Manpower replacement policy (staffing policy)
All organizations face the problem of initial recruitment and filling up of vacancies caused by promotions, transfer, employee quitting their jobs or retirement and deaths. The principle of replacement used in industry for replacement of parts etc an also be used for recruitment and promotion policies, which
are laid down as personnel policy of an organization. The assumption made in such case is that the destination of manpower is already decided. Few examples will illustrate this point.
Example 8.18 An army unit requires 200 men, 20 junior commission officers (JCO’s) and 10 officers. Men are recruited at the age of 18 and JCO’s and officers are selected out of these. If they continue in service, they retire at the age of 40. At present there are 800 jawans and every year 20 of them retire. How many jawans should be recruited every year and at what age promotions should take place?
Sol. If 800 jawans had been recruited for the past 22 years (age of recruitment 40 years – age of entry 18 years), the total number of them serving up to age of 39 years = 20 × 22 = 440
Total number of jawans required = 200 +20 +10 = 230
Total number of jawans to be recruited every year in order to maintain a strength of 230
=800/440 × 230=418
Let a jawan be promoted at age of X the up to X – 1 year number of jawans recruited is 200 out of 230.
Hence out of 800, jawans required
= 200/230 × 800
= 696.
696 will be available up to 5 years as 20 retire every year and (800 – 20 × 5) = 700.
Hence promotion of jawans is due in 6th year.
Out of 230 jawans required, 20 are JCO’s, therefore if recruited 800, number of JCO’s
= 20/230 × 800 =70 approximately.
In, a recruitment of 800, total number of men and JCO’s
= 697 + 70
= 766
Number of officers required = 800 – 766 = 34
The number will be available in 20 years of service, so promotion of JCO’s to officers is due in21 year of service.
Example 8.19 College X plans to raise the strength of its faculty to 150 and then keep it at that level. The wastage off acuity due to retirement, quitting, deaths etc based on the length of service of the faculty member is as given below.
Block years 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
% of teachers 
0 
5 
10 
15 
20 
30 
35 



Those who live up to end of year 
0 
5 
35 
60 
65 
70 
85 
100 


(i) Find the number off acuity members to be recruited evelY year.
(ii) If there are 10 posts of Head of Departments (HODS) for which length of service is the only criterion of promotion, what is the average length of service after which a new faculty member should expect promotion?
Sol. Let us assume that the recruitment per year is 100. These 100 teachers join initially in the block of 05 years, will serve for 35 years and will become 0 in 7th block of 5 years i.e. at the service of 35 years.
Those 100 who join between the block of year 5 – 10 will serve for 30 years and become 15, the third set of teachers will become 30 after 25 years of service and so on.
Year 
No of faculty members 
0 
100 
5 
95 
10 
65 
15 
40 
20 
35 
25 
30 
30 
15 
35 
0 
Hence if 100 faculty members are required every year, the total number of staff members after 35 years (7 block of 5 years) of service = 380
To maintain staff strength of 150, the number to be recruited every year
= 100/380 × 150
= 40
If the college recruits 40 every year, then they want 10 as HOD’s. Hence if the college recruits 100 every year then they will need HOD’s = 10/40 × 100 = 25.
It can be seen from the above table that 0 + 15 + 30 ≥ 25 faculty member as HOD’s can be done after 20 years of service.
REVIEW
 1. What is replacement problem? When does it arise?
 2. Describe various types of replacement situations.
 3. Enumerate various replacement problems.
 4. What are the situations which make the replacement of items necessary?
 5. Give a brief account of situations of which the replacement problems arise. What does the theory of replacement establish?
 6. Discuss in brief replacement procedure for the items that deteriorate with time.
 7. The cost of maintenance of a machine is given as a function increasing with time and its scrap value is constant. Show that the average annual cost will be minimized by replacing the machine when the average cost to date becomes equal to the current maintenance cost.
 8. Discuss the replacement problem where items are such that maintenance costs increase with time and the value of money also changes with time.
 9. Find the optimum replacement policy which minimizes the total of all future discounted costs for an equipment which costs Rs A and which needs maintenance costs of Rs C_{1}, C_{2}, …. C_{n} etc. (C_{n}_{+}_{1}>C_{n}) during the first year, second year etc., and further D is the depreciation value per unit of money during a year.
 10. State some of the simple replacement policies.
 11. Construct the cost equation reflecting the discounted value of all future costs for a policy of replacing equipment after every n periods. Hence establish the following:
(i) Replace if the next period cost is greater than the weighted average of previous costs.
(ii) Do not replace if the next period’s cost is less than the weighted average of previous costs.
 12. What is group replacement? Give an example.
 13. Write a short note on group replacement and individual replacement policies.
 14. The cost per item for the individual replacement is C_{1} and the cost per item of group replacement is C_{2}. If only individual replacement is more economical than the group replacement along with the individual, find relation between C_{1} & C_{2.}
 15. Consider a group replacement model. There are N items in the group. Replacement is made after every / periods. Assume that all the failures in a group are replaced at the end of the period. Further
C_{1} is the cost of replacing a unit in the group
C_{2} is the cost of replacing a failure (C_{2} > C_{1} ).
Find an expression for the least cost associated with group replacement.
 16. A large population is subject to a given mortality curve for a long period of time. All deaths are immediately replaced by births and there are no other entries or exists, Show that the age distribution becomes stable and that the number of deaths per unit time becomes constant.
 17. In a machine shop, a particular cutting tool costs Rs 6 to replace. If a tool breaks on the job, the production disruption and associate costs amount to Rs 30. The past life of a tool is given as follows.
Job No: 
1 
2 
3 
4 
5 
6 
7 
Proportion of broken tools on Job: 
.01 
.03 
.09 
.13 
.25 
.55 
.95 
After how many jobs should the shop replace a tool before it breaks down?
 18. It has been suggested by a data processing firm that they adopt a policy of periodically replacing all the tubes in a certain piece of equipment. A given type of tube is known to have the mortality distribution shown in the table:
Tube failures/ week 
1 
2 
3 
4 
5 
Probability of broken tools on job 
.01 
.03 
.09 
.13 
.25 
There are approximately 1000 tubes of this type in all the combined equipment. The cost of replacing the tubes on an individual basis is estimated to be Re 1·00 per tube and the cost of a group replacement policy average Rs 0·30 per tube. Compare the cost of preventive replacement with that of remedial replacement.
 19. A truck has been purchased at a cost of Rs 1,60,000. The value of the truck is depreciated in the first three years by Rs. 20,000 each year and Rs .. 16,000 per year thereafter Its maintenance and operating costs for the first three years are Rs. 16,000, Rs. 18,000 and Rs. 20,000 in that order and increase by Rs. 4,000 every year. assuming an interest rate of 10% find the economic life of the truck.
 20. A manual stamper currently valued at Rs. 10,000 is expected to last 2 years and costs Rs 4,000 per year to operate. An automatic stamper which can be purchased for Rs 3,000 will last 4 years and can be operated at an annual cost of Rs 3,000. If money carries the rate of interest 10% per annum. Determine which stamper should be purchased.
 21. The cost of a new machine is Rs. 5,000. The maintenance cost of nth year is given by
R_{n}=500(n1); n=1, 2…
Suppose that the discount rate per year is .05. After how many years will it be economical to replace the machine by new one?
 22. A machine costs Rs 10,000 operating costs are Rs 500 per year for the first five years. Operating costs increase by Rs. 100 per year in the sixth and succeeding years. Assuming a 10 percent discount rate of money per year, find the optimum length of time to hold the machine before it is replaced. State clearly the assumptions made.
 23. A pipeline is due for repairs. It will cost s 10,000 and last for 3 years. Alternatively, a new pipeline can be laid at a cost of Rs 30,000 and lasts for 10 years. Assuming cost of capital to be 10% and ignoring salvage value, which alternative should be chosen?
 24. The cost pattern for two machines M_{1} and M_{2} when money value is not considered is given below:
Cost at the beginning of the year (in Rs)
Year 
M_{1} 
M_{2} 
1 2 3 
900 600 700 
1400 100 700 
Find the cost pattern for each machine when money is worth 10% per year and hence find the machine which is less costly.
 25. An engineering company is offered two types of material handling equipment A and B. A is priced at Rs. 60,000 including cost of installation, and the costs for operation and maintenance are estimated to be Rs 10,000 for each of the first five years, increasing by Rs. 3,000 per year in the sixth and subsequent year equipment B with a rated capacity same as A, requires an initial investment of Rs. 30,000 but in terms’ of operation and maintenance costs more than A. These costs for B are estimated to be Rs. 13,000 per year for the first six years, increasing by Rs. 4,000 per year for each year from the 7th year onwards. The company expects a return of 10 percent on all its investments. Neglecting the scrap value of the equipment at tile end of its economic life, determine which equipment the company should buy.
 26. An individual is planning to purchase a car. A new car will cost Rs. 1,20,000. The resale value of the car at the end of the year is 85% of the previous year value. Maintenance and operation costs during the first year are s. 20,000 and they increase by 15% every year. The minimum resale value of the car can be Rs. 40,000.
(i) When should the car be replaced to minimum average annual cost (ignore interest)?
(ii) If interest of 12% is assumed, when should the car be replaced?
 27. A truck owner estimates that the running costs and the salvage values of thicks for various years will be as tabulated below. If the purchase price of a truck is Rs 80,000, estimate the optimum replacement age for the truck. Assume that the rate of return on the capital invested in transportation business is 15% per year:
 28. A large computer installation contains 2,000 components of identical nature which are subject to failure as per probability distribution given below:
Week end 
1 
2 
3 
4 
5 
Percentage failure to date 
10 
25 
50 
80 
100 
Components which fail have to be replaced for efficient functioning of the system. If they are replaced as an when failure occur, the cost of replacement per unit is Rs 3. Alternatively if all components are replaced in one lot at periodical intervals and individually replaced only as such failures occur between group replacement, the cost of component replaced is Re. 1.
(a) Access which policy of replacement would be economical.
(b) If group replacement is economical at current costs, then assess at what cost of individual replacement would group replacement be uneconomical.
(c) How high can the cost per unit in group replacement be to make a preference for individual replacement policy?
 29. Let p (t) be the probability that a machine in a group of 30 machines would break down in period t. The cost of repairing a broken machine is Rs. 200. Preventive maintenance is performed by servicing all the 30 machines at the end of T unit of time. Preventive maintenance cost is Rs. 15 per machine. Find optimum T which will minimize the expected total cost per period of servicing, given that
0.03  for t=1  
p (t)= 
p(t1)+0.01  for t=2, 3 … , 10 

0.13  for t=11, 12, 13 … 
 30. An electric company which generates and distributes electricity conducted a study on the life poles. The appropriate life data are given in the following table:
Years after installation 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
Percentage poles failing 
1 
2 
3 
5 
7 
12 
20 
30 
16 
4 
(i) If the company now installs 5,000 poles and follows a policy of replacing poles only when they fail, how many poles are expected to be replaced each year during the next ten years?
To simplify the computation assume that failures occur and replacements are made only at the end of a year.
(ii) If the cost of replacing individually is Rs. 160 per pole and if we have a common group replacement policy, it costs Rs 80 per pole, find out the optimal period for group replacement.
 31. A computer contains 10,000 resistors. When any resistor fails, it is replaced. The cost of replacing a resistor individually is Re. 1. If all the resistors are replaced at the same time, the cost per resistor would be reduced to 35 paise. The percentage surviving say S (t) at the end of month t and P (t) the probability of failure during the month t, are
t 
0 
1 
2 
3 
4 
5 
6 
S (t) 
100 
97 
90 
70 
30 
15 
0 
P (t) 
– 
0.03 
0.07 
0.20 
0.40 
0.15 
0.15 
What is the optimum replacement plan?
 32. Given that cost of failure replacement is 3 times compared to the cost of common preventive replacement of Rs. 100 per item in a system of 1000 items and the following data, suggest the strategy between failure replacement and common preventive replacement that should be followed for cost reduction
Week 
0 
1 
2 
3 
4 
5 
% of item surviving at the end of the week 
100 
90 
75 
40 
15 
6 
 33. A manufacturer wants to replace a machine. The purchase price of the machine is Rs. 10,000. Following other details are available.
Year 
Maintenance (Rs.) 
Resale Price (Rs.) 
1 2 3 4 5 6 7 8 
1200 1300 1500 2000 2200 3000 3200 3800 
6000 3000 2000 1000 800 500 400 300 
Suggest in which year the machine may be replaced, if the supplier of the machine is prepared to prove 3 years insitu maintenance free of cost.
 34. Preventive replacement.
 35. Present value of investment.
 36. Group replacement.
 37. Individual replacement.
 38. Present value of investment.
 39. What is replacement? Write a note on preventive replacement.
 40. Describe briefly the model to explain the importance of Replacement Policy in
an organisation.  41. Describe briefly the replacement model to explain the importance of Replacement Policy in an organisation.
 42. Write note on group replacement policy.
 43. What is replacement? Write a note on preventive replacement?
 44. Explain how the theory of replacement is used in the following problems:
(i) Replacement of items that fail completely.
(ii) Replacement of items whose maintenance cost varies with time.
 45. A certain transport company operating in Rajasthan has a fleet of 20 trucks. A truck costs Rs. 1,75,000. From last experience, the transporting company has following information:
Age of Truck in years 
1 
2 
3 
4 
5 
6 
Operating cost (Rs.) 
15600 
20000 
27000 
45000 
85000 
98000 
Salvage value (Rs.) 
11000 
95000 
88000 
70000 
60000 
45000 
What is the optimum replacement period?
 46. The original cost of a machine is Rs, 45,000, Operating cost varies as follows:
Year 
Op. Cost 
1 2 3 4 5 6 7 
3400 4500 7500 10000 12500 18000 21000 
If 10% is the discount rate of money what should be the optimum replace
interval?
 47. Determine the optimal replacement period for the following. The maintenance cost and resale value of an equipment whose purchase prize is 3,00,000 is as follows:
Year 
Maintenance Cost Rs. in lacs. 
Resale price Rs. in lacs 
1 2 3 4 5 6 7 8 
0.36 0.48 0.60 0.72 0.84 0.96 1.08 1.20 
2.00 1.50 1.00 0.80 0.70 0.60 0.50 0.40 
 48. Cost price of a machine is Rs. 50,000 the maintenance cost and scrap value are given in the following table: Calculate the economic life and minimum average cost:
Year 
Maintenance cost 
Scrap value 
1 2 3 4 5 6 7 
2500 2800 3500 5000 6000 7500 10000 
30000 25000 22000 18000 12000 10000 10000 
 49. Determine the Optimum Replacement Period for a machine costing Rs. 6,000. Maintenance cost is Rs. 1,0000 in each of the first 4 years and increases by Rs.
200 every year thereafter. Salvage is NIL. The time value of money is 10 %.  50. Obtain the economic life of the machine and minimum average cost from the following data. Purchase price is Rs. 20,000/
Year 
Maintenance cost 
Resale Price 
1 2 3 4 5 6 
1500 1700 2000 2500 3500 5500 
17000 15300 14000 12000 8000 3000 
 51. Original cost of equipment is Rs. 1,20,000, The year wise maintenance cost is given as under, Find the optimum replacement period if discount rate of money is 10 %, scrap value is Rs. 20,000.
Year 
Maintenance cost 
1 2 3 4 5 6 7 8 
10000 13000 15000 20000 25000 32000 40000 50000 
 52. The probability (P_{n}) of failure just before age (n) are shown below, In individual replacement costs Rs. 1.25 and group replacement costs Rs. 0.50 per item, find the optimal groupreplacement policy (assuming that there are 1000 bulbs in use).
(n)  1  2  3  4  5  6  7  8  9  10  11 
.03  .03  .05  .07  .10  .15  .20  .15  .11  .08  .06 
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