Cost Optimization and Sensitivity Analysis of the Machine Repair Problem with Variable Servers and Balking

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Procedia - Social and Behavioral Sciences 00 (2011) 000–000 Procedia - Social and Behavioral Sciences 25 (2011) 178 – 188

Procedia Social and Behavioral Sciences www.elsevier.com/locate/procedia

International Conference on Asia Pacific Business Innovation & Technology Management

Cost Optimization and Sensitivity Analysis of the Machine Repair Problem with Variable Servers and Balking Kuo-Hsiung Wanga*, Yuh-Ching Lioub and Dong-Yuh Yangc a Department of Business Administration, Asia University, Taichung, Taiwan Department of Applied Mathematics, National Chung-Hsing University,Taichung, Taiwan c Institute of Information and Decision Sciences, National Taipei College of Business, Taipei, Taiwan b

Abstract

We consider a machine repair model with M identical machines and variable servers considering balking concept. A recursive method is applied o develop analytical steady-state solutions. We also calculate various system performance measures, such as the expected number of failed machines, the expected number of operating machines, the expected number of busy and idle servers, average balking rate, machine availability, operative utilization, and so on. A cost model is constructed to determine the optimal values of the number of busy servers, and maintain the balking rate at a certain level. Numerical results are provided in which various system performance measures are evaluated under optimal operating conditions. Sensitivity analysis is performed to investigate the effect of changes in the system parameters on the expected cost function. Finally, numerical results are also given for illustrative purposes. © 2011 2011 Published Published by byElsevier ElsevierLtd. Ltd.Selection Selectionand/or and/orpeer-review peer-reviewunder underresponsibility responsibilityofofthe theAsia Pacific Business Innovation and Technology Management Society Asia Pacific Business Innovation and Technology Management Society (APBITM).” keywords: Balk; cost; machine repair problem; optimization; variable servers.

* Corresponding author. Tel.: +886-4-2232-3456 ext 5541; fax: +886-4-2332-1157. E-mail address: [email protected]

1877-0428 © 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of the Asia Pacific Business Innovation and Technology Management Society doi:10.1016/j.sbspro.2011.10.539

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1. Introduction Queueing models with balking or reneging (or both) have been investigated by many researchers. For models involving balking, Haight (1957) first studied an M/M/1 queue with balking. For models involving the combined effects of balking and reneging, Ancker and Gafarian (1963a, 1963b) first analyzed the limited waiting room in an M/M/1 queue. An M/G/1 queue with balking and customers departing without service (reneging) was proposed by Jaiswal [7] and Rao [9]. Analytical solutions of the single-server Markovian overflow queue with balking, reneging and an additional server for longer queues were developed by Abou-El-Ata and Shawky (1922). Shawky (1997) derived steady-state solutions of the single-server machine repair problem with balking, reneging and an additional server for longer queues. Moreover, Ke and Wang [8] examined an M/M/R machine repair problem with balking, reneging and server breakdowns for which cost analysis is provided. They developed matrix-analytic solutions and perform sensitivity analysis. For the M/M/R machine repair problem with balking, reneging, and standby switching failures, Wang et al.(2007) utilized the direct search with steepest decent method to determine the global maximum value until the availability, balking, and reneging constraints are satisfied. Recently, for the M/M/1 machine repair model with working vacation, Wang et al. (2009) applied the direct search method and the Newton’s method to find the global minimum value until the system availability constraint is satisfied. The purpose of this paper is threefold. Firstly, we use a recursive method to develop analytical steady-state solutions which is used to perform system performance measures, Tseng et al. (2009) Secondly, an expected cost function per unit time is constructed to determine the joint optimal values of R at the minimum cost until the balking rate constraint is satisfied. Thirdly, a sensitivity analysis is carried out to verify the effect of the system parameters  and  on the cost function. 2. Model Descriptions and Assumptions We consider a machine repair model with M identical operating machines which are maintained by variable number of servers. The detailed descriptions and assumptions of this model are described as follows: 1. Each of the operating machines fails according to a Poisson process with parameter  . Each of the operating machines fails independently of the state of the others. 2. The behavior of impatient failed machines which upon arrival may or may not enter the queue for service depending on the number of failed machines in the system. 3. The minimum number of busy servers is denoted by  . We assume that  is one, i.e.,   1. 4. A failed machine which on arrival finds n  s failed machines in the system, either decides to enter the queue with probability bn or balks (do not enter) with probability 1  bn , where bn is defined as follows:

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1, if n s 0;  bn   , if n  s  1;  0, otherwise. 

It is noted that  is the probability of a failed machine entering the queue. 5. When a machine fails, it is immediately sent to a server. It is served in order of breakdown, with identical service rate  . 6. Service time distribution of the server is assumed to be exponentially distributed. Once a machine is repaired, it is as good as new and goes into operating state. 7. Failed machine arrives at the server form a single waiting line. The server can repair only one failed machine at a time. 8. When all of these servers are busy and the number of failed machines waiting in the queue reaches a specific number N, then one additional server will be active instantly and the queue length immediately reduces to N-1. 9. If while   1 servers are working, the queue length should again reach N, another server actives and the queue length again reduces to N-1. 10. The maximum number of busy servers is R. This process of increasing the number of servers may continue until all R servers are busy, then no increase in the number of servers is possible and the queue is allowed to grow without bound. 11. When there are  failed machines in the queue and services are completed, the servers that are emptied are cancelled. This reduction process may continue until the minimum allowable number of servers is reached. For simplicity, we only consider one case for this machine repair model: R  N  1  M . 3. Steady-state Results For the machine repair problem with balking and variable servers, the states of the system are presented by pairs {(n, s)| n=0,1,2, … , M-R; s=0,1,2, … , R}, where n is the number of failed machines waiting in the queue, s is the number of busy servers, and R is the maximum number of busy servers. We define the following steady-state probabilities: Pn,s ≡ probability that there are n failed machines in the queue when there are s servers are working, where n=0,1, … , M-R and s=0,1, … , R. Steady-state equations Applying Markov process, the steady-state equations for the machine repair problem with variable servers and balking are given as follows. M  P0,0   P0,1 , n s 0,

(1)

(M  n  1)  Pn,1  Pn 1,1 , 0  n    1, s 1,

(2)

(M  n  1)  Pn,1  Pn 1,1  (M  N ) PN 1,1 ,   n  N  2,  s 1,

(3)

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(M  n  s) Pn, s  s P , s s Pn 1, s  (M  N  s  1) PN 1, s ,   n  N  2, 2  s  R  1,

s P , s  (M  N  s  2) PN 1, s 1 , n  N  1, 2  s  R,

(5)

R P , R  M  n  R   Pn, R 

(6)

 M  n  R   Pn, R

R Pn 1, R ,   n  N  2, s R,

(4)

(7)

R Pn 1, R , N  1  n  M   R  1, s R.

Steady-state solutions From (2), it follows that

 M  n  1

P n 1,1



Pn,1   M  n  1 Pn,1 , 0  n    1, 

(8)

where    / . . Solving (8) recursively and using (1) yields M!  n 1  n P0,0 , 1  n   . Pn,1  M  n  1 !

(9)

Using (3) and (9), we finally obtain n   M  n  j  2 !  n  j  Pn,1  n   M  k    P0,0       M  N  P ,   1  n  N  1. (10)  M  n  1 ! N 1,1 j 1  k 0 

Likewise, we get from (10) that  N 1



 N 1   M  k   

 (11) P  M  N  j  1 ! 0,0 M  N ! j 1 Solving (4)-(7), recursively, and we obtain analytic solutions Pn , s in neat closed-form as follow: PN 1,1 

P , s 

Pn, R

1

 k 0

N 1 

     M  N  j

 M  N  s  2 s

      k 0  R  n 

k

 PN 1, s 1 , 2  s  R, n  N  1,

M  R  n  k ! P ,   1  n  N  1,  M  R  n !  ,R

(12) (13)

n

Pn, R

  M  k  R  1 k N

R

n  N 1

  

n  N 1

PN 1, R , N  n  M  R.

(14)

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We should note that steady-state solutions Pn,s (n=0,1, … , N-1 and s=0,1, … , R) for case 1 can be found because the number of states are finite. From (8)-(15), P0,0 is obtained from the normalization condition N 1

R

N 1

M R

s 2 n 

 n N

P0,0   Pn,1    Pn, s   n 0

P

n, R

 1.

(15)

4. System Performance Measures Our analysis is based on the following system performance measures of the machine repair problem with variable servers and balking. L1  the expected number of failed machines in the system when one server is working, Ls  the expected number of failed machines in the system when s servers are working, where s = 2, 3, 4, … , R-1, LR  the expected number of failed machines in the system when all the servers are working,

L  the expected number of failed machines in the system, Lq  the expected number of failed machines in the queue, E[O]  the expected number of operating machines in the system, E[ I ]  the expected number of idle servers in the system, E[ B]  the expected number of busy servers in the system, BR  average balking rate, MA  machine availability (the fraction of the time that the machines are working), OU  operative utilization (the fraction of the busy servers). The expressions for L1 , Ls , LR , L , Lq , E[O] , E[ I ] , E[ B] and BR are given by  L1

 Ls  LR

N 1

  n  1 P

n ,1

n0

N 1

  n  s  P n

(16)

,

n, s

, 2  s  R 1 ,

M R

  n  R  P

n, R

n

R 1

L  L1  Ls  LR , s 2

,

(17) (18) (19)

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N 1

R 1 N 1

 n 0

s 2 n 

Lq   nPn,1   nPn,s 

M R

 nP

 n 

n, R

,

(20)

E O  M  L, E I  

(21)

N 1

R 1 N 1

 n 0

s 2 n 

  R  1 Pn,1    R  s  Pn,s ,

(22)

E  B R  E  I  ,  BR

(23)

N 1

R 1 N 1

  M  n  1  1    P    M  n  s   1    P

 n 0

n ,1

n, s

s 2 n



M R

  M  n  R   1    P

n, R

 n 

. (24)

The machine availability and the operative utilization for two cases are defined as: MA 

and

OU 

E O 

(25)

M

E  B

(26) . R The MAPLE computer program would be implemented to compute various system performance measures in the above. 5. Cost Optimization and Sensitivity We develop a steady-state expected cost function per unit time and impose a constraint on the balking rate in which R is a decision variable. One may notice that the discrete variable R is required to be natural number. Our main objective is to determine the optimum number of busy servers R , say R* , so as to minimize this function until the balking rate constraint is satisfied. We select the following cost elements:

C1  cost per unit time when one failed machine is in the system, C2  cost per unit time when one server is idle, C3  cost per unit time when one server is busy, C4  loss cost per unit time when one failed machine balks. Using the definitions of these cost elements listed above, the total expected cost function per unit time is given by

F ( R)  C1L  C2 E  I   C3 E  B  C4 BR.

(27)

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The cost minimization problem can be presented mathematically as

Minimize F ( R) , R

subject to: 0    1 . The cost parameters in (27) are assumed to be linear in the expected number of the indicated quantity. Since the expected cost function is highly non-linear and complex, it would be a hard task to develop analytic results for the optimum value R* . Subsequently, we give a numerical example to illustrate how to obtain the optimum R to minimize the cost. We consider the number of operating machines M  15 . Due to the discrete property of R, we first use direct substitution of successive values of R into the cost function until the minimum value of F ( R) say F ( R* ) is achieved and the balking constraint 0    1 is satisfied. The following numerical results are provided by considering cost parameters as follows:

C1  $140 / day, C2  $100 / day, C3  $130 / day, C4  $120 / day. We first fix N  4 ,   0 ,   0.8 ,   1.0 , vary the number of busy servers R from 2 to 11, and choose different values of  = 0.3, 0.6, 0.8. One observes from Table 1 that a minimum expected cost (a) of $1130.07 is achieved at R*  3 for   0.3 ; (b) of $1634.96 is achieved at R*  4 for   0.6 ; and (c) of $1863.86 is achieved at R*  5 for   0.8 . Next, we fix N  4 ,   0 ,   0.8 ,   0.4 , vary the number of busy servers R from 2 to 11, and choose various values of  = 0.5, 1.0, 1.5. It reveals from Table 2 that a minimum expected cost (a) of $1794.32 is achieved at R*  5 for   0.5 ; (b) of $1324.38 is achieved at R*  4 for   1.0 ; and (c) of $1087.06 is achieved at R*  3 for   1.5 . Cost Sensitivity Analysis With the developed cost function, we perform a sensitivity analysis for the expected cost function with respect to changes in specific values of the system parameters. Differentiating the expected cost function with respect to  and  . The sensitivity of the expected cost function is calculated as follows:

E  I  E  B  F L BR ,  C1  C2  C3  C4      where    and  . A graphical analysis is presented to study the effects of  and  on the expected cost function. We choose R  4, 6, 8 and consider the following two cases. Case (i): M  15 , N  4 ,   0 ,   0.8 ,   1.0 ,  varies from 0.1 to 0.8. Case (ii): M  15 , N  4 ,   0 ,   0.8 ,   0.5 ,  varies from 0.6 to 2.0. We observe from Figures 1-2 that F /   0 and F /   0 , respectively. It is noted that  has a positive sign of sensitivity, which means that incremental changes in 

(28)

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increase the expected cost. In contrast,  has a negative sign of sensitivity, which means that an increase in  improves the expected cost. From Figure 1, one can easily see that (i) F /  decrease as  increases for all R ; and (ii) as  is fixed, F /  decreases as R increases. Figure 2 gives that (i) F /  monotonically increases as  increases for all R ; and (ii) as  is fixed, F /  increases as R increases. In summary, it reveals from Figures 1-2 that  and  affect the expected cost significantly. 6. Conclusions In this paper, we proposed to study the machine repair model with M identical machines and variable servers considering balking concept. We first used a recursive method to develop analytical steady-state solutions. Next, an efficient computer program (MAPLE) was implemented to calculate various system performance measures. An expected cost function was constructed to determine the optimal value of R* until the balking rate constraint is satisfied. Finally, a sensitivity analysis of the cost function was performed for specific values of the system parameters  and  . Moreover, numerical investigations show that  and  affect the expected cost significantly.

References [1] Abou-El-Ata, M. O.; Shawky, A. I. (1992). The single-server Markovian overflow queue with balking, reneging and an additional server for longer queues. Microelectronics and Reliability, 32, 1389-1394. [2] Ancker, C. J., Jr.; A. V. Gafarian (1963a). Some queueing problems with balking and reneging: I. Operations Research, 11, 88-100. [3] Ancker, C. J., Jr.; A. V. Gafarian (1963b). Some queueing problems with balking and reneging: II. Operations Research, 11, 928-937. [4] Haight, F. A. (1957). Queueing with balking, Biometrika, 44, 360-369. [5] Ke, J.-C.; Wang, K.-H. Cost analysis of the M/M/R machine-repair problem with balking, reneging, and server breakdowns. Journal of the Operational Research Society, 50, 275-282 (1999). [6] Rao, S. S. (1968). Queueing with balking and reneging in M/G/1 systems, Metrika, 12, 173-188. [7] Shawky, A. I. (1997). The single-server machine interference model with balking, reneging and an additional server for longer queues. Microelectronics and Reliability, 37, 355-357. [8] Tseng, M.L., Lin, Y.H., and Chiu, A.S.F. (May 2009). FAHP based study of cleaner production implementation in PCB manufacturing firms, Taiwan. Journal of Cleaner Production 17(14),1249-1256

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[9] Wang, K.-H.; Ke, J.-B.; Ke, J.-C. (2007). Profit analysis of the M/M/R machine repair problem with balking, reneging, and standby switching failures. Computers & Operations Research, 34, 835-847. [10] Wang, K.-H.; Chen, W.-L.; Yang, D.-Y. (2009). Optimal management of the machine repair problem with working vacation: Newton’s method. Journal of Computational and Applied Mathematics, 233, 449-458. Table 1. The expected cost F ( R) for   0.3, 0.6, 0.8.

R/

0.3

0.6

0.8

2 3 4 5 6 7 8 9 10 11

1286.60 1130.07 1157.30 1240.09 1336.72 1436.42 1536.68 1637.03 1737.39 1837.74

1836.74 1711.56 1634.96 1641.51 1705.36 1794.12 1891.33 1990.82 2090.77 2190.79

1982.50 1924.28 1875.53 1863.86 1903.30 1978.54 2070.60 2168.64 2268.31 2368.27

Table 2. The expected cost F ( R) for   0.5, 1.0, 1.5.

R/

0.5

1.0

1.5

2 3 4 5 6 7 8 9 10 11

1952.50 1879.33 1816.54 1794.32 1828.05 1900.94 1992.25 2090.11 2189.74 2289.71

1548.86 1355.74 1324.38 1383.15 1472.31 1570.00 1669.70 1769.79 1869.91 1970.04

1203.12 1087.06 1134.87 1223.83 1322.10 1422.25 1522.72 1623.22 1723.73 1824.24

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Kuo-Hsiung Wang et al. / Procedia - Social and Behavioral Sciences 25 (2011) 178 – 188

Figure 1. Sensitivity analysis of the expected cost function with respect to  for different R ( M  15 , N  4 ,   0 ,   0.8 and   1.0 ).

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Figure 2. Sensitivity analysis of the expected cost function with respect to  for different R ( M  15 , N  4 ,   0 ,   0.8 and   0.5 ).