# Technical application for inspection sampling for repairable systems in an economic system

- Emmanuel Hagenimana
^{1}Email authorView ORCID ID profile, - Song Lixin
^{1}Email author and - Patrick Kandege
^{1}

**Received: **13 June 2016

**Accepted: **26 October 2016

**Published: **10 November 2016

## Abstract

In this article we develop a model for determining the appropriate level of inspection sampling for any manufacturing process. The model is useful for manufacturers, who naturally are concerned with profits and therefore with minimizing the cost of production. The model design aims to reduce total manufacturing cost and has general applicability to various manufacturing operations. The model considers the interests of consumers, who wish to minimize the cost of production while simultaneously ensuring the final product is of high quality. The cost parameters for production, the acceptance test, and admissible strategy are applied in the model. The cost components are formulated along with the minimization of the expected cost, and we used the repairable systems to guarantee the maintenance and sustainability of the economic system. We also discuss the assumptions and their appropriateness, as well as the application of the model to burn-in of system components.

## Keywords

## Background

The performance of the economic system is measured and fine tuned using inspection sampling. As discussed in Hamaker (1980), Hamaker (1951), to maintain a desirable level of product performance, the product value is based not only on characteristic designs but also on whether it performs to the associated specifications (Roeloffs 1967; Anscombe 1967). Although the product can be appreciated or accepted by customers according to fixed specifications as indicated above, some organizations use the test to verify the acceptance of their products in different markets.

In the case considered here, consumers and producers reach an agreement on the price of a product before it reaches the market or is delivered. This procedure helps manufacturers and their clients in that the former has the opportunity to produce more goods of a higher quality, while the latter can buy the product safe in the knowledge that it will perform as desired. Different researchers such as Kaplan and Strömberg (2004), Anderhub et al. (2002), Fehr and Gächter (2007) argue that the procedure helps producers invest enough capital in product quality.

Market competition offers advantages to both manufacturers and consumers, and forces manufacturers to communicate with customers who want quality goods at a reasonable price. While producers seek to attract customers to buy their products, customers are looking for quality products at the lowest price. Therefore, market competition benefits consumers through lower prices and improved quality of goods. For more details of how market competition benefits manufacturers and consumers refer to Mills et al. (2016) and Acharya and Lambrecht (2015).

As discussed by Berger and Udell (1998), Loss and Renucci (2012) the global investment economy depends entirely on private domestic funds, something well known to many researchers. Also, Stantcheva (2014) reveals that investments are economically significant, being a delay in wealth consumption as wealth is instead used for the manufacture of other products and for services related to the manufacturing process. Examples of investments include a factory manufacturing construction equipment, a construction business, or any company involved in production.

We often see references in the literature to investment in the organization, yet few researchers mention gross private household investment. The literature on investment discusses the financial investment of cash so as to generate income. However, it is also possible to expand the value of economic expertise or utilize the term investment to characterize all actions related to capital investment that utilizes savings, a range of activity known as financial investment savings. Interested readers can refer to Duncan (1956), van der Waerden (1960) for more details.

In this article, to test some interesting arguments such as those in Hill (1960), Singh (1966), we analyze insurance payments, and consider the acceptance test such that the results also incorporate detailed study of the mechanism of the technics in van der Waerden (1960). From the perspectives of both consumers and manufacturers, the right product will likely depend on the cost of the materials involved in its production. Consequently, test control requirements are very high. The minimum price of the product is constant and dependent on the products effectiveness or efficiency. The exchange of a defective product for a fully functional one is possible only with the payment of insurance.

The producer receives a premium if they can improve the cost of the efficient product. From Anderhub et al. (2002), Murthy and Asgharizadeh (1999) a contract where insurance is paid against the return of a defective product achieves harmony between customer and producer, which implies that the acceptance test is right, and also confirms the good quality of the product.

The study by Acemoglu et al. (2008) reveals that product quality is subjective. That is, the product characteristics that cause users to attribute a high value to a product depend on many factors. Also, the study by Flehinger and Miller (1964) envisages that the different levels of a products quality characteristics should result in various levels of consumer satisfaction.

The procedures involved in examining product quality are also discussed in Flehinger and Miller (1964). This study deals with the situation where the agreement or contract is not satisfied, but process tests are required and associated with the payment of insurance against all test outcomes. Based on the local economic situation, the manufacturer must provide just enough to maximize the product improvement. In this case, all parties benefit from product quality. Everyone must be party to a contract that helps all parties obtain profits easily.

The purpose of this article is to properly understand the relationship between customer and producer with regard to product characteristics, and the role of acceptance inspection in the economic system, something also dealt with in Acemoglu and Verdier (2000), Koch and Peyrache (2011). To achieve these objectives, we use the combination of principles and methods used in the acceptance inspection system. To improve our results, we also consider the implications of a repairable system, and so obtain a good understanding of the intervals of the product lifetime and product repair time under such a system. In this case, the customer and producer guarantee the quality and price of the goods.

The remainder of this article is organized as follows: In “A note on assumptions” section, we consider some assumptions that allow us to contribute to the main proofs provided in “Main proofs” section. In “Repairable system introduction” section, we introduce the implications of a repairable system, specifically in relation to how such a system can serve as a useful aid in model construction. In “Main proofs” section, we provide the main proofs that show how the set of admissible strategies becomes exhausted for decreasing. In “Numerical examples for clarification” section, we consider an example that clarifies aspects of our paper. In “Application to burn-in” section we discuss the application of burn-in. Finally in “Conclusion” section, we present conclusions and acknowledgments.

## A note on assumptions

In this article, we use the following assumptions in solving the problem below

\(H_0\): Let \(\mu\) be the composite of vectors \(\mu _1, \mu _2, \ldots ., \mu _n\), that describes the *n* product parameters that collectively are known as product quality. Also, we assume that for the system state, the product parameters determine the rate of failure and the mean interval of repair. The system state increases the number of components. This implies that the proportions of the product parameters are wrong. That the proportions are wrong is supported by the acceptance test, showing that the value of the product parameters is assured.

\(H_1\): We consider the function \(\Theta (\mu )\) of the given parameters \(\mu _i\). This function represents the motivation of the customer as the given functions increase monotonically.

\(H_2\): The fundamental parameter vectors \(\mu _0 ={\mu _{10}, \mu _{20}, \ldots ,\mu _{n0}}\) characterize the improvement of product quality in the case where the producer has not taken it into consideration. Therefore, by considering \(\Phi (\mu )\) which is measured as a cost continuous function for the given parameters \(\mu _i < \mu _{i0}\).

Because \(\Phi (\mu _0)=0,\) the product improvement can help us to decrease or minimize the values of \(\mu _i\) through \(\mu _{i0}\). This implies that \(\Theta (\mu )\) and \(\Phi (\mu )\) are assigned to customers and producers.

\(H_4\): It is also assumed that a test of cost outcomes depends on testing procedure. Thus, this originates from the consumer or producer or both simultaneously. We also let \(\xi _\beta (\tau )\) and \(\xi _\gamma (\tau )\) denote the producer and consumer shares associated with the parameters, respectively.

To maximize profit, we assume that any given strategy \(\Upsilon _\beta (\tau )\) must have a maximum strategy \(\mu _i\le \mu _{i0}\); we achieve this by increasing the capital. It is known that the investment payment and the producer are tested enough to calculate the product of the given parameters \(\mu _1^\star ,\ldots \ldots ..,\mu _n^\star\) that have been formulated and described as the function of the parameter strategy.

From this, we cannot find any investment strategy where there exists a unique application point in \((\Upsilon _\beta ^{\prime},\Upsilon _\beta ^\star )\) such that \(\Upsilon _\beta ^{\prime} >\Upsilon _\beta ^\star\) and \(\Upsilon _\gamma ^{\prime}\star >\Upsilon _\gamma ^\star\) or \(\Upsilon _\gamma ^{\prime}\ge \Upsilon _\gamma ^\star\) and \(\Upsilon _\beta ^{\prime}\ge \Upsilon _\beta ^\star\).

\(G_1\): It is also assumed that the term \(\Upsilon _\beta (\mu ,\tau ,\delta )\) is maximized for \(\mu ^\star\) where \(\mu _i\le \mu _{i0}\).

\(G_2\): It is proved that for the above terms \(\Upsilon _\beta (\mu ^\star ,\tau ,\delta )>0\) and \(\Upsilon _\gamma (\mu ^\star ,\tau ,\delta )>0\), respectively.

## Repairable system introduction

Most maintenance models consider comprehensive support where a system becomes as good as new after each maintenance action, as detailed in Duncan (1956), Endrenyi et al. (1998), Hagenimana et al. (2016). However, in reality, system performance deteriorates over time, which is why we investigate the performance of a system that is subject to imperfect repair, something also discussed in Scarf (1997). We present two cases, namely maintenance by repair and replacement, and maintenance by probabilistic repair and replacement. The objective is to assess the systems long-term behavior by deriving clues related to the expressions of its operational probability behavior.

In our case, the repairable system is applied to the economic system. To guarantee the systems sustainability, we use mechanical components or processing equipment such that experiments can be performed and the computational results given by financial parameters. For further details refer to Percy and Kobbacy (2000), Kobbacy and Murthy (2008).

Based on this, we establish some assumptions of a repairable system that are helpful in our proofs.

\(M_0\): We assume that the distribution of mean \(\mu \tau\) represents the number of equipment or system component failures that occur during time \(\tau\). In this case, we consider only the lifetime of the system and neglect the repair time and \(\mu\) gives the essential characteristics of the product material.

\(M_2\): We assume that the number of systems failures at any given time interval \(\tau\) is 0 and that the system lifetime includes the procedures involved in the acceptance test. We also consider that the number of tests is represented by \(\alpha _i\) as outcomes of a trial and that (*i*) denotes the number of failures. Finally, we assume that \(\delta _i\) gives the insurance payment for all *i*.

\(M_3\): We expect that the delay time to the deficiency follow the same probability density function denoted \(\gamma (\eta )\) with cumulative distribution function indicated as \(\Gamma (\eta )\)

\(M_4\): Correction of repairs at failure are taken to be minimal repairs which bring the material equipment to become as good as before.

\(M_6\): The test interval distance is directly proportional to test cost, which is determined by the customer and the test procedures.

Considering the previous assumptions, the necessary and sufficient conditions for the acceptable procedures are summarized here. Accordingly, we have two conditions as follows:

\(C_1\): \(\delta _0=\delta {'}, \delta _i=0\), where \(1\le i\)

\(C_2\): \(\delta _0=\delta _0^{\prime}, \delta _1=\delta _1^{\prime}\), \(\delta _i=0\), where \(2\le i\)

This implies that the insurance payment is treated using the assumptions of \(C_1\). This case is used to show that there is no failure of the given interval test, and the case based on the assumptions of \(C_2\) is also used, thus satisfying the small insurance payment and failure in the range test.

## Main proofs

Besides this, as the maximum value \(\tau ^{\prime}\) is expressed by the Eqs. (30) or (32) as given above, it shows that all admissible strategies lie in assumption \((C_1)\), which has a greater value than the maximum given under condition \((C_1)\) because \(\tau >\tau _2^{\prime}\). Furthermore, we analyze the strategy points given in assumption \((C_2)\) determined by the following

## Numerical examples for clarification

###
*Example 1*

###
*Example 2*

By this example, we consider the model that expanded early and evaluated the validity of the product materials. From this model notation, we have a significant number of cost and downtime parameters which required to be carefully taken into consideration. In this example we use three kinds of production options which help us to analyse and understand the process of the inspection sampling such as: In product option one, the producer takes out controls and repairs to correct the deficiency product materials recognized at the inspection of the respectively reasonable interval of time. The manufacturer accomplishes correction of repair to the failure up to the end of the service based on the agreement of the period.

The customer compensates cost and needs to meet a particular level of reliability and availability of the product materials. Then if the price is fixed, producer accomplishes failure of the repairs and then inspection to the system where he corrects all types of failures and deficiency found over agreement of the period without considering the extra cost to the customer. Since failure is not amended in agreement time, the producer includes punishment which gives guarantee to the producer that the client has all right to use the product materials for too long.

In the product option 2, the customer takes out inspections and repairs to the deficiency product content noticed at time of inspections on the respectively reasonable interval of time.

The manufacturer accomplishes correction of repair to failure up to the end of the service due to the agreement of deadline. Here as failure takes place, the customer addresses the problem of the producer to repair the failed product item. The producer request for payment of the debt amount for all repairs without considering extra parts and act of punishment related to the term and conditions for product item remains in failed specification time.

Hence the customer thinks reasonable to get the same level to the reliability and availability of product materials respectively.

In product option three the customer executes correction to the repairs of failures throughout of agreement period to the inspections where repair at the time of inspections of product materials at a reasonable interval of time are taken out by the producer.

Here customers consider reasonably to get the required level to the reliability and availability which is preserved. Based on the introduction to these three product options we proceed as follows,

The relation exists between some of the cost and downtime parameters must be reasonably assigned. We assume that: The extra part to the average cost is taken as \(\$ 500,\) and the working time per person cost of the repair staff to the customer is \(\$ 80\) per working day. The working hours per individual cost for the repair crew to the producer is \(\$ 120\) per day per parson. We suppose that the failure downtime is taken as four times of inspection downtime per repair of the deficiency considered at an inspection is given as \(\frac{1}{20}\)th of the failure repair downtime.

The additional is made according to the inspection sampling of repair to the product materials arranged. Consequently, there are some persons and extra parts who are prepared for the work where the repair crew consists of the individuals for both producer and customers.

For the option two and three, the manufacturer only replaces the workforce cost with a profit margin (20% of the employment cost) since the customer compensates for the extra part cost. Based on the different background and efficiency, the customers pass more time on inspection and repairs materials.

We suppose that the downtime caused by the inspections and repair done under the control of the customers is three times compared to that one observed in product option one. Therefore for the product option three, the downtime for failure repair by the client to that one done by the producer is similar to that one observed in product option two.

*C*is known as to the production cost of the equipment materials.

From the above relation (41) producer or customer have the required information, in such that the other party has to decide the offer done based on the working time per individual on their availability.

The basic information is the estimated value of the arrival of deficiency product materials to the delay time distribution work and its parameters, and also different downtime and cost information are also considered.

Assume that the producer has the whole information and attempts all options as the following.

Product of option one finishing maintenance service with the cost of \(\$ 50.00\) to the agreement period.

Product of option two, the failure is based only on maintenance and cost of \(\$ 864\) per failure without the extra part’s cost.

And hence, product in option three, inspection plus repair have cost of \(\$ 216\) per inspection and \(\$ 43\) for deficiency product materials amended during the inspection without the extra part’s cost.

However, the inspection must be accomplished within 10 days of interval time. It was noticed that, if the customers were able to evaluate the best option for them, they should also maintain and control all information which is an obstacle to them while they only have had limited information.

Therefore customers are obliged to guess the estimated rate to the arrival of deficiency product materials. Let the parameter to the delay time distribution be assumed as exponential distribution based on the given above information:

The availability and reliability are required and then producer propose the inspection at interval time and then guess the values of the downtime information based on the cost charged to customer under the given product of option 2 and three respectively.

Assume that customer has found the workforce required to accomplish work. The present employer rate to the producer together with the market profit margin, and he may compute the down times evaluated by manufacturer using the relation above in (41). And then if down times have evaluated, it is easy to get the estimated value of \(\alpha\) and \(\lambda\) by also using expressions above in 6 due to the availability and reliability levels to the producer proposed in inspection sampling related to the time interval.

The computation of \(\sigma\) and \(\lambda\) are not taken to be exact while the significance value employed by the producer on the required availability and reliability levels. Therefore, the customer could estimate downtime, which is larger that producer’s ones. Assume that customer his determined to producer’s downtime by using the above relation given in (41) times three times to obtain his assessment values and then parameters \(\lambda\) and \(\sigma\) are founded.

Using the given above relation in (4) \(E(\Lambda _{\gamma }(10))\le 0.8,\) is closed related to \(0.8\le e^{-E(\Lambda _{\gamma }(10))}\) which has an estimated value of \(E(\Lambda _{\gamma }(10))\le 0.222\).

Therefore by the given information to the expected downtime, the reliability level required with helped by the expression given in (6). We have the assessed value of \(E(\Lambda _{\gamma }(10))\) that is \(E(\Lambda _{\varsigma }(10))\le 0.531\).

The condition that \(\lambda \le 0.05,\) motivates customers to run the same model due to those two evaluated parameters. And conclude that the given product materials in option one are taken as the best to be considered on their side while the producer on his side there is no profit, only loss without even considering the inspection interval time used in his inspection sampling process.

We noticed that it is totally impossible to understand how producer can get profit which is similar to the chosen from three products options.

Therefore to obtain this, the customers have to repeat the identical model with essay and error, for getting the exact combination to the \(\sigma\) with \(\lambda\). It can supply logical answer which must be closed related to the producer’s estimation of \(\sigma\) and \(\lambda\) where they have been estimated as 0.01 and 0.01 respectively.

From the estimated parameters designed above, the customers are now able to run their model and emphasize to the inspection interval time needs to assess the agreement accepted by the product option one. Therefore the producer on his side is a pleasure to agree on the contract since he is only marginally worse off but still evaluated as better than that of product option two and three respectively.

For more details regarding numerical example of this model see Golmakani and Moakedi (2012, 2013), Murthy and Asgharizadeh (1999), Zhao et al. (2012)

## Application to burn-in

*b*in this section results from the selection criterion. Assume that cost is not a consideration and we simply want to maximize mean life. Consequently, we want to determine

*b*such that the mean residual life is minimized because only those items that survive with a fixed burn-in time are replaced by the given services, which are established in Watson and Wells (1961). Consequently, we want to find \(b^{\star }\) such that the following statement is true,

*F*has a bathtub-shape curve \(h(\tau )\) in the following situations:

- 1.
\(\tau _{1}=0,\) in this case, there is little or no need to burn-in, and it follows that \(b^{\star }=0\). This statement is always taken as true.

- 2.
\(\tau _{2}=\infty\) and \(\tau _{1}>0,\) in this situation, we are always allowed to choose \(b^{\star }=\tau _{1}\).

- 3.
\(\tau _{1}=\tau _{2}=\infty ,\)

*F*is known as the decreasing failure rate function, and in this case the cost should be taken into consideration. - 4.
\(0<\tau _{1}\le \tau _{2}<\infty ,\) Thus, the value of \(b^{\star }\) is equivalent to the unique change point \(\tau ^{\star }\) of \(\mu (\tau )\).

Accordingly, we do not need to burn-in products for a long time because of the first change point \(t_{1}\) where the failure rate function *F* is decreasing.

Another application is motivated by Mi (1995). Let us consider the cost component that has a lifetime \(X_{1}\) with a cumulative distribution function. Suppose that burn-in for this kind of component occurs at a given time *b*. Further suppose that the component survives the burn-in. In this case the component is allowed to pass into field operations, in contrast to the assumption above. A new component with lifetime \(X_{2}\) (i.i.d) as \(X_{1}\) is taken or considered for the field operation. Our target now is to find the optimal burn-in time by minimizing the mean life of the cost components, which are finally used in field operation after a long delay.

Admissible strategies \(\xi _f=300,L=10000,\xi =35000,\mu _0=0.1, \xi _\gamma =200,\xi _\beta =0\)

\(\tau\) | \(\delta\) | \(\Upsilon _\beta ^\star\) | \(\Upsilon _\beta (\mu _0)\) | \(\Upsilon _\gamma ^\star\) | \(\Upsilon _\beta ^\star +\Upsilon _\gamma ^\star\) |
---|---|---|---|---|---|

11.42 | 300,100 | 187,500 | 95,800 | 0 | 187,500 |

15 | 238,200 | 124,800 | 53,100 | 62,000 | 186,800 |

17.83 | 207,200 | 93,100 | 34,800 | 93,100 | 186,200 |

20 | 189,400 | 74,800 | 25,600 | 111,000 | 185,800 |

25 | 160,600 | 44,800 | 13,200 | 140,000 | 184,800 |

30 | 141,900 | 24,800 | 7100 | 159,000 | 183,800 |

35 | 128,900 | 10,500 | 3900 | 172,300 | 182,800 |

38.57 | 122,000 | 2600 | 2600 | 179,500 | 182,100 |

## Conclusion

In this paper, we developed a model based on the cost of producing an effective or defective item, our aim being to minimize the production cost of that item. Minimization of production cost thus is the primary objective of this article. By associating the probabilities of generating effective or defective items with the cost of production, we can obtain the real total cost of production. The economic design of a single sampling attribute inspection plan is the purpose of the development of the cost model presented in this article. It is easy for an individual to adjust to problem solving because all the cost components are integrated into the design model. Trial and error can obtain the acceptance sampling plan that results from the lowest cost. It is then necessary to know the distribution of the entire process; we use the binomial distribution for the whole process of determining defective items along with arbitrary cost data. We observed that it is impossible to find other admissible strategies similar to the previous one, which proves the uniqueness of problem-solving strategies by using the implications of repairable systems. In this case, we neglect the repair time by considering only the lifetime of repairable systems. The numerical application results show that the test was performed in the range 17.83 over a 60-min period during which insurance was purchased at a cost of \(\pounds 207200\), and the profit expected to be available for equal sharing by the customers and the manufacturer was \(\pounds 13100\) for each failure. In this article we also discussed the burn-in as an application. Furthermore, inspection is necessary to safeguard system components against destruction. Future research will investigate different maintenance conditions.

## Declarations

### Authors’ contributions

All authors contributed equally and significantly to this work. All authors read and approved the final manuscript.

### Acknowledgements

SL Conceived, designed and guided to the development of the study of the article; EH reviewed literature, wrote proofs, drafted the manuscript and made revisions; PK made a substantial contribution to the structural evolution of development of the article, discussion of results and changes. All authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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