs = profit
$9,375  $5,420 = $3,955

Bottleneckbased approach: Product A has the higher contribution margin/unit at the bottleneck

Product A

Product B

Margin

50.00

55.00

Time at bottleneck

10 min

20 min

Contribution margin per minute

5.00

2.75

Work Station

Minutes at Start

Mins. Left after Making 90 As

Mins. Left after Making 85 Bs

Can Only Make 75 Bs

W

2400

1500

310


X

2400

1500


1500/20 = 75

Y

2400

1050

115


Make 90 units of A (900 minutes used – leaves 1500 minutes) can make 75 units of B
Product

Overhead

Raw Mat’l

Labor

Purchase Parts

Total Costs

Revenues

A


90 x 2 = 180


90 x 3 = 270


90 x $55 = 4950

B


75 x 5 = 375


75 x 5 = 375


75 x $65 = 4875

Totals

3500

555

3 x $6 x 40 hrs = 720

640

5415

9825

Profit=Revenue – costs
$9,825 – $5,415 = $4,410

$4,410 $3,955 = $455 increase using TOC, which is a 12% increase

Student answers will vary  this is one possible solution. Assemblyline balancing with longest work element rule to produce 40 units per hour.



S1 = {A, C, E}, S2 = {B}, S3 = {G, D}, S4 = {H, F, I}, S5 = {J, K}



Work Element

Cumulative

Idle Time

Station

Candidate(s)

Choice

Time (sec)

Time (sec)

( sec)

S1

A

A

40

40

50


C

C

30

70

20


E

E

20

90

0

S2

B

B

80

80

10

S3

D, F, G

G

60

60

30


D, F, I

D

25

85

5

S4

F, H, I

H

45

45

45


F, I

F

15

60

30


I

I

10

70

20

S5

J

J

75

75

15


K

K

15

90

0
 