^{1}

^{*}

^{2}

^{2}

Several researches have been done to provide better alternative to the existing replacement models, but the research works did not adequately address the replacement problem for items that fail suddenly. Hence, a modified replacement model for items that fail suddenly has been proposed using the knowledge of probability distribution of failure times as well as that of variable replacement cost. The modified cost functions for implementing both individual and group replacements were derived. The modified cost functions were minimized using the principle of classical optimization in order to find the age at which replacement of items would be appropriate. Conditions under which the individual and group replacement policies should be adopted were derived. Two real data sets on failure time of LED bulbs and their replacement costs were used to validate the theoretical claims of this work. In essence, goodness-of-fit test was used to select appropriate probability distribution of failure times as well as that of replacement costs for data sets I and II respectively. The goodness-of-fit results showed that failure times of LED bulbs follow the Smallest Extreme Value and Laplace distributions for data sets I and II respectively. Similarly, it was observed that individual replacement cost followed the two-parameter Gamma and Largest Extreme Value distributions for data sets I and II respectively. Further, the group replacement cost was found to follow the log-normal and two-parameter Weibull distributions for data sets I and II respectively. Based on the empirical study, we observed that individual replacement policy is better than group replacement policy in terms of cost minimization for both existing model and the proposed model. In view of the results, the proposed replacement policy was recommended over the existing one because it yielded lower replacement costs than the existing replacement model.

In many organizations, several job performing units like men, machines, equipment, parts etc. are used for carrying out day-to-day activities. When any job performing unit is new, it works with full operating efficiency and due to usage or of time, it may become old and some of its components wear out and the operating efficiency of the job performing unit falls down. In order to regain the efficiency, maintenance is carried out. The act of maintenance consists of replacing the worn out part, or oiling or overhauling, or repair etc. Once maintenance is attended, the efficiency may not be regained to the previous level but a bit less than that of the previous level. For example, if the operating efficiency is 95 per cent and due to deterioration, the efficiency reduces to the level of 90 per cent, after maintenance, it may regain to the level of 93 percent. Once again due to usage, the efficiency falls down and the maintenance is to be attended. After some time, the efficiency reduces to such a level that the maintenance cost will become very high and due to low efficiency, the unit production cost will be very high and at this time, the management has to think of replacing the job performing unit. According to [

The appropriate age at which replacement should be implemented with minimal cost constitutes a large class of problem in organizations. However, several works have been carried out in this direction. To this effect, two replacement models exist in literature, namely: replacement models for items that fail gradually with the passage of time and that for items which fail suddenly. Considerable efforts have been made in addressing the problem of replacement of items that fail gradually with the passage of time [

The ultimate objective of this paper is to propose an improved model for optimal replacement of items that fail suddenly by putting forward a replacement model that will modify [

The average cost of individual replacement, A ( n ) i is given in [

A ( n ) i = N E ( X ) C i (1)

where C i is the cost per item for individual replacement, E ( X ) is the expected life of the item, and N is the total number of items in the system.

The expression for computing E(X) of Equation (1) is given as:

E ( X ) = ∑ j = 1 K j P j (2)

where P j is the probability of items that fail at the end of jth period and k is the end of the period of each replacement. Works by [

P x = M ( t − 1 ) − M ( t ) N , M ( t − 1 ) > M ( t ) (3)

where M ( t − 1 ) is the number of survivors at any time, t − 1, M ( t ) is the number of survivors at time, t and N is the initial number of items in the system. The individual replacement policy is concerned with replacing an item as at when it fails and Equation (1) is the average individual replacement cost per period.

The average cost of group replacement per period A ( n ) g is given in [

A ( n ) g = C ( n ) n = N C g + C i ∑ X = 1 n − 1 N ( X ) n (4)

where C ( n ) is the total cost of group replacement, N ( X ) is the number of failures (or replacements) at the end of the jth period, C i is the cost per item for individual replacement, C g is the cost of replacing an item when all the items in that group are replaced simultaneously, and n is the age of replacement of items that fail suddenly.

The expression for computing N ( X ) of Equation (4) is as given below

N ( X ) = ∑ j = 1 X N X − j P j , X = 1 , 2 , ⋯ , n (5)

In Equation (1) and Equation (4) respectively, [_{i} and C_{g} are fixed (or constant) over time. However, in reality, rarely do costs of replacement of items appear to be fixed over time due to the dynamic nature of the world’s economy. Since the values of C_{i} and C_{g} cannot be predicted with certainty, it suffices to view C_{i} and C_{g} as random variables that can be governed by some probability laws. Based on the probability distributions of C_{i} and C_{g} respectively, we now obtain the expected values E(C_{i}) and E(C_{g}) [

C i m = C V i (6)

where C i m is the modified cost of replacing an individual item on failure C V i is the variable cost of replacing an individual item on its failure.

Substituting Equation (6) into Equation (1), we obtain the modified cost function for individual replacement of items that fail suddenly as:

A m ( n ) i = N E ( X ) C V i (7)

where A m ( n ) i is the modified average cost of individual replacement per period.

Since variable cost component, C V i has been incorporated into the replacement function in Equation (1), it is worthwhile to determine a probability distribution for C V i so that the expected value of C V i is E ( C V i ) . If we take the expectation of both sides of Equation (7), we obtain

E [ A m ( n ) i ] = N E ( X ) E ( C V i ) (8)

Let C ( n ) m be the modified cost of replacing items as a group, C V i is the variable cost of replacing an individual item on its failure and C V g is the variable cost per item when all items are replaced as a group. Then substituting C V i for and C V g into Equation (4), we obtain the modified cost function for group replacement as shown in Equation (9).

A m ( n ) g = C ( n ) m n = N C V g + C V i ∑ X = 1 n − 1 N ( X ) n (9)

Taking expectation of both sides of Equation (9), we obtain;

E [ A m ( n ) g ] = E [ C ( n ) m ] n = N E ( C V g ) + E ( C V i ) ∑ X = 1 n − 1 N ( X ) n (10)

We shall obtain the value of n that minimizes Equation (10) using classical optimization.

Recall from the numerator of Equation (4.9) that,

E [ C ( n ) m ] = N E ( C V g ) + E ( C V i ) ∑ X = 1 n − 1 N ( X ) (11)

If we replace n by n + 1 in Equation (4.10), we obtain

E [ C ( n + 1 ) m ] = N E ( C V g ) + E ( C V i ) ∑ X = 1 n N ( X ) E [ C ( n + 1 ) m ] = N E ( C V g ) + E ( C V i ) [ N ( 1 ) + N ( 2 ) + ⋯ + N ( n − 1 ) + N ( n ) ] E [ C ( n + 1 ) m ] = N E ( C V g ) + E ( C V i ) [ ∑ X = 1 n − 1 N ( X ) + N ( n ) ] E [ C ( n + 1 ) m ] = N E ( C V g ) + E ( C V i ) ∑ X = 1 n − 1 N ( X ) + E ( C V i ) N (n)

E [ C ( n + 1 ) m ] = E [ C ( n ) m ] + E ( C V i ) N ( n ) (12)

If we replace n by n − 1 in Equation (11), we obtain;

E [ C ( n − 1 ) m ] = N E ( C V g ) + E ( C V i ) ∑ X = 1 n − 2 N ( X ) E [ C ( n − 1 ) m ] = N E ( C V g ) + E ( C V i ) [ N ( 1 ) + N ( 2 ) + ⋯ + N ( n − 2 ) ]

E [ C ( n − 1 ) m ] = N E ( C V g ) + E ( C V i ) [ N ( 1 ) + N ( 2 ) + ⋯ + N ( n − 2 ) + N ( n − 1 ) − N ( n − 1 ) ]

E [ C ( n − 1 ) m ] = N E ( C V g ) + E ( C V i ) [ ∑ X = 1 n − 1 N ( X ) − N ( n − 1 ) ]

E [ C ( n − 1 ) m ] = N E ( C V g ) + E ( C V i ) ∑ X = 1 n − 1 N ( X ) − E ( C V i ) N ( n − 1 )

E [ C ( n − 1 ) m ] = E [ C ( n ) m ] − E ( C V i ) N ( n − 1 ) (13)

⇒ E [ C ( n ) m ] = E [ C ( n − 1 ) m ] + E ( C V i ) N ( n − 1 ) (14)

From Equation (10), we define

E [ A m ( n + 1 ) g ] = E [ C ( n + 1 ) m ] n + 1 (15)

Substituting Equation (14) into Equation (15), we obtain

E [ A m ( n + 1 ) g ] = E [ C ( n ) m ] + E ( C V i ) N ( n ) n + 1 (16)

From Equation (10), we define

E [ A m ( n − 1 ) g ] = E [ C ( n − 1 ) m ] n − 1 (17)

Substituting Equation (13) into Equation (17), we obtain

E [ A m ( n − 1 ) g ] = E [ C ( n ) m ] − E ( C V i ) N ( n − 1 ) n − 1 (18)

and from definition, we know that

Δ E [ A m ( n ) g ] = E [ A m ( n + 1 ) g ] − E [ A m ( n ) g ] (19)

Substituting Equation (10) and Equation (16) into Equation (19), we obtain

Δ E [ A m ( n ) g ] = E [ C ( n ) m ] + E ( C V i ) N ( n ) n + 1 − E [ C ( n ) m ] n Δ E [ A m ( n ) g ] = n E [ C ( n ) m ] + n E ( C V i ) N ( n ) − ( n + 1 ) E [ C ( n ) m ] n ( n + 1 ) Δ E [ A m ( n ) g ] = n E [ C ( n ) m ] + n E ( C V i ) N ( n ) − n E [ C ( n ) m ] − E [ C ( n ) m ] n ( n + 1 ) Δ E [ A m ( n ) g ] = n E ( C V i ) N ( n ) − E [ C ( n ) m ] n ( n + 1 ) Δ E [ A m ( n ) g ] = n { E ( C V i ) N ( n ) − E [ C ( n ) m ] n } n ( n + 1 )

Δ E [ A m ( n ) g ] = E ( C V i ) N ( n ) − E [ C ( n ) m ] n ( n + 1 ) (20)

From definition, we know that

Δ E [ A m ( n − 1 ) g ] = E [ A m ( n ) g ] − E [ A m ( n − 1 ) g ] (21)

Substituting Equation (10) and Equation (17) into Equation (21), we obtain

Δ E [ A m ( n − 1 ) g ] = E [ C ( n ) m ] n − { E [ C ( n ) m ] − [ E ( C V i ) ] N ( n − 1 ) } n − 1 Δ E [ A m ( n − 1 ) g ] = ( n − 1 ) E [ C ( n ) m ] − n { E [ C ( n ) m ] − [ E ( C V i ) ] N ( n − 1 ) } n ( n − 1 ) Δ E [ A m ( n − 1 ) g ] = n E [ C ( n ) m ] − E [ C ( n ) m ] − n E [ C ( n ) m ] + n [ E ( C V i ) ] N ( n − 1 ) n ( n − 1 )

Δ E [ A m ( n − 1 ) g ] = n [ E ( C V i ) ] N ( n − 1 ) − E [ C ( n ) m ] n ( n − 1 ) (22)

Substituting Equation (14) into Equation (22), we obtain

Δ E [ A m ( n − 1 ) g ] = n E ( C V i ) N ( n − 1 ) − E [ C ( n ) − 1 m ] + E ( C V i ) N ( n − 1 ) n ( n − 1 ) Δ E [ A m ( n − 1 ) g ] = n E ( C V i ) N ( n − 1 ) − E ( C V i ) N ( n − 1 ) − E [ C ( n − 1 ) m ] n ( n − 1 ) Δ E [ A m ( n − 1 ) g ] = ( n − 1 ) E ( C V i ) N ( n − 1 ) − E [ C ( n − 1 ) m ] n ( n − 1 ) Δ E [ A m ( n − 1 ) g ] = ( n − 1 ) { E ( C V i ) N ( n − 1 ) − E [ C ( n − 1 ) m ] n − 1 } n ( n − 1 )

Δ E [ A m ( n − 1 ) g ] = E ( C V i ) N ( n − 1 ) − E [ C ( n − 1 ) m ] n − 1 n (23)

Thus, according to [

Δ E [ A ( n − 1 ) ] < 0 < Δ E [ A ( n ) ] (24)

The condition stated in Equation (24) stem from the fact that the function, E [ A ( n ) ] is said to be achieve its minimum value at a point, n, if

Δ E [ A ( n − 1 ) ] < 0 or Δ E [ A ( n ) ] > 0

where Δ E [ A ( n − 1 ) ] = E [ A ( n ) ] − E [ A ( n − 1 ) ] and Δ E [ A ( n ) ] = E [ A ( n + 1 ) ] − E [ A ( n ) ] .

From Equation (24), the condition Δ E [ A m ( n − 1 ) g ] < 0 gives

[ E ( C V i ) ] N ( n − 1 ) − E [ C ( n − 1 ) m ] n − 1 n < 0 [ E ( C V i ) ] N ( n − 1 ) − E [ C ( n − 1 ) m ] n − 1 < 0

[ E ( C V i ) ] N ( n − 1 ) < E [ C ( n − 1 ) m ] n − 1 (25)

Equation (25) states that group replacement should not be made at the end of nth period if the expectation of average cost of individual replacement at the end of (n − 1)th period is not less than the overall expectation of average cost per unit period by the end of (n − 1) periods.

Similarly, from Equation (24) the condition 0 < Δ E [ A m ( n ) g ] gives

0 < E ( C V i ) N ( n ) − E [ C ( n ) m ] n ( n + 1 ) 0 < E ( C V i ) N ( n ) − E [ C ( n ) m ] n E [ C ( n ) m ] n < E ( C V i ) N (n)

⇒ E ( C V i ) N ( n ) > E [ C ( n ) m ] n (26)

Equation (26) states that group replacement should be made at the end of nth period if the expectation of average cost of individual replacement for the nth period is greater than the overall expectation of average cost per unit time period through the end of n periods.

In this section we shall make use of real-life data collected from two different hotels to validate the theoretical results of this work and the results are presented in

In order to apply Equation (5), we need to estimate the probability of failure times, P_{j}. In this regard, we cannot just employ any kind of probability distribution by mere guesses. According to [

The goodness of fit tests measures the compatibility of data with a theoretical probability distribution function. In other words, these tests show how well the distribution we selected fits the research data. For this purpose Kolmogorov-Smirnov (K-S), Anderson-Darling (AD) and Chi-Squared tests may be utilized. In chi-square test, data is grouped and intervals need to be determined to evaluate the goodness-of-fit. This is an important limitation of chi-square test since there are no clear guidelines for selection of the intervals and test results may change depending on the selection of intervals. The Kolmogorov-Smirnov (K-S) and Anderson-Darling (AD) tests on the other hand, do not require

Results | Existing Method [ | Proposed Method | ||
---|---|---|---|---|

Data Sets | Data Set I | Data Set II | Data Set I | Data Set II |

Fitted Probability Distribution for failure times | … | … | Smallest Extreme Value (or Gumbel) with: μ = 34.25888 σ = 34.11878 | Laplace with: θ = 5183.0120 ϕ = 94.2625 |

Fitted Distribution for |Individual Replacement Cost | … | … | Gamma with: α = 13.68094 β = 42.42969 | Largest Extreme Value with: μ = 501.57496 σ = 55.02559 |

Fitted Distribution for Group Replacement Cost | … | … | Lognormal with: μ = 5.76867 σ = 0.16455 | Weibull with: α = 159.14436 β = 1.68840 |

Expected Cost of Replacement | C i = N 700.00 C g = N 400.00 | C i = N 700.00 C g = N 400.00 | E ( C V i ) = N 580.38 E ( C V g ) = N 324.47 | E ( C V i ) = 533.34 E ( C V g ) = 354.76 |

Average Cost of Individual Repl. Policy per period | A ( n ) i = N55,300.00 | A ( n ) i = N 54,600.00 | E [ A ( n ) i ] = N 46,427.20 | E [ A ( n ) i ] = N 41,600.52 |

Average Cost of Group Repl. Policy per period | A ( n ) g = N 57,000.00 | A ( n ) g = N 54,450.00 | E [ A ( n ) g ] = N 49,4538.00 | E [ A ( n ) g ] = N 50,441.00 |

Appropriate time to replace failed LED bulbs | After every 8^{th} period (i.e., after every 39,420 burning hours) | After every 6^{th} period (i.e., after every 30,660 burning hours) | After every 7^{th} period (i.e., after every 35,040 burning hours) | After every 6^{th} period (i.e., after every 30,660 burning hours) |

Expected Life of an LED bulb | 9.1109 hours | 7.5307 hours | 9.03971 hours | 7.53074 hours |

Average No. of replaced bulbs | 79 bulbs | 78 bulbs | 80 bulbs | 78 bulbs |

Average cost of individual replacement per hour | N 12.63 | N 12.47 | N 10.60 | N 9.50 |

grouping of the data or determination of intervals. One of the major limitations of Kolmogorov-Smirnov (K-S) test is that it does not detect the discrepancies at tails very well; however the Anderson-Darling (AD) test is mainly designed to detect the discrepancies in tails [

The Anderson-Darling procedure is a general test to compare the fit of an observed cumulative distribution function to an expected cumulative distribution function. This test gives more weight to the tails than the Kolmogorov-Smirnov test. Anderson-Darling statistic measures how well the data follow a particular distribution. The better the distribution fits the data, the smaller this statistic will be. Further, the Anderson-Darling statistic is used to compare the fit of several distributions to see which one is best or to test whether a sample of data comes from a population with a specified distribution. The hypotheses for the Anderson-Darling test are:

H_{0}: The data follow a specified distribution;

H_{1}: The data do not follow a specified distribution. (27)

The Anderson-Darling statistic (A^{2}) is defined as

A 2 = − n − 1 n ∑ i = 1 n ( 2 i − 1 ) [ ln F ( X i ) + ( 1 − F ( X n + 1 − i ) ) ] (28)

where F ( X i ) is the cumulative distribution function of the specified distribution and X i are the ordered data.

If the p-value for the Anderson-Darling (AD) test is lower than the chosen significance level α , we reject the null hypothesis, H_{o} and conclude that the data do not follow the specified distribution. Alternatively, the hypothesis regarding the distributional form is rejected at the chosen significance level α , if the test statistic, A^{2}, is greater than the critical value obtained from a table. In general, critical values of the Anderson-Darling test statistic depend on the specific distribution being tested.

In goodness-of-fit test, several probability distribution(s) may appear to fit the data well and there would be need to choose the best probability distribution for modeling the failure times. To select the best probability distribution from amongst the fitted distributions, we shall select a distribution with the largest p-value. Among extremely close p-values, we shall select a distribution that has been used previously for a similar data set [

From

Similarly from

A replacement cost is the cost of replacing an item of an organization at the same value. The replacement cost can change, depending on changes in the market value of the item and any other costs required for preparing the item for use.

Our argument in this study stems from the fact that a careful examination of the cost function due to [

Consequently, since individual replacement cost, C_{i} and group replacement cost, C_{g} may not remain fixed over time, it is worthwhile to view them as random variables and then determine their respective probability distributions. Again, it is essential to choose the correct probability distribution for the cost data. [

In line with the process taken by [_{i}) and the expectation of group replacement cost, E(C_{g}). With these expected costs, we then modify [

As

data are lognormal, Gamma, Weibull, Largest Extreme Value and many others. To determine among the aforementioned distributions, the one that best fit the data on individual replacement cost, we conducted the goodness-of-fit test and the results show that the Gamma distribution best fits the individual replacement cost for data set 1.

As shown in

As

Finally, as shown in

From

β = 42.42969. Also, results in

Similarly, from

As shown in

Also, from ^{th} period (i.e., between 35,041 - 39,420 hours) but immediately after the 8^{th} period (i.e., after every 39,420 burning hours), a group replacement is required and the average cost of group replacement would stand at about N 57,000.00. In a similar fashion, ^{th} period (i.e., after every 35,040 burning hours), a group replacement is required with an average group replacement cost of N 49,458.00. Here, the period of replacement obtained using the replacement model by [

From

Finally, as shown in

This work discusses the construction of replacement model for items that fail suddenly. The ultimate objective is to propose a replacement model which may be used to improve the existing replacement models. In this paper, a modified replacement model for items that fail suddenly was proposed following [

In conclusion, the proposed replacement model provides a better model for replacement of items that fail suddenly than the replacement model by [

Based on the results of this work, the following recommendations have been made:

The proposed replacement model should be used in the optimal replacement of items that fail suddenly until further studies prove otherwise.

Hotels, transport companies, filling stations, electrical companies, government agencies and other policy and decision makers are encouraged to make use of the proposed replacement policies for proper planning, policy formulations and implementation, as this will give a more reliable policy in the replacement of various items whose failure is sudden.

Finally, we recommend that researchers should address this kind of replacement problem from a simulation study angle so as to generalize further in this area.

The authors are grateful to the reviewers for their constructive comments and suggestions, which have helped to significantly improve both the content and exposition of this paper.

The authors declare no conflicts of interest regarding the publication of this paper.

Enogwe, S.U., Oruh, B.I. and Ekpenyong, E.J. (2018) A Modified Replacement Model for Items That Fail Suddenly with Variable Replacement Costs. American Journal of Operations Research, 8, 457-473. https://doi.org/10.4236/ajor.2018.86026