The objective function at Equation

The objective function at Equation 8.31 aims to minimize the total cost TC of maintenance for all periods in the planning horizon. Constraints at Equations 8.32 and 8.33 respectively balance the workforce size and the maintenance workload between adjacent periods. Constraints at Equations 8.34 and 8.35 respectively relate regular time and overtime work hours to the number of regular maintenance workers in each period. Constraints at Equation 8.36 ensure that the number of assigned subcontract work hours does not exceed the available limit in each period.

Example 8.10: Maintenance workload for the next five months is 2,500, 1,500, 1800, 2800 and 2200 man-hours. This workload can be met by employees on regular time at a cost of $10 per hour, employees on overtime at a cost of $15 per hour, or subcontractors at a cost of $18 per hour. The initial workforce size is 10 employees. Each employee works for 150 regular time hours and a maximum of 60 overtime work hours per month. Maximum capacity of subcontract workers is 200 h per month. Early maintenance costs $8 per hour per month, while late maintenance costs $14 per hour per month. For each employee, hiring cost is $800 and firing cost is $1000. Assuming zero starting and ending backlog, model and solve this capacity planning problem using mathematical programming.
The integer programming model is given by
T
min TC 10Rt 15Ot 18St 8At 14Bt 800H t 1000Ft
t 1
subject to
W1 = 10 + H1 – F1
Wt = Wt – 1 + Ht – Ft, t = 2, . . , 5
A1 – B1 = R1 + O1 + S1 – 2500
A2 – B2 = A1 – B1 + R2 + O2 + S2 – 1500 A3 – B3 = A2 – B2 + R3 + O3 + S3 – 1800 A4 – B4 = A3 – B3 + R4 + O4 + S4 – 2800 0 = A4 – B4 + R5 + O5 + S5 – 2200
Rt = 150Wt, t = 1, . . , 5
Ot 60Wt, t = 1, . . , 5
St 200, t = 1, . . , 5
The optimum solution of the above model was obtained by the optimization software package LINDO. The minimum total cost TC is $120,920. Decision variables with non-zero values are shown in Table 8.11.

Table 8.11. Integer programming optimal solution of Example 8.10

Month t 1 2 3 4 5
Workforce size Wt 11 11 12 15 15
Regular time hours Rt 1650 1650 1800 2250 2250
Overtime hours Ot 660 0 0 500 0
Subcontract hours St 40 0 0 0 0
Backlogged hours Bt 150 0 0 50 0
Hired employees Ht 1 0 1 3 0

182 H.K. Al-Fares and S.O. Duffuaa


8.9 Stochastic Techniques for Capacity Planning

Stochastic models for capacity planning consider various uncertainties ever present in real-life maintenance systems. Uncertainties in maintenance surround both maintenance workload or demand (i.e., timing and severity of equipment failure) and maintenance capacity (i.e., availability and effectiveness of maintenance resources). Usually, uncertainties are represented by probability distributions with specified values of the means and variances. Stochastic models for maintenance capacity planning include queuing models, simulation models, and stochastic programming. Stochastic programming models are mathematical programming models similar to the deterministic models discussed in the previous section, except that some of their elements are probabilistic. Although these models have been used for maintenance capacity planning (e.g., Duffuaa and Al-Sultan, 1999), they are beyond the scope of this chapter, and thus will not be discussed further. In the remainder of this section, queuing theory models and computer simulation models are presented.

8.9.1 Queuing Models

Queuing models deal with systems in which customers arrive at a service facility, join a queue, wait for service, get service, and finally depart from the facility. Queuing theory is used to determine performance measures of the given system, such as average queue length, average waiting time, and average facility utilization (Taha, 2003). In addition, queuing models can be used for cost optimization by minimizing the sum of the cost of customer waiting and the cost of providing service. In applying queuing theory to maintenance systems, the maintenance jobs or required maintenance tasks are considered as the customers, and maintenance resources such as manpower and equipment are considered as the servers.
Queuing systems differ from each other in terms of several important characteristics. To define clearly the characteristics of the given queuing situation, a standard notation (Taha, 2003) is used in the following format:
(a/b/c):(d/e/f)

where


a = customer inter-arrival time distribution
b = service time (or customer departure) distribution
c = number of parallel servers
d = queue discipline, i.e., order or priority of serving customers
e = maximum number of customers allowed in the system (queue plus service)
f = size of the total potential customer population


Standard symbols are used to represent individual elements of the above notation (symbols a and b). Arrival and service distributions (symbols a and b) are represented by the symbols M (Markovian or Poisson), D (deterministic or constant), E (Erlang or Gamma), and G (general). The queue discipline (symbol d) is represented by the symbols: FCFS (first come, first served), LCFS (last come, first served), SIRO (service in random order), and GD (general discipline). The

Maintenance Forecasting and Capacity Planning 183


symbol M corresponds to the exponential or Poisson distributions. If the inter- arrival time is exponential, then the number of arrivals during a specific period is Poisson. These complementary distributions have a significant role in queuing theory because they have the Markovian (or forgetfulness) property, which makes them completely random In order to introduce specific queuing models for maintenance capacity planning, the following notation is defined:
n = number of customers in the system (queue plus service)
n = customer arrival rate with n customers in the system
n = customer departure rate with n customers in the system
 = server utilization = n /n
pn = probability of n customers in the system
Ls = expected number of customers in the system Lq = expected number of customers in the queue Ws = expected waiting time in the system
Wq = expected waiting time in the queue

Waiting time and the number of customers are directly related by Little’s Law, one of the most fundamental formulas in queuing theory:
Ls = eff Ws, or Lq = eff Wq (8.38)

where


eff = effective customer arrival rate at the system

Most queuing models are applicable to maintenance capacity planning. Two of
these models are presented below, namely the (M/M/c):(GD//) system and the (M/M/R) (GD/k/k) system.

8.9.1.1 The (M/M/c):(GD//) System
This queuing system has Markovian inter-arrival and service times, c parallel servers (repairmen), and general service disciplines. Since there are no limits on the number of customers in the system, then = eff. Defining = /, the steady- state performance measures for this system are given by
c1

Lq  p0
(c 1)!(c )2

(8.39)


where

Ls = Lq +  (8.40)

⎪⎧c1  n
p0 ⎨ 

1
c ⎪⎫
⎬,

1


(8.41)

⎪⎩n0 n!

c!(1 / c) ⎪⎭ c

The expected number for waiting time in the queue Wq and expected total time in the system Ws are respectively obtained by dividing Lq and Ls by .

The above model can be used in maintenance capacity planning to determine the optimum number of servers c (maintenance workers). In this case, the objective would be to minimize the total cost TC of waiting (i.e., cost of equipment downtime) plus the cost of providing maintenance (i.e., cost of maintenance workers). For example, this objective can be expressed as follows:

184 H.K. Al-Fares and S.O. Duffuaa





where

min TC(c) = cM c + cW Ls(c) (8.42)

cM = cost of maintenance workers per employee
cW = cost of waiting time in the queue

It should be noted that Equation 8.42 is only a typical example of a relevant
objective in maintenance capacity planning. Several alternative objective functions are possible; for instance, c could be replaced by , while Ls could be replaced by Lq, Ws, or Wq.

Example 8.11: A maintenance department repairs a large number of identical machines. Average time between failures is 2 h and 40 min, and average repair time is 5 h; both are exponentially distributed. The hourly labor cost is $15 per maintenance employee, while the hourly cost of downtime is $40 per waiting machine. Use queuing theory to determine the optimum number of maintenance employees.

 = 1/2.6667 = 0.375
 = 1/5 = 0.2
 = 0.375 /0.2 = 1.875
Since /c = 1.875/c < 1, then c > 1.875, or c 2
For c = 2, the average number of waiting machines Ls(2) and associated total cost
TC(2) are calculated by Equations 8.39–8.42 as follows:

p0 (2) ⎨1.875 

1.8752

1
⎫
⎬  0.03226

⎩n0 n!

2!(1 1.875 / 2) ⎭ 31


Lq (2) 


1.875


21

( 1 ) 421.875 13.60887

(2 1)!(2 1.875) 2 31 31
Ls(2) = 13.60887 + 1.875 = 15.48387
TC(2) = 15(2) + 40(15.48387) = 649.35
For c 3, the average number of waiting machines Ls(c) and associated total cost TC(c) are similarly calculated by Equations 8.39–8.42. Because TC(c) is convex, we should start with c = 2 and increment c by one employee at a time until the total cost TC(c) begins to increase. The calculations are summarized in Table 8.12, showing that the optimum number of maintenance employees is equal to 4.

Table 8.12. Queuing model solution of Example 8.11

c p0(c) Ls(c) TC(c)
2 0.03226 15.48387 649.35
3 0.13223 2.52066 145.83
4 0.14924 2.00265 140.11
5 0.15255 1.90328 151.13

Maintenance Forecasting and Capacity Planning 185


8.9.1.2 The (M/M/R):(GD/K/K) System
This queuing system is called the machine repair or machine servicing model. It has Markovian inter-arrival and service times, R parallel servers (repairmen), and a general service discipline.

Kết quả (Việt) 1:

Hàm mục tiêu tại phương trình 8.31 nhằm mục đích giảm thiểu tổng chi phí TC bảo trì trong tất cả thời gian ở chân trời lập kế hoạch. Các hạn chế tại 8,32 cho phương trình và 8.33 tương ứng cân bằng lực lượng lao động kích thước và khối lượng công việc bảo trì giữa thời kỳ lân cận. Các hạn chế tại phương trình 8.34 và 8,35 tương ứng liên quan thường xuyên thời gian và thời gian bù giờ làm việc giờ để số lượng công nhân bảo trì thường xuyên trong từng thời kỳ. Các hạn chế tại 8.36 phương trình đảm bảo rằng số lượng subcontract được chỉ định giờ làm việc không vượt quá giới hạn có sẵn trong từng thời kỳ.Ví dụ 8,10: Khối lượng công việc bảo trì cho năm tháng tiếp theo là 2.500, 1.500, 1800, 2800 và 2200 giờ. Khối lượng công việc này có thể được đáp ứng bởi các nhân viên thường xuyên thời gian chi phí $10 cho mỗi giờ, nhân viên trên giờ làm thêm chi phí $15 cho mỗi giờ, hoặc nhà thầu phụ chi phí $ 18 cho giờ. Kích thước của lực lượng lao động ban đầu là 10 nhân viên. Mỗi nhân viên làm việc cho 150 thường xuyên thời gian giờ và tối đa là 60 giờ làm việc thêm giờ mỗi tháng. Sức chứa tối đa của người lao động subcontract là h 200 mỗi tháng. Bảo trì đầu chi phí $8 mỗi giờ mỗi tháng, trong khi chi phí bảo trì cuối $14 cho giờ mỗi tháng. Đối với mỗi nhân viên, thuê chi phí là $800 và bắn chi phí là $1000. Giả sử không bắt đầu và kết thúc backlog, mô hình và giải quyết vấn đề lập kế hoạch khả năng này bằng cách sử dụng toán học lập trình.Mô hình lập trình số nguyên được cho bởiTMin TC 10Rt 15Ot 18St 8At 14Bt 800H t 1000Ftt 1tùy thuộc vàoW1 = 10 + H1-F1Wt = Wt-1 + Ht-Ft, t = 2,. . , 5A1-B1 = R1 O1 ++ S1-2500A2-B2 = A1-B1 + R2 + O2 + S2-1500 A3-B3 = A2-B2 R3 + O3 ++ S3-1800 A4-B4 = A3-B3 R4 + O4 ++ S4-2800 0 = A4-B4 R5 + O5 + S5-2200RT = 150Wt, t = 1,. . , 5OT 60Wt, t = 1,. . , 5St 200, t = 1,. . , 5Các giải pháp tối ưu của các mô hình ở trên thu được bằng cách tối ưu hóa phần mềm gói LINDO. Tối thiểu tổng chi phí TC là $120,920. Các biến quyết định với giá trị không được hiển thị trong bảng 8.11.8.11 bảng. Số nguyên lập trình các giải pháp tối ưu của ví dụ 8,10Tháng t 1 2 3 4 5Lực lượng lao động kích thước Wt 11 11 12 15 15Thường xuyên thời gian giờ Rt 1650 1650 1800 2250 2250Thêm giờ giờ Ot 660 0 0 500 0Subcontract giờ St 40 0 0 0 0Trệ giờ Bt 150 0 0 50 0Thuê nhân viên Ht 1 0 1 3 0 182 H.K. Al-giá vé và so Duffuaa8.9 các kỹ thuật ngẫu nhiên lập kế hoạch nâng cao năngCác mô hình ngẫu nhiên năng lực lập kế hoạch cho xem xét sự không chắc chắn khác nhau đã từng có trong cuộc sống thực bảo trì hệ thống. Sự không chắc chắn trong duy trì bao quanh cả hai khối lượng công việc bảo trì hoặc nhu cầu (tức là, thời gian và mức độ nghiêm trọng của thiết bị thất bại) và khả năng bảo dưỡng (tức là, tính khả dụng và hiệu quả của tài nguyên bảo trì). Thông thường, sự không chắc chắn được đại diện bởi phân bố xác suất với các giá trị được chỉ định của các phương tiện và chênh lệch. Các mô hình ngẫu nhiên để bảo trì năng lực lập kế hoạch bao gồm xếp hàng các mô hình, mô hình mô phỏng, và lập trình ngẫu nhiên. Mô hình lập trình ngẫu nhiên là toán học lập trình mô hình tương tự như các mô hình xác định, thảo luận trong phần trước, ngoại trừ rằng một số yếu tố của họ là xác suất. Mặc dù những mô hình đã được sử dụng để bảo trì năng lực lập kế hoạch (ví dụ như, Duffuaa và Al-Sultan, 1999), họ vượt ra ngoài phạm vi của chương này, và do đó sẽ không được thảo luận thêm. Trong phần còn lại của phần này, xếp hàng các mô hình lý thuyết và mô hình mô phỏng máy tính cũng được trình bày trong phòng.8.9.1 mô hình xếp hàngQueuing models deal with systems in which customers arrive at a service facility, join a queue, wait for service, get service, and finally depart from the facility. Queuing theory is used to determine performance measures of the given system, such as average queue length, average waiting time, and average facility utilization (Taha, 2003). In addition, queuing models can be used for cost optimization by minimizing the sum of the cost of customer waiting and the cost of providing service. In applying queuing theory to maintenance systems, the maintenance jobs or required maintenance tasks are considered as the customers, and maintenance resources such as manpower and equipment are considered as the servers.Queuing systems differ from each other in terms of several important characteristics. To define clearly the characteristics of the given queuing situation, a standard notation (Taha, 2003) is used in the following format:(a/b/c):(d/e/f) where a = customer inter-arrival time distributionb = service time (or customer departure) distributionc = number of parallel serversd = queue discipline, i.e., order or priority of serving customerse = maximum number of customers allowed in the system (queue plus service)f = size of the total potential customer population Standard symbols are used to represent individual elements of the above notation (symbols a and b). Arrival and service distributions (symbols a and b) are represented by the symbols M (Markovian or Poisson), D (deterministic or constant), E (Erlang or Gamma), and G (general). The queue discipline (symbol d) is represented by the symbols: FCFS (first come, first served), LCFS (last come, first served), SIRO (service in random order), and GD (general discipline). The Maintenance Forecasting and Capacity Planning 183symbol M corresponds to the exponential or Poisson distributions. If the inter- arrival time is exponential, then the number of arrivals during a specific period is Poisson. These complementary distributions have a significant role in queuing theory because they have the Markovian (or forgetfulness) property, which makes them completely random In order to introduce specific queuing models for maintenance capacity planning, the following notation is defined:n = number of customers in the system (queue plus service)n = customer arrival rate with n customers in the systemn = customer departure rate with n customers in the system = server utilization = n /npn = probability of n customers in the systemLs = expected number of customers in the system Lq = expected number of customers in the queue Ws = expected waiting time in the systemWq = expected waiting time in the queueWaiting time and the number of customers are directly related by Little’s Law, one of the most fundamental formulas in queuing theory:Ls = eff Ws, or Lq = eff Wq (8.38) where eff = effective customer arrival rate at the system Most queuing models are applicable to maintenance capacity planning. Two ofthese models are presented below, namely the (M/M/c):(GD//) system and the (M/M/R) (GD/k/k) system.8.9.1.1 The (M/M/c):(GD//) SystemThis queuing system has Markovian inter-arrival and service times, c parallel servers (repairmen), and general service disciplines. Since there are no limits on the number of customers in the system, then = eff. Defining = /, the steady- state performance measures for this system are given byc1 Lq  p0(c 1)!(c )2 (8.39) where Ls = Lq +  (8.40) ⎪⎧c1  np0 ⎨  1c ⎪⎫⎬, 1 (8.41) ⎪⎩n0 n! c!(1 / c) ⎪⎭ c The expected number for waiting time in the queue Wq and expected total time in the system Ws are respectively obtained by dividing Lq and Ls by .The above model can be used in maintenance capacity planning to determine the optimum number of servers c (maintenance workers). In this case, the objective would be to minimize the total cost TC of waiting (i.e., cost of equipment downtime) plus the cost of providing maintenance (i.e., cost of maintenance workers). For example, this objective can be expressed as follows:
184 H.K. Al-Fares and S.O. Duffuaa





where

min TC(c) = cM c + cW Ls(c) (8.42)

cM = cost of maintenance workers per employee
cW = cost of waiting time in the queue

It should be noted that Equation 8.42 is only a typical example of a relevant
objective in maintenance capacity planning. Several alternative objective functions are possible; for instance, c could be replaced by , while Ls could be replaced by Lq, Ws, or Wq.

Example 8.11: A maintenance department repairs a large number of identical machines. Average time between failures is 2 h and 40 min, and average repair time is 5 h; both are exponentially distributed. The hourly labor cost is $15 per maintenance employee, while the hourly cost of downtime is $40 per waiting machine. Use queuing theory to determine the optimum number of maintenance employees.

 = 1/2.6667 = 0.375
 = 1/5 = 0.2
 = 0.375 /0.2 = 1.875
Since /c = 1.875/c < 1, then c > 1.875, or c 2
For c = 2, the average number of waiting machines Ls(2) and associated total cost
TC(2) are calculated by Equations 8.39–8.42 as follows:

p0 (2) ⎨1.875 

1.8752

1
⎫
⎬  0.03226

⎩n0 n!

2!(1 1.875 / 2) ⎭ 31


Lq (2) 


1.875


21

( 1 ) 421.875 13.60887

(2 1)!(2 1.875) 2 31 31
Ls(2) = 13.60887 + 1.875 = 15.48387
TC(2) = 15(2) + 40(15.48387) = 649.35
For c 3, the average number of waiting machines Ls(c) and associated total cost TC(c) are similarly calculated by Equations 8.39–8.42. Because TC(c) is convex, we should start with c = 2 and increment c by one employee at a time until the total cost TC(c) begins to increase. The calculations are summarized in Table 8.12, showing that the optimum number of maintenance employees is equal to 4.

Table 8.12. Queuing model solution of Example 8.11

c p0(c) Ls(c) TC(c)
2 0.03226 15.48387 649.35
3 0.13223 2.52066 145.83
4 0.14924 2.00265 140.11
5 0.15255 1.90328 151.13

Maintenance Forecasting and Capacity Planning 185


8.9.1.2 The (M/M/R):(GD/K/K) System
This queuing system is called the machine repair or machine servicing model. It has Markovian inter-arrival and service times, R parallel servers (repairmen), and a general service discipline.