OM Numerical 2012

Post Graduate Diploma in ManagementOperations Management – Numerical Problems

Productivity1. Two types of passenger cars – Deluxe & Limited versions - were produced by a car manufacturer

in 2004.Quantities sold, price per unit and labour hours are as follows. What is the labour productivity for each car?

Type of Car Quantity Rupees per unit

Deluxe car 4000 units sold Rs. 300,000

Limited car 6000 units sold Rs. 225,000

Labour, Deluxe 20,000 hours Rs. 250 per Hr

Labour, Limited 30,000 hours Rs. 100 per Hr

2. Student Tuition at ABC University is Rs. 100.00 per semester credit hour. The ABC Foundation supplements school revenue by matching student tuition, rupee for rupee. Average class size for a typical three-credit course is 50 students. Labour costs are Rs 4000.00 per class, material costs are Rs 20.00 per student per class, and overhead costs are Rs 25000.00 per class.

(a) What is the multifactor productivity ratio?

(b) If instructors work an average of 14 hours per week for 16 weeks for each three-credit class of 50 students, what is the labour productivity ratio?

(Ans: Multifactor productivity 1.00; Labour Rs. 133.93 per Hour)

3. During a particular week workers at a garment-manufacturing unit worked for 360 hours to produce a batch of 132 garments, of which 52 were “seconds” (meaning that they were flawed). Seconds are sold for Rs 90.00 each at the factory outlet store. The remaining 80 garments are sold to retail distribution at Rs 200.00 each. What is the labour productivity ratio?

(Ans: Rs. 57.44 sales/hour)

Breakeven Analysis4. The operations manager of a renowned brand of fruit Juice Company has narrowed the search for

a new facility location to three towns. Annual fixed costs (land, property taxes, insurance, equipment and building) and variable costs (labour, materials, transportation and variable overhead) are shown in following table

Fixed & variable costs of the fruit juice company

TownFixed Cost per

year (Rs. ‘000)

Variable costper crate (Rs.)

Meerut 2000 12Delhi 3000 10

Ludhiana 1200 15

Plot the total cost curves for the three towns on a single graph. Identify on the graph the approximate range over which each town provides the lowest cost.

Revised Nov 20121 Dr. A. K. Dey

Using break-even analysis, calculate the break-even quantities to determine the range over which each town provides the lowest cost.

5. An orthodontist estimates that adding two new chairs will increase fixed costs by Rs 140000, including the annual equivalent cost of the capital investment and the salary of one more technician. Each new patient is expected to bring in Rs 2500 per year in additional revenue, with variable costs estimated at Rs 500 per patient. The two new chairs will allow the orthodontist to expand the practice by as many as 200 patients annually. How many patients would have to be added for the new process to break even?

6. Two different processes are being considered for making a new product. The first process is less capital intensive, with fixed cost of only Rs 60000 per year and variable cost of Rs 650 per unit. The second process has fixed costs of Rs 300000 but variable costs of only Rs 2.50 per unit

(a) What is the break-even quantity beyond which the second process becomes more attractive than the first?

(b) If the expected annual sale for the product were 800 units, which process would you choose?

7. Calculate the break even product value of using Blue Dart Express Mail (overnight delivery) versus Government Speed Post Service (three day delivery) for sending a package from New Delhi to Mumbai. The following table contains appropriate costs. Assume that the inventory cost is 25% per year of the product value and that there are 365 days in a year.

WeightCost - Blue Dart

Express MailCost - Government Speed

Post ServiceOvernight delivery Three day delivery

500 gms 180.00 25.001 Kg 290.00 45.005 Kg 600.00 150.00

8. JAIPUR Express is setting up a delivery service near or in Delhi. JAIPUR is unsure whether to locate in the city, in the near suburb, or the far suburb. The city requires higher rent (fixed cost) but lower transportation expenses (variable costs). The costs are as follows:

City Near Suburbs Far Suburbs

Fixed Cost, 1000 (Rs) 1000 800 500

Variable cost Rs/job 10 25 50

(i) Find the break-even point.(ii) JAIPUR estimates the distribution of jobs for the coming year as follows

Number of Jobs 4000 12000 16000 20000

Probability 0.2 0.4 0.3 0.1

Which of the three locations should be selected?

Quality & Acceptance SamplingRevised Nov 2012

2 Dr. A. K. Dey

9. Backwoods American Inc., produces expensive water repellent, down-lined parkas. Company implemented a total quality management program in 2000. Following are quality related accounting data that have been accumulated for the five year period after the program’s start.

Year2000 2001 2002 2003 2004

Quality Costs (000s) in USDPrevention 3.2 10.7 28.3 42.6 50.0Appraisal 26.3 29.2 30.6 24.1 19.6Internal Failure

39.1 51.3 48.4 35.9 32.1

External Failure

118.6 110.5 105.2 91.3 65.2

Accounting Measures (000s) in USDSales 2700.6 2690.1 2705.3 2810.2 2880.7Manufacturing Cost

420.9 423.4 424.7 436.1 435.5

i) Compute the company’s total failure costs as a percentage of total quality costs for each of the five years. Does there appear to be trend to this result? If so, speculate on what might have caused the trend.

ii) Compute prevention costs and appraisal costs, each as a percentage of total costs, during each of the five years. Speculate on what the company’s quality strategy appears to be.

iii) Compute quality sales indices and quality cost indices for each of the five years. Is it possible to assess the effectiveness of the company’s quality management program from these index values?

10. The Colonial House Furniture Company manufactures two-drawer oak file cabinets that are sold unassembled through catalogues. The company initiates production of 150 cabinet packages each week. The percentage of good quality cabinets averages 83% per week, and the percentage of poor quality cabinets that can be reworked is 60%.

i) Determine the weekly product yield of the file cabinets.

ii) If the company desires a product yield of 145 units per week, what increase in the percentage of good quality products must result?

11. Burger Doodle is fast food restaurant that processes an average of 680 food orders each day. The average cost of each order is USD 6.15. Four percent of the orders are incorrect, and only 10% of the defective orders can be corrected with additional food items at an average cost of USD 1.75. The remaining defective orders have to be thrown out.


i) Compute the average product cost.

ii) In order to reduce the number of wrong orders, Burger Doodle is going to invest in a computerized ordering and cash register system. The cost of the system will increase the average order cost by USD 0.05 and will reduce defective orders to 1%. What is the annual net cost effect of this quality improvement initiative?

iii) What other indirect effects on quality might be realized by the new computerized order system?

12. From the data of 16 Kg rice filling operations displayed in the table below, compute lower and upper control limits of x bar and R charts. After the process has settled down, hourly samples of 20 sacks have been drawn, averages and ranges were computed. Total

12 times samples were drawn. Values of A = 0.180, = 0.414 and = 1.586 are

collected from tables.

Sample No.

Sample Mean

Sample Range

Sample No.

Sample Mean

Sample Range

Kgs Kgs Kgs Kgs1 16.20 2.00 7 16.00 2.902 15.90 2.10 8 16.10 1.803 16.30 1.80 9 16.30 1.504 16.40 3.00 10 16.30 1.005 15.80 3.50 11 16.40 1.006 15.90 3.10 12 16.50 0.90

13. An automated production process consists of assembling four components to make a 5 amp fuse. Every two hours a sample of 75 fuses is collected and each fuse is tested. Typically in the past, 1.8 fuses on average have failed the test. Compute 3 sigma control limits for a p chart.

14. From the data of 300 ml bottle filling operations displayed in the table below, compute lower and upper control limits of x bar and R charts. After the process has settled down, hourly samples of 20 bottles have been drawn, averages and ranges were computed. Total

12 times samples were drawn. Values of A = 0.180, = 0.414 and = 1.586 are

collected from tables.

Sample No.

Sample Mean

Sample Range

Sample No.

Sample Mean

Sample Range

ml ml ml ml1 301.0 2.0 7 300.0 1.32 300.5 2.1 8 301.0 1.83 299.8 1.8 9 300.8 1.54 299.2 1.8 10 299.4 1.0


5 300.5 1.5 11 298.8 1.06 300.8 1.0 12 298.1 0.9

15. An automated production process consists of assembling three components to make a sub assembly. Every two hours a sample of 100 sub assemblies is collected and each sub assembly is tested. Typically in the past, 2.8 sub assemblies on average have failed the test. Compute 3 sigma control limits for a p chart.

16. Dept of Transport must accept sections of paving constructed by private contractors. These sections are usually 10 KM long. They are accepted based on the cores drilled on the concrete pavement at a random interval. The standard specification requires a compression strength of 12500 kg per square cm. The head of inspection wishes to establish a statistical acceptance plan wherein sections of pavement would be accepted based on the mean compression strength of a sample of 50 cores removed from each section. He knows from experience that the standard deviation of compression strength of thousands of cores from hundreds of KMs is 1625 Kg/Sq cm.

i) Set an acceptance criterion if one tailed alpha is 0.005.

ii) Fifty cores are pulled from a 10KM section. The mean compression strength of the sample is 11500 Kgs/Sq cm. Should the section of pavement be accepted?

Capacity17. Mehta & Mehta own a small accounting service and one computer. If their customers keep

organized records, either of the owners can use the computer to prepare one tax return per hour, on average. During the first two weeks of April, both Mehtas work seven twelve-hour shifts. This allows them to use their computer around the clock.

(a) What is the peak capacity, measured in tax returns per week? (168)(b) The Mehtas normally operate from 9 A. M. to 7 P. M., five days per week. What is their

effective capacity, measured in tax returns per week? (50)(c) During the last week of December, the Mehtas processed 40 tax returns. What is their

utilization, as a percentage of effective capacity? (80%)

18. A unit manufacturing voltage stabilizers expects the annual demand in the next four years to be:

Year 1 2 3 4Demand 10,000 12,000 16,000 20,000Probability 0.4 0.3 0.2 0.1

Sales revenue per stabilizer is Rs. 300. The existing unit has fixed one time operating cost of Rs. 6,00,000. Existing variable costs & estimated variable cost after expansion are as follows:

Variable costs ( Rs. Per unit)Annual Output level Existing After capacity expansion

10,000 200 25012,000 150 15516,000 160 13020,000 190 145


Fixed one time operating cost after affecting capacity expansion is Rs. 8,00,000. To maximize net profit, should the unit go in for expansion of capacity or continue with existing capacity?(Ans: Capacity should not be expanded)

19. An automobile brake supplier operates on two eight-hour shifts, five days per week, fifty-two weeks per year. Following table shows the time standards, lot sizes and demand forecasts for three components. Because of demand uncertainties, the operations manager obtained three demand forecasts (pessimistic, expected & optimistic). The manager believes that a 20 percent capacity cushion is best.

(a) What is the minimum number of machines needed? The expected number? The maximum number?(2, 3 and 4 machines)

(b) If the operation currently has three machines and the manger is willing to expand capacity by 20 percent through short-term options in the event that the optimistic demand occurs, what is the capacity gap?

Component

Time StandardDemand ForecastProcessing Set up Lot Size

Hr/unit Hr/lot Units/lot Pessimistic Expected OptimisticA 0.05 1.0 60 15000 18000 25000B 0.20 4.5 80 10000 13000 17000C 0.05 8.2 120 17000 25000 40000

20. Manvinder Singh operates Singh’s Garage & Manhole Cover Recycling Center at Okhla Phase II. The Garage has one bay dedicated to wheel alignments. Although the recycling center is open at night, the garage normally is open only on weekdays from 7 am to 6 pm and on Saturdays from 7 am to noon. An alignment takes an average of 40 mins to complete, although Singh charges customers for two hours according to the norms set by the association. During May, before the rains set in, the road repairs and man hole cover demand peaks and Singh’ garage is open from 7 am to 9 pm on weekdays and from 7 am to 5 pm on Saturdays.

(a) What are the garage’s peak and effective capacities in wheel alignments per week? (120 & 90)

(b) During the second week of May Singh’s Garage completed 90 alignments. What is the utilization as a percentage of effective capacity? As a percentage of peak capacity?(Ans: Utilization as a percentage of Effective capacity is 100% and as a percentage of peak capacity is 75%)

21. Create a capacity plan for a critical bottleneck operation at the Akash Footwear Company. You have to decide the number of machines required. Three products (men’s, women’s children’s footwear) are manufactured. The time standards (processing & set up), lot sizes and demand forecasts are given in the following table. The firm operates two eight-hour shifts, 5days per week, 50 weeks per year. Experience shows that a capacity cushion of 5% is sufficient.

ProductTime Standards

Lot sizes Demand ForecastProcessing Set up(Hrs/pair) (Hrs/lot) (Pairs/lot) (Pairs/year)

Men’s Footwear 0.05 0.5 240 80,000Women’s Footwear 0.10 2.2 180 60,000


Children’s Footwear 0.02 3.8 360 120,000 (a) How many machines are needed? (Ans: 3.83 i.e., 4 machines)(b) If the operation currently has two machines, what is the capacity gap? ( 2 machines)

22. Given the following information on maximum production rate and hours worked for an oven that cooks pizzas, determine the capacity per week.

Description Value UnitMaximum production rate 40 Pizzas/hourNumber of hours 84 Hours/WeekOvertime Zero Hours/WeekPreventive maintenance (performed after closing) Zero Hours/WeekEquipment failure (Unplanned downtime) 2% Of planned hours

(Ans: 3292.8 pizzas per week)23. Assume that each contact for life insurance takes three hours, house insurance takes two hours,

and automobile insurance takes one hour. What are the capacities of an agent that works 40 hours per week under Mix 1 and Mix 2?

Type Mix 1 Mix 2Life Insurance 0.20 0.40House Insurance 0.30 0.40Auto Insurance 0.50 0.20

(Ans: Mix - 1 28.67 contacts/week & Mix - 2 21.33 contacts/week; As the mix shifts away from auto to life and house, which require more time per contact, the capacity declines)

24. To assemble the frames of 25 rocker/recliner chairs, each assembler takes his or her work order to the inventory clerk to pick the parts required to make the chairs. This takes about thirty minutes. After returning to the work area, each assembler completes twenty-five chair frames in three an a half hours. To increase the capacity of assembling chair frames, a separate stock picker could be hired to gather inventory for all the assemblers. Then each assembler would be able to increase production by one-seventh because the thirty minutes out of every four hours formerly used to pick stocks could now be used to assemble chairs. Determine the percentage increase in the capacity per assembler.

(Ans 14.28%)

25. Annual demand for a manufacturing company is expected as follows:

Units demanded 8000 10000 15000 20000Probability 0.5 0.2 0.2 0.1

Revenues are Rs. 35 per unit. The existing manufacturing facility has annual fixed operating costs of Rs. 200,000. Variable manufacturing costs are Rs. 7.75 per unit at the 8000 unit output level; Rs. 5.00 per unit at the level of 10000 units; Rs. 5.53 per unit at the 15000 unit level and Rs. 7.42 at the 20000 unit output level.


An expanded facility under consideration would require Rs. 250,000 fixed operating cost annually. Variable costs would average Rs. 9.40 per unit at 8000 unit level; Rs. 5.20 at 10000; Rs. 3.80 at 15000 and Rs. 4.90 at the 20000 level.

To maximize net earnings which size facility should be selected?

26. Integrated manufacturing company is a producer of kites and windsocks. Relevant data on a bottleneck operation in the shop floor for the next year are given in the following table

Item Kites WindsocksDemand forecast 30000 units/year 12000 units/yearLot size 20 units 70 unitsStandard processing time 0.3 hour/unit 1.0 hour/unitStandard set up time 3.0 hours/lot 4.0 hours/lot

The shop works two shifts per day, eight hours per shift, 200 days per year. There currently are four machines, and a 25 percent capacity cushion is desired. How many machines should be purchased to meet next year’s demand without resorting to any short term capacity solution?

Location27. The following table gives the map coordinates and the shipping loads for a set of cities that we

wish to connect though a central “hub.” Near what map coordinates should the hub be located?

City Map Coordinate (x,y) Shipping Load

A (5,10) 5

B (6,8) 10

C (4,9) 15

D (9,5) 5

E (7,9) 15

F (3,2) 10

G (2,6) 5

28. Our main distribution center in Phoenix, AZ is due to be replaced with a much larger, more modern facility that can handle the tremendous needs that have developed with the city’s growth. Fresh produce travels to the seven store locations several times a day making site selection critical for efficient distribution. Using the data in the following table, determine the map coordinates for the proposed new distribution center.


Store Locations Map Coordinates (x,y)

Truck Round Trips per Day

Mesa (10,5) 3

Glendale (3,8) 3

Camelback (4,7) 2

Scottsdale (15,10) 6

Apache Junction (13,3) 5

Sun City (1,12) 3

Pima (5,5) 10

29. A multinational company has to take decision regarding the location of its production capacity in India. It can choose from five possible alternatives. Data for these five locations are given below:

Economic factors LocationsParticulars A B C D ELabour costs 120 110 160 85 75Tranportation cost 10 8 7 12 14Local Taxes 17 20 25 19 17Cost of power 21 29 25 18 23Other costs 16 11 12 16 18

Other Factors A B C D EAttitude of community Very Good Fair Good Fair Very GoodAvailability of labour Good Very Good Fair Outstanding AcceptableTransport arrangement Fair Acceptable Outstanding Acceptable FairQuality of life Acceptable Fair Good Very Good Outstanding

(a) Which site would you recommend on the basis of operating costs?(b) Devise a method of quantifying the other factors and determine the best site on the basis of

both the operating costs & other factors.

30. An electronics manufacturer must expand by building a second facility. The search has been narrowed to four locations; all acceptable to management in terms of dominant factors Assessment of theses sites in terms of seven location factors is shown below. For example location A has a factor score of 5 (excellent) for labour climate; the weight for this factor (20) is the highest of any.

Sr. No.

Location FactorFactor weight

Factor Score for each LocationA B C D


1 Labour Climate 20 5 4 4 52 Quality of life 16 2 3 4 13 Transportation system 16 3 4 3 24 Proximity to market 14 5 3 4 45 Proximity to materials 12 2 3 3 46 Taxes 12 2 5 5 47 Utilities 10 5 4 3 3

Calculate the weighted score for each location. Which location should be recommended?

31. Cost data in Rupees for two sites – Cochin and Chennai are given below:

Descriptions Cochin ChennaiVariable costs 100.46 95.77Annual overheads 2,100,000 1,900,000Initial investments 175,000,000 168,000,000Incentives – one time 25,000,000 10,500,000

(a) If the facility is scheduled to produce 45,000 units per year, over a five year period determine which site has a lower total cost?

(b) At what point of time will the costs of two sites be equal?

(Ans: (a) Total costs Cochin – 183,103,500 and Chennai – 188,548,250 (b) 18.25 years)

Facility Layout32. A manufacturing company wishes to put up a new production facility, which would have seven

departments. Two alternative layouts are being considered as per data given below:

Distance between departments 1 to 7Trips betweenDepartments

Distance between departmentsLayout A Layout B

1-2 28 521-3 24 321-4 40 502-3 45 202-4 25 603-4 40 484-5 48 365-6 52 465-7 54 666-7 50 50

Sequence of processing of products through departments and production batches per year are given below:

Sr. No. Product Sequence Production per year (batches)

A 1-2-3-4-5-6-7 1500B 1-4-5-6-7 300C 1-2-4-5-6-7 200


D 1-3-4-5-7 400E 1-3-4-5-6-7 1400

Use load distance analysis to determine which layout minimizes the total annual distance that the batches of products travel through the proposed production facilities.(Layout B 851000 & Layout A 858000; B is preferred)

33. Following figure shows two layout options of a facility. The distance between any two adjacent departments is 10 m. No diagonal movement is possible, e.g., if a load has to be moved from department 7 to department 5 in layout A, it can be either through departments 8, 9 & 6 or through departments 3, 1 & 2 by traveling a distance of 40 m. Accompanying table shows the department processing sequence of various products and their quantity produced per month. Which layout is better in terms of lower total load-distance value?

Layout A

1 2 5

3 4 6

7 8 9

Layout B

5 3 4

9 6 1

2 7 8

Sr. No. Product Processing sequence Quantity per month1 V 5-7-2-9 40002 W 4-6-3-7-8-9 10003 X 1-2-7-8 20004 Y 5-2-1-7-9 30005 Z 3-4-7-8-9 1000

34. As in most kitchens, the baking ovens in Lori’s Kitchen in New Orleans are located in one area near the cooking burners. The refrigerators are located next to each other as are the dishwashing facilities. A work area of tabletops is set aside for cutting, mixing, dough rolling, and assembling of final servings, although different table areas may be reserved for each of these functions.

Given the following Interdepartmental Activity Matrix, develop an appropriate layout for Lori’s Kitchen.

Interdepartmental Activity Matrix

Cooking Burners (A)

Refrigerators (B) Dishwashing (C) Work Area (D)

Cooking burners (A) - 7 193 12

Refrigerator (B) - 4 82


Dishwashing (C) - 222

Work Area (D) -

With a distance of 10 feet between adjacent areas the present layout is:

A B C D

Computing the Load * Distance measure:

Load * DistanceA to B 7 * 10 70A to C 193*20 3860A to D 12*30 360B to C 4*10 40B to D 82*20 1640C to D 222*10 2220Total 8190

Develop a preferred layout. What is the sum of the loads * distance of your new layout?

35. The activity matrix of a research laboratory is exhibited below:

The explanations of different departments are: A – Executive offices, B – Stress laboratory, C – Group Dynamics laboratory, D – Research report writing office, E - Marketing Office and F – Word processing and mail room. All departments are of equal area.

To departments

Fro

m d

epar

tmen

ts

A B C D E F

A 15 35 20 40 50

B 10 20 20 50 10

C 15 0 40 40 20

D 10 15 30 10 10

E 30 0 40 10 10

F 50 20 0 0 30

Evaluate following two proposed layouts and recommend one of these.

Layout A B C E


D A F

Layout B

B E C

D A F

(Ans: Layout B is preferred)

36. An initial solution has been given to the following process layout problem. Given the flows described and a cost of Rs. 20 per unit per meter, compute the total cost for the layout. Each location is 100 meter long and 50 meter wide as shown in the following figure. Use centers of departments for distances and measure distances using rectilinear method.

To departmentsF

rom

dep

artm

ents A B C D

A 0 10 25 55

B 0 10 5

C 0 15

D 0

(Ans Rs. 4,45,000)

PERT/CPM

29. In the interest of serving the public in Benjamin County, St. Adolf’s Hospital has decided to relocate from Christofer to Northville, a large suburb that at present has no primary medical facility. The move to Northville will involve constructing a new hospital and making it operational. Judy Kramer, executive director of the board of St. Adolf’s Hospital, must prepare for a hearing that is scheduled for next week, before the Central Ohio Hospital Board on the proposed project, including time & cost estimates for its completion.

With the help of her staff Kramer has identified 11 major project activities. She also has specified the immediate predecessors for each activity, as shown in the following table:

Activity Description Immediate predecessor(s)

Estimated time (weeks)

A Select Administrative & Medical staff -- 12B Select site & do site survey -- 9


100x50 mts

D

100x50 mts

C100x50 mts

B100x50 mts

A

C Select Equipment A 10

DPrepare final construction plans & layout

B10

E Bring utilities to the site B 24

FInterview applicants & fill positions in nursing & support staff

A10

GPurchase & take delivery of equipment

C35

H Construct the hospital D 40I Develop an information system A 15J Install the equipment E, G, H 4K Train nurses & support staff F, I, J 6

(a) Draw the network diagram.(b) Determine the critical path.(c) Calculate the slack for the activities.

30. For the St Adolf’s Hospital project the indirect costs are $ 8000 per week. After week 65, St Adolf’s Hospital incurs a penalty cost of $ 20,000 per week if the hospital is not fully operational. With critical path completion time of 69 weeks the hospital faces potentially large penalty costs. Judy wants to calculate estimated cost by considering all possible measures of time reduction. She has collected following data. Help her to estimate the minimum cost schedule.

Activity Normal Time

Normal Cost

Crash Time

Crash Cost

Max Time Reduction (wk)

Cost of crashing per week

A 12 12,000 11 13,000 1 1,000

B 9 50,000 7 64,000 2 7,000

C 10 4,000 5 7,000 5 600

D 10 16,000 8 20,000 2 2,000

E 24 120,000 14 200,000 10 8,000

F 10 10,000 6 16,000 4 1,500

G 35 500,000 25 530,000 10 3,000

H 40 1,200,000 35 1,260,000 5 12,000

I 15 40,000 10 52,500 5 2,500


J 4 10,000 1 13,000 3 1,000

K 6 30,000 5 34,000 1 4,000

Totals 1,992,000 2,209,500

31. For the same hospital project as given in question numbers 1 & 2 the probabilistic time estimates are shown below

ActivityOptimistic Time

aMost likely Time

mPessimistic Time

bA 11 12 13

B 7 8 15

C 5 10 15

D 8 9 16

E 14 25 30

F 6 9 18

G 25 36 41

H 35 40 45

I 10 13 28

J 1 2 15

K 5 6 7

(a) Calculate the probability that the hospital will become operational in 72 weeks(b) What is the probability that the project will be completed in 65 weeks?

32. Consider the following project information

ActivityActivity

Time (Week)Immediate

predecessorsA 4 --B 3 --C 5 --D 3 A,BE 6 BF 4 D,CG 8 E,CH 12 F,G

(a) Draw the network diagram for this project.(b) Specify the critical path(s).(c) Calculate the total slack for the activities A & D.


33. An advertising manager has developed the network diagram shown below for a new advertising campaign. In addition the manager has gathered time estimates for each activity as shown in the table below.

Activity

Time estimates (weeks)Immediate

Predecessor(s)OptimisticMost

LikelyPessimistic

A 1 4 7 --B 2 6 7 --C 3 3 6 BD 6 13 14 AE 3 6 12 A, CF 6 8 16 BG 1 5 6 E, F

(a) Calculate the expected time and variance for each activity.(b) Calculate the activity slacks and determine the critical path using the expected activity

times.(c) What is the probability of completing the project within 23 weeks?

34. Your company has just received an order from a good customer for a specially designed electric motor. The contract states that, starting on the thirteenth day from now, your firm will experience a penalty of Rs 100 per day until the job is completed. Indirect project


A

D

C

E

B

Start

Finish

G

F

costs amount to Rs 200 per day. The data on direct costs and activity precedence relationships are given in the following table:

Sr. No.

ActivityNormal Time (days)

Normal cost (Rs)

Crash Time (days)

Crash cost (Rs)

Immediate Predecessor(s)

1 A 4 1000 3 1300 None2 B 7 1400 4 2000 None3 C 5 2000 4 2700 None4 D 6 1200 5 1400 A5 E 3 900 2 1100 B6 F 11 2500 6 3750 C7 G 4 800 3 1450 D, E8 H 3 300 1 500 F, G

(a) Draw the network project diagram.(b) What completion time would you recommend?

35. You have constructed a network diagram depicting the various activities in the project that you are managing. In addition, you have asked various managers and subordinates to estimate the amount of time that they would expect each of the activities to take. Their responses are shown in the following table:

Sr. No. ActivityTime Estimates (days)

Optimistic Most likely Pessimistic1 A 5 8 112 B 4 8 113 C 5 6 74 D 2 4 65 E 4 7 10

(a) What is the expected completion time of the project?(b) What is the probability of completing the project in 21 days?

36. A project consists of nine activities whose time estimates (in weeks) and other characteristics are given below:


A

C

D

EB

Start Finish

ActivityPreceding activities

Time estimates (weeks)Most

OptimisticMost

LikelyMost

PessimisticA -- 2 4 6B -- 6 6 6C -- 6 12 24D A 2 5 8E A 11 14 23F B, D 8 10 12G B, D 3 6 9H C, F 9 15 27I E 4 10 16

(a) Show the PERT network for the project.(b) Identify the critical activities. (A-D-F-H : 35 weeks) (c) What are the expected project completion time and its variance? (Expected 35 weeks;

SD 3.23)(d) What is the probability of completing the project one week before the expected time?

(0.3783)(e) If the project is required to be completed by December 31 of given year and the

manager wants to be 95% sure of meeting the deadline, when should he start the project work? (X= 282 days; Start date 25th March)

(f) A penalty of Rs 15000 per week is to be imposed on the contractor if the project is not completed in 36 weeks. What is the probability that he has to pay a penalty? (P = 0.378) What is the probability that he has to pay a penalty of Rs 45000? (P = 0.0687)

37. The following table gives the activities in a construction project and other relevant information:

ActivityImmediate predecessors

Time (Days) Direct Cost (Rs)Normal Crash Normal Crash

A -- 4 3 60 90B -- 6 4 150 250C -- 2 1 38 60D A 5 3 150 250E C 2 2 100 100F A 7 5 115 175G D, B, E 4 2 100 240

Indirect costs vary as follows:

Days : 15 14 13 12 11 10 9 8 7 6Cost (Rs): 600 500 400 250 175 100 75 50 35 25


(a) Draw a network diagram.(b) Determine the project duration, which will return in minimum total project cost. (10

Days, Rs 943)

38. The data for a project are:

ActivityPreceding activity

Time (in weeks) Cost (in Rupees)Normal Crash Normal Crash

A None 3 2 18,000 19,000B None 8 6 600 1,000C B 6 4 10,000 12,000D B 5 2 4,000 10,000E A 13 10 3,000 9,000F A 4 4 15,000 15,000G F 2 1 1,200 1,400H C, E, G 6 4 3,500 4,500I F 2 1 7,000 8,000

(a) Draw a project network diagram and find the critical path.(b) If a deadline of 17 weeks is imposed for completion of the project, which activities

will be crashed? What would be the additional costs? What would be the critical activities of the network after crashing?

(Ans: Activities to be crashed: A-one week, B-one week, E-two weeks and H-two weeksTotal cost = Normal + Crashing = 62,300 + 6,200 = Rs 68,500Activities that are critical after crashing = A, B, C, E & H)

39. The following table lists the activities of a maintenance project:

ActivityDuration (months)


1-2 2 4-7 31-3 2 5-8 11-4 1 6-8 42-5 4 7-9 53-6 5 8-9 33-7 8

(a) Draw the project network.(b) Find the critical path & duration of the project.

40. For a project following information is given:

Activity Node Time Activity Node TimeA 1-2 4 G 3-6 10B 1-3 8 H 4-6 8


C 1-4 12 I 5-6 6D 1-5 6 J 6-7 12E 2-6 6 K 6-8 14F 3-7 4 L 7-8 8

The firm loses Rs. 500.00 every week beyond a total of 34 weeks. Consider each of the proposals given here individually and state whether the firm should accept it.(a) Activity J can be reduced to 9 weeks by taking resources from activity K, thereby

raising its expected time to seventeen weeks, at no extra cost.(b) Activity H can be reduced to 5 weeks by using overtime, and extra labour at a cost of

Rs. 1000.(c) Activity G can be reduced to 7 weeks at a total cost of Rs 500.(d) Activity L can be reduced to 5 weeks by taking resources from activity K, thereby

raising its expected time to sixteen weeks, at no additional cost.

41. The following table gives the list of various activities involved in the production of wireless communication equipment, their immediate predecessor, their normal time and cost estimates and their crash time and cost estimates.

Act Id

Activity Description

Time (Months) Cost (Rs) Immediate PredecessorNormal Crash Normal Crash

A System Calculation

3 1 45000 63000 --

B Release of Drawing

2 1 15000 23000 --

C Procure PSU 12 10 85000 105000 A,BD Procurement of

RM8 6 450000 500000 A,B

E Documentation 4 3 40000 49500 A,BF Time study 3 1 25000 41000 EG PCB Manufacture 6 5 50000 59000 D,FH Mechanical parts 7 6 200000 220000 D,FI Electronics

assembly2 2 43000 43000 C,G,H

J Testing 4 2 50500 75000 I

Indirect costs per month are Rs 20,000. Determine the optimal time versus cost schedule.

42. A company is about to begin a project to design a new product. Management has estimated that the project should require 45 days to complete. Although 45 days seemed to be very short to the process engineers at first, after some discussions it was concluded that they probably could meet the deadline because the product and the processes were so similar to the present processing technologies in use at their plant. These activities, their precedence relationships, and their durations were estimated by the engineers:

Act Id Activity Description Activity Immediate Revised Nov 2012

20 Dr. A. K. Dey

Duration (days)

Predecessor

A Initial study 12 --B Process Tech study 10 --C Vendor capability survey 8 --D Modification of facility 14 BE Intermediate facility redesign 6 CF Intermediate product redesign 18 B,AG Specific process machinery design 11 D,EH Vendor involvement & integration 21 CI Final facility, product & process design 7 F,G

(a) Construct a CPM network.(b) Compute & display ES, EF, LS & LF and slack values in a table(c) Calculate the critical path and estimate the duration of completion.

43. A government tender that appeared on page four of today’s newspaper has caught the attention of Mr. R. Mathur, the General Manager of Rosset Manufacturing Company. The tender calls for short listing of vendors for possible supply of Space Signaling systems. Vendors who can successfully demonstrate a Space Signaling system, would be considered.

In a staff meeting it is unanimously decided to bid for the tender. Only problem being whether there would be sufficient time to prepare a model without delaying any of the existing commitments and how much money should be spent? Mr. Mathur estimated that if the project can be completed in 41 days and within Rs 2,00,000, it is worth the try.

A project team is formed by drawing persons with suitable skills from different departments. It is felt that in case the company succeeds in executing some orders for the signaling system, these project team members will go back to their departments with adequate knowledge.

The team developed the following list of project activities, time estimates, and precedence relationships as part of the project plan:

ActivityPrecedence Relationship

Present duration (Days)

Accelerated Duration (Days)

Present Cost (Rs)

Accelerated Cost (Rs)

A -- 7 5 14000 14800B A 9 7 18000 24000C A 6 4 20000 30000D A 12 8 10000 12000E B,C 11 10 13000 15000F C 14 13 15000 19000G D 16 10 25000 31000


H C,B 16 15 23000 25700I H 12 10 27000 31000J E,F,G 18 18 29000 29000

If the time of completion is of utmost importance, will Mr. Mathur take up the project? What will be the total cost? (Show detailed calculations to justify your decision.)

44. A company is about to begin a project to design a new product. Management has estimated that the project should require 45 days to complete. Although 45 days seemed to be very short to the process engineers at first, after some discussions it was concluded that they probably could meet the deadline because the product and the processes were so similar to the present processing technologies in use at their plant. These activities, their precedence relationships, and their optimistic, most likely & pessimistic time durations were estimated by the engineers:

Act Id

Activity DescriptionOptimistic Time

Most likely

time

Pessimistic time

Immediate Predecess

orA Initial study 10 12 14 --B Process Tech study 10 10 10 --C Vendor capability survey 6 8 10 --D Modification of facility 10 14 18 BE Intermediate facility redesign 3 6 9 CF Intermediate product redesign 14 18 22 B,AG Specific process machinery

design10 11 12 D,E

H Vendor involvement & integration

18 21 24 C

I Final facility, product & process design

5 7 9 F,G

(a) Construct a PERT/CPM network.(b) Compute & display ES, EF, LS & LF and slack values in a table(c) Calculate the critical path and estimate the duration of completion.(d) What is the probability that the project will take longer than 45 days? How confident

are you that the team can complete the project within 45 days?45. A work project consists of twelve activities labeled A through L. Upon being asked to

specify the order in which the jobs had to be done, the manager answered as follows:A, B & C are the first activities of the project and can start simultaneously and immediately; A & B precede D while B precedes E, F & H. Activities F & C precede G while E & H precede I & J. C,D,F & J precede K, which in turn, precedes L. Further, I,G & L are the terminal activities of the project.The completion times of various activities are listed here:

Act A B C D E F G H I J K LRevised Nov 2012

22 Dr. A. K. Dey

Days

6 4 10 1 1 3 14 6 9 2 7 5

(c) Draw a network diagram corresponding to this project(d) Obtain lengths of all the paths and determine critical path.

46. The following table lists the activities of a maintenance project:



1-2 2 4-7 31-3 2 5-8 11-4 1 6-8 42-5 4 7-9 53-6 5 8-9 33-7 8

(a) Draw the project network.(b) Find the critical path & duration of the project.

47. Given an auditing project with following activities:

ActivityStandard Deviation

CriticalDuration

Weeksa. Add 2 yes 2b. Balance 1 3c. Count 0 Yes 4d. Deduct 3 2e. Edit 1 Yes 1f. Finance 2 6g. Group 2 Yes 4h. Hold 0 yes 2

Find:(a) The probability of completing this project in 12 weeks (or less) as the

client desires(b) The probability of completing this project in 13 weeks (or less).(c) The probability of completing this project in 16 weeks (or less), the

client’s drop dead date.(d) The number of weeks required to assure a 92.5 percent chance of

completion, as guaranteed by the auditing firm.(Ans: a. 0.369, b. 0.500, c. 0.841 & d. 17.32)

48. Before erection of a building, the project manager evaluated the project using PERT/CPM analysis. The activities with their corresponding times were recorded in the following table.Find the critical path, the slack times and the expected completion time.


Sr. No.

ActivityTimes Preceding

tasksOptimistic Most Likely Pessimistic1 Lay foundation 8 10 13 --2 Dig hole for scale 5 6 8 --3 Insert scale bases 13 15 21 24 Erect frames 10 12 14 1,35 Complete building 11 20 30 46 Insert scales 4 5 8 57 Insert display cases 2 3 4 58 Put in office equip 4 6 10 79 Finishing touches 2 3 4 8,6

(Ans: Critical path 2 – 3 – 4 – 5 – 7 – 8 – 9; Activity 1 is 11.66, activity 6 is 4, all others are zero; 66.34)

49. Given a network for an engineering project with normal times and crash times (in parentheses), find the lowest cost time-cost plan. Assume indirect costs for facilities and equipment are Rs. 10,000 per day. The data are:

Activity Time Reduction, Direct Cost per day1-2 Rs 3000 first day, Rs 5000 second day2-3 Rs 80003-4 Rs 2500 first day, Rs 6000 second day

2-4Rs 3000 first day, Rs 7000 second day, Rs 9000 third day

6(4) 8(5)

5(4) 6(4)

(Ans: Completion days – Total Cost: 17 days – Rs 170,000; 16 days – 162,500; 15 days – 155,500; 14 days – 150,500; 13 days – 146,500 and 12 days – 144,500)

50. The events of the project below are designated as 1, 2, and so on. Find the critical path.

ActivityPreceding

EventSucceeding

EventTime Estimate

(weeks)Preceding activity

A 1 2 3 NoneB 1 3 6 NoneC 1 4 8 NoneD 2 5 7 AE 3 5 5 BF 4 5 10 C


1 2

3

4

G 4 6 4 CH 5 7 5 D, E, FI 6 7 6 G

The project must be completed in 16 weeks. The activities C, F, H & I may be crashed as follows. Assume partial crashing. Find the best schedule and its cost.

ActivityCrash Time

(weeks)Additional cost per week (Rs.)

C 7 4000F 6 2000H 2 1000I 3 3000

(Ans: Critical path 1- 4 – 5 – 7, Crash C by 1 week, F by 3, H by 3 & I by 1 for a cost of Rs 16000 extra)

51. Rajiv’s dilemma

Rajiv is bidding for a market research project. He has prepared a list of activities and made a network diagram. The organization for which he is preparing the bid is in a great hurry to finish the market research project. They can not wait because the selling season has almost started.

Rajiv is facing a dilemma whether to quote

1. The duration of the project with minimum total cost or 2. The minimum possible duration of the project

Help him to find out the time & cost for the above two alternatives. Data for the project are as given below:

Activity NodesNormal duration (days)

Crash duration (days)

Incremental cost of crashing

(Rs/day)A 1-2 5 3 40B 1-3 4 2 10C 1-4 7 5 20D 2-3 4 2 30E 4-3 3 2 50F 2-5 2 2 -G 3-5 7 5 20H 3-6 3 1 10I 4-6 4 3 30J 5-6 6 5 40

Revised Nov 2012

25 Dr. A. K. Dey

The total normal cost is Rs. 500, cost of supervision is Rs. 20 per day, penalty is Rs. 10 per day over 20 days, reward is Rs. 10 per day below or equal to 18 days.

52. Following are the manpower requirements for each of the activity in a project:

ActivityNormal Time

(days)

Manpower required per

day1-2 10 21-3 11 32-4 13 42-6 14 33-4 10 14-5 7 34-6 17 35-7 13 56-7 9 87-8 1 11

The contractor stipulates that during the first 26 days only 5 men and for the remaining 8 days maximum 11 men can be made available. Re arrange the activities suitably for leveling the manpower resources, satisfying the above condition.


OM Numerical 2012

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