**Applied Mathematics**

Vol.07 No.08(2016), Article ID:66696,25 pages

10.4236/am.2016.78071

Minimization of the Expected Total Net Loss in a Stationary Multistate Flow Network System

Kristina Skutlaberg, Bent Natvig

Department of Mathematics, University of Oslo, Oslo, Norway

Copyright © 2016 by authors and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

Received 15 March 2016; accepted 21 May 2016; published 24 May 2016

ABSTRACT

In the present paper, a three-component, stationary, multistate flow network system is studied. Detailed costs and incomes are specified. The aim is to minimize the expected total net loss with respect to the expected times the components spend in each state. This represents a novelty in that we connect the expected component times spent in each state to the minimal total net loss of the system, without first finding the component importance. This is of interest in the design phase where one may tune the components to minimize the expected total net loss. Due to the complex nature of the problem, we first study a simplified version. There the expected times spent in each state are assumed equal for each component. Then a modified version of the full model is presented. The optimization in this model is completed in two steps. First the optimization is carried out for a set of pre-chosen fixed expected life cycle lengths. Then the overall minimum is identified by varying these expectations. Both the simplified and the modified optimization problems are nonlinear. The setup used in this article is such that it can easily be modified to represent other flow network systems and cost functions. The challenge lies in the optimization of real life systems.

**Keywords:**

Reliability, Nonlinear Optimization, Multistate Flow Network

1. Introduction

A series of challenges concerning reliability engineering is presented in [1] . Some of these challenges are connected to the representation and modeling of complex systems, such as multistate systems, and their operational tasks, for instance maintenance optimization.

Over the past decades various measures of component importance have been studied. The use of such measures permits the reliability analyst to prioritize the system components in order to allocate resources efficiently. In [2] a new theory for measures of importance of system components is presented. Generalizations of the Birnbaum, Barlow-Proschan and Natvig measures (see [3] - [5] respectively) from the binary to the multistate case, both for unrepairable and repairable systems are covered. A numerical study of the above mentioned multistate measures of component importance is also covered in [2] . Loss of utility due to the system leaving the different sets of better states are introduced in that study. However, no detailed costs or incomes are specified. Recently, work has been done to also include costs in the determination of component importance for binary systems. In [6] and [7] the Birnbaum measure is extended to also include both failure induced and maintenance costs, while [8] and [9] introduce other cost-effective importance measures.

In maintenance optimization studies one is often interested in choosing a maintenance plan which minimizes life cycle costs, maximizes net present value or maximizes system reliability for a given system. See for instance [10] - [14] for some recent work on these subjects.

In this article we will look at one particular type of maintenance action, the complete repair. As the components reach the complete failure state, they are repaired to what we will denote the perfect functioning state. The aim is to include both costs and incomes in the study of a repairable multistate flow network system. To achieve this, we will define incomes and cost functions for the purpose of minimizing the expected total net loss over a time period with respect to the expected component times in the different states. This represents a novelty in that we connect the expected component times spent in each state to the minimal total net loss of the system, without first finding the component importances.

It would of course have been nice to optimize with respect to probability distributions instead of expectations, but this is not trivial even for a simple three-component system. However, the optimization problem considered in this article is particularly interesting in a design or re-design phase, where one may tune the components in such a way that the expected total net loss is minimized.

With the optimization problem considered in this article we are facing complex dependencies. We therefore study both a simplified version and a modified version of the optimization problem. In the simplified version we see that the optimal expected time spent in each state increases with increasing operational time for all three cost function types considered. However, the extent of the increase differs with the different basic cost function types. Due to basic investment costs this is not a trivial result. In the modified version of the optimization problem we only find approximate solutions. We observe that the different types of cost functions influence the end results significantly. For instance one of the functioning states is redundant for two of the three cost function types when the cost function parameter is increasing. For both problems we see that the minimum expected total net loss is increasing with increasing component cost per repair.

The rest of the article is organized as follows: Section 2 introduces the basic model, the three different types of cost functions and the three-component system of interest. The simplified version of the optimization problem with results is presented in Section 4. Section 5 presents the modified optimization problem with results, and concluding remarks are found in Section 6.

2. Basic Model

Let S be the set of possible system states, and, the set of possible component states. Throughout this article we will assume that. Since we are regarding the system as a flow network, the system state is the amount of flow that can be transported through the network. In the same way, the component state is the amount of flow that can be transported through each component. Let be the vector of component states at time t. That is if component is in state at time t.

A binary minimal cut set is a minimal set of components which upon failure will break the connection between the endpoints of the network. Let, , be the binary minimal cut sets of the network. Then, by applying the max-flow-min-cut theorem (see [15] ), we get that the system state is given by

(1)

Thus, the system state equals the smallest total flow through the minimal cut sets of the system. Assume now that no components are in series with the rest of the system. Then there must be at least two components in every minimal cut set. If all components are in the perfect functioning state, M, the system state will be at least 2M, and therefore we must have that. Thus, the assumption of equality between the set of system states and the set of component states, , implies that at least one component is in series with the rest of the system. For this reason, we will in Sections 4 and 5 focus on the three component system given in Figure 1.

Assume that the components deteriorate by going through all states in, from the perfect functioning state M to the complete failure state 0, before being repaired back to M.

Let be the expected time component spends in state, and let the vector of positive expected component times in each state be denoted.

Assume for that the basic investment costs of component i spending the expected amount of time in state are given by the cost functions for and for. These basic costs appear once in the time interval for each combination of and.

For any given functioning state, it seems natural that these basic expenses grow when the expected times become large. If however, no time is spent in a functioning state, there will not be any basic costs of keeping the component in this state. Similarly, the shorter the expected time spent in the complete failure state, the more expensive it should be. In other words, the faster a complete repair is executed, the more

expensive it should be. Therefore, we assume that the cost function is increasing and the cost function is assumed to be decreasing; moreover, both functions are assumed to be twice differentiable with. Throughout this article we assume the cost functions, and, to be of one of the following types:

Type 1: and

Type 2: and

Type 3: and,

where are constants. These cost functions are constructed by the authors according to the above mentioned criteria to represent a variation in the potential basic cost development.

In this article we only consider perfect repairs. Let denote the cost per repair from the complete failure state to the perfect functioning state of component. The total number of repairs of component in the interval is denoted for.

Let be the fixed income per unit of time when the system is in state, and assume that

This means that the income decreases, starting from the perfect functioning state, from one state to the next until the system reaches the complete failure state, where the income is non-positive. Thus, there is a loss per time unit that the system spends in the complete failure state. Such negative income might correspond to interest rate expenses connected to system building investments. The presence of such costs will increase the incentive for repairing the failed components.

Figure 1. A system with three components.

and for and, and then a modification of the original problem where the

optimization was done in two steps (see Section 5). This method found an approximate solution. The indication of lack of constructive conclusions is mainly due to that we are facing complex dependencies.

The variables and for in the simplified problem, and the variables and for and in the modified full problem, were varied one at a time with three different types of cost functions.

For the simplified problem we were able to find expressions for the optimal for cost functions of type 1. For cost functions of type 1 and 3, the objective function, (8), turned out to be a convex function. With cost functions of type 2 the objective function is neither convex nor concave. The type 2 cost functions are logarithmic, and hence concave while the other two types are convex. For this cost function, we saw in Figure 7 that the optimal has a minimum as increased, which seems unnatural.

In both the simplified problem and the modified full model, the minimum expected net loss was increasing with increasing for every cost function type (as seen in Figure 4, Figure 10 and Figure 11 respectively).

As the operational time T increased we saw a decrease in the minimum expected net loss in the modified full model for all three cost functions (as seen in Figure 8). This is in contrast to the results with the simplified model when the exponential cost functions were used. Then, the minimum expected net loss increased at first, before it started to decrease (as seen in Figure 3).

For every cost function parameter, , we varied, we saw in Figures 17-19 that the optimal was constant for cost functions of type 2. The same observation of constant for cost functions of type 2 was done in Figures 23-25 where were varied. The values were also close to

zero. Hence, it was, for cost functions of type 2, optimal to spend as little time as possible in state 1 independent

of the values of the parameters. With cost functions of type 1 we observed the same, except from when was increasing where decreased from around 15 to close to 0 as increased from 0 to 2. For

stayed constant and close to 0. Thus, it seems like the functioning component state 1 is in a way redundant for cost functions of type 1 and 2. This was not the case with cost functions of type 3.

The general objective function (5) can quite easily be modified to represent the expected net loss of other network flow systems, and to include other types of cost functions. However, with larger systems, with more components and possibly more component states, the optimization problem quickly becomes large. Hence, the real challenge lies in the optimization of real life systems.

Acknowledgements

The authors thank Professor Geir Dahl for the idea on how to modify the full optimization problem and Ph.D Olav Skutlaberg for valuable feedback throughout the process.

Cite this paper

Kristina Skutlaberg,Bent Natvig, (2016) Minimization of the Expected Total Net Loss in a Stationary Multistate Flow Network System. *Applied Mathematics*,**07**,793-817. doi: 10.4236/am.2016.78071

References

- 1. Zio, E. (2009) Reliability Engineering: Old Problems and New Challenges. Reliability Engineering and System Safety, 94, 125-141.

http://dx.doi.org/10.1016/j.ress.2008.06.002 - 2. Natvig, B. (2011) Multistate Systems Reliability Theory with Applications. Wiley, New York.

http://dx.doi.org/10.1002/9780470977088 - 3. Birnbaum, Z.W. (1969) On the Importance of Different Components in a Multicomponent System. In: Krishnaiah, P.R., Ed., Multivariate Analysis—II, Academic Press, Waltham, 581-592.
- 4. Barlow, R.E. and Proschan. F. (1975) Importance of System Components and Fault Tree Events. Stochastic Processes and their Applications, 3, 153-173.

http://dx.doi.org/10.1016/0304-4149(75)90013-7 - 5. Natvig, B. (1979) A Suggestion of a New Measure of Importance of System Components. Stochastic Processes and their Applications, 9, 319-330.

http://dx.doi.org/10.1016/0304-4149(79)90053-x - 6. Gao, X., Barabady, J. and Markeset, T. (2010) Criticality Analysis of a Production Facility Using Cost Importance Measures. International Journal of Systems Assurance Engineering and Management, 1, 17-23.

http://dx.doi.org/10.1007/s13198-010-0002-0 - 7. Wu, S. and Coolen, F. (2013) A Cost-Based Importance Measure for System Components: An Extension of the Birnbaum Importance. European Journal of Operational Research, 225, 189-195.

http://dx.doi.org/10.1016/j.ejor.2012.09.034 - 8. Xu, M., Zhao, W. and Yang, X. (2011) Cost-Related Importance Measure. Proceedings IEEE International Conference on Information and Automation (ICIA), Shenzhen, 6-8 June 2011, 644-649.
- 9. Gupta, S., Bhattacharya, J., Barabady, J. and Kumar, U. (2013) Cost-Effective Importance Measure: A New Approach for Resource Prioritization in a Production Plant. International Journal of Quality and Reliability Management, 30, 379-386.

http://dx.doi.org/10.1108/02656711311308376 - 10. Nourelfath, M., Chatelet, E. and Nahas, N. (2012) Joint Redundancy and Imperfect Preventive Maintenance Optimization For Series—Parallel Multi-State Degraded Systems. Reliability Engineering and System Safety, 103, 51-60.

http://dx.doi.org/10.1016/j.ress.2012.03.004 - 11. Gomes, W.J.S., Beck, A.T. and Haukaas, T. (2013) Optimal Inspection Planning for Onshore Pipelines Subject to External Corrosion. Reliability Engineering and System Safety, 118, 18-27.

http://dx.doi.org/10.1016/j.ress.2013.04.011 - 12. Marais, K.B. (2013) Value Maximizing Maintenance Policies Under General Repair. Reliability Engineering and System Safety, 119, 76-87.

http://dx.doi.org/10.1016/j.ress.2013.05.015 - 13. Doostparast, M., Kolahan, F. and Doostparast, M. (2014) A Reliability-Based Approach to Optimize Preventive Maintenance Scheduling for Coherent Systems. Reliability Engineering and System Safety, 126, 98-106.

http://dx.doi.org/10.1016/j.ress.2014.01.010 - 14. Mendes, A.A., Coit, D.W. and Ribeiro, J.L.D. (2014) Establishment of the Optimal Time Interval between Periodic Inspections for Redundant Systems. Establishment of the Optimal Time Interval between Periodic Inspections for Redundant Systems, 131, 148-165.
- 15. Ford, L.R. and Fulkerson, D.R. (1956) Maximal Flow through a Network. Canadian Journal of Mathematics, 8, 399-404.

http://dx.doi.org/10.4153/cjm-1956-045-5 - 16. Magnus, J.R. and Neudecker, N. (1999) Matrix Differential Calculus with Applications in Statistics and Economics. Wiley, New York.
- 17. Ye, Y. (1987) Interior Algorithms for Linear, Quadratic, and Linearly Constrained Non-Linear Programming. PhD Thesis, Department of ESS, Stanford University, Stanford.
- 18. Ghalanos, A. and Theussl, S. (2012) Rsolnp: General Non-linear Optimization Using Augmented Lagrange Multiplier Method. R Package Version 1.12.