EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 10, No. 4, 2017, 877-889 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Optimal reliability allocation for redundancy series-parallel systems Saad Abbas Abed1, Constantin Udriste1,∗, Ionel Tevy1 1 Department of Mathematics and Informatics, Faculty of Applied Sciences University Politehnica of Bucharest, Romania Abstract. This paper examines the optimal reliability approaches to allocate the reliability values based on minimization of the total cost for a series-parallel systems. The problem is approached as a nonlinear programming problem and general costs formulas were suggested. The original results include: (i) submersion of a ”series-parallel system” into a ”series system”, (ii) detailed analyse of a series-parallel system whose components of each subsystem have the same reliability; (iii) designing series-parallel systems by similarities with other engineering problems; (iv) dualities between reliability systems and electric circuits. 2010 Mathematics Subject Classifications: 90B25, 90C26, 90C46. Key Words and Phrases: Reliability Allocation, Optimality Conditions, Duality, Similarities 1. Introduction Redundancy is the provision of alternative means or parallel paths in a system for accomplishing a given task such that all means must fail before causing a system failure. System reliability and mean life can be increased by additional means by applying redun- dancy at various levels. The problem of reliability allocation and optimization has been widely treated by many authors. Although most of the attention to this issue has been given to the redundancy allocation problem [9],[10], a different approach to the problem is taken in this paper. A series-parallel system can be improved by four methods (Wang, 1992): (1) use more reliable components; (2) increase redundant components in parallel; (3) utilize both (1) and (2); and (4) enable repeatedly the allocation of entire system framework. Prasad and Kuo (2000) pointed out that Misra algorithm sometimes cannot yield an optimal solution, and suggested a method of searching for the upper limit of reliability’s objective function. The total cost of a system can be minimized subject to the resource constraints to determine the optimum number of redundant components for each ∗Corresponding author. Email addresses: saadaaa2013@yahoo.com (S. A. Abed), udriste@mathem.pub.ro (C. Udriste), vasca@yahoo.fr (I. Tevy) http://www.ejpam.com 877 c© 2017 EJPAM All rights reserved. S. A. Abed , C. Udriste, Ionel Tevy / Eur. J. Pure Appl. Math, 10 (4) (2017), 877-889 878 stage, when the reliability of each component is known. In other situations, the reliabil- ity of the system can be maximized subject to the resource constraint to determine the reliability of the components in the system when the number of redundant units in each stage is known. As a result, addition of redundant components or increase components reliability leads to the increase of the system reliability. The main objective of the reliability optimization is maximizing the reliability cost ratio, developing, mostly, two directions, as follows: (1) minimizing the system cost, but insuring a minimal reliability level; (2) maximizing the system reliability with respect of some costs constraints. The redundancy allocation problem has previously been analyzed for many different system structures, objective functions and time-to-failure distributions. Generally, the problem domain has been limited to series-parallel systems with active redundancy or ’k- out-of-n’ systems consisting of a single subsystem. A number of studies have examined such problems [1], [4]. The parameters of the proposed cost function can be altered, allowing the mathematicians/engineers to investigate different allocation scenarios. Thereafter, designers can decide and plan on how to achieve the assigned minimum required reliabilities for each of the components. 2. Optimization of series-parallel system Until recently, for the purpose of modelling equilibrium flow reliability systems, it has often been assumed that all the particular cases must be solved separately. Now we underline that there are inter-related systems governed by a unique set of equilibrium criteria. Hypothesis: components have two states: working or failed; the reliability of each component is known and is deterministic; failure of individual components are indepen- dent; failed components do not damage other components or the system, and the compo- nents are not repaired. The theory in this Section can be applied to any simple or complex systems, and any number of redundant components can be added to the system to have maximum possible improvement. 2.1. From ”series-parallel system” to ”series system” To a series system of n components characterized by reliabilities 0 ≤ Ri ≤ 1, i = 1, ..., n, and by the admissible level of reliability of the whole system RG, we associate the program P: Find min R C(R1, ..., Rn) subject to Rs = n∏ i=1 Ri ≥ RG, 0 < Ri ≤ 1, i = 1, 2, ..., n. S. A. Abed , C. Udriste, Ionel Tevy / Eur. J. Pure Appl. Math, 10 (4) (2017), 877-889 879 Such kinds of programs were studied and solved in [14] and [15]. In order to formulate an optimal problem associated to a series-parallel system, we need additional notations: n number of subsystems; ki number of different types of avail- able components for the i-th subsystem, i = 1, ..., n; 0 ≤ rij ≤ 1 reliability of the j-th component for the i-th subsystem, i = 1, ..., n, j = 1, ..., ki; RG admissible level of relia- bility of the whole system; xij ∈ N number of j components used in the i-th subsystem, i = 1, ..., n, j = 1, ..., ki. Double-indexed numbers do not form matrices as ”lines” would not have the same number of elements; they will be arranged like vectors. The associated program is p: Find min r C(r11, ..., r1k1 , ..., rn1, ..., rnkn), r = (r11, ..., r1k1 , ..., rn1, ..., rnkn), subject to Rs = n∏ i=1 Ri = n∏ i=1 [ 1− ki∏ j=1 (1− rij)xij ] ≥ RG, 0 < rij ≤ 1, i = 1, 2, ..., n; j = 1, 2, ..., ki. Our goal is to determine the mutual relations between the program p and the program P . Passing from a series-parallel system to a series system is important when we need to decide if a given series-parallel system is optimal or not. Theorem 1. The image of the program p via a submersion R is the program P . Proof. The function R of components Ri = 1− ∏ki j=1(1−rij)xij , i = 1, 2, ..., n, changes the program p into the program P . It is a submersion from Rk1+...+kn to Rn since dRi Ri − 1 = ki∑ j=1 xij drij rij − 1 and hence the differential is everywhere surjective. Also R([0, 1]k1+....+kn) = [0, 1]n. Moving from p to P is possible only when for C there exists a cost C : [0, 1]n ⊂ Rn → R such that the following costs diagram to be commutative: [0, 1]n ⊂ Rn C−→ R R↖ ↗C = C ◦R [0, 1]k1+...+kn ⊂ Rk1+...+kn Remark 1. A submersion locally looks like a projection Rn × Rm−n → Rn, while an immersion locally looks like an inclusion Rm → Rm × Rn−m. Corollary 1. To each optimal program P there corresponds an infinity of optimal programs p. Additional assumptions lead to more accurate design on subsystems. S. A. Abed , C. Udriste, Ionel Tevy / Eur. J. Pure Appl. Math, 10 (4) (2017), 877-889 880 2.2. Example of series-parallel system with additional assumptions Let us analyse a series-parallel system with additional assumption that, in each sub- system, all components have the same reliability. We use the notations: n number of subsystems; 0 ≤ ri ≤ 1 is the reliability of each component in subsystem i; 0 ≤ Ri ≤ 1 is the reliability of subsystem i; Ci(Ri) is the cost of each subsystem i; xi = number of components in stage i; C(r1, ..., rn) = C ◦R(r1, ..., rn) =∑n i=1 aiCi(Ri) is the total system cost, where ai > 0. In general the functionality of each subsystems can be unique, however there can be several choices for, many of the subsystems providing the same functionality, but differently reliability levels. The objec- tive is to allocate reliability to all or some of the components of that system, in order to meet that goal with a minimum cost. An important problem P is formulated as a non- linear programming problem, with additively decomposable cost function and a nonlinear constraint: P: Find min r C(r1, ..., rn) = n∑ i=1 aiCi(Ri), ai > 0, r = (r1, ..., rn), subject to Rs = n∏ i=1 Ri = n∏ i=1 [ 1− (1− ri)xi ] ≥ RG, 0 < ri ≤ 1, i = 1, 2, ..., n, where Rs is the system reliability; RG is the system reliability goal and Ri = 1−(1−ri)xi . The foregoing formulation is designed to achieve a minimum total system cost, subject to RG, a lower limit on the system reliability. The diffeomorphism Ri = 1 − (1 − ri)xi , i = 1, ..., n, changes a program p into a program P , and p will have unique solution. The inverse is ri = 1− (1−Ri) 1 xi . Remark 2. The Kuhn-Tucker necessary conditions for previous program P were discussed in [13] and [14]. 3. Designing series-parallel systems by similarities 3.1. Similar to waste treatment plant The problem of design a reliable system is similar to the design of an industrial waste treatment plant. Indeed the plant design incorporates several treatment processes in series (see, [3], pp. 494-495). The cost formula for the i-th components has the form Ci = ci[1− (1− ri)xi ]ai , i = 1, ..., n, S. A. Abed , C. Udriste, Ionel Tevy / Eur. J. Pure Appl. Math, 10 (4) (2017), 877-889 881 where Ci is the total annual cost, ci > 0, ai is a fixed negative exponent, ri is reliability of i-th component. For a design problem involving n components and r = (r1, ..., rn), the minimization problem becomes min r n∑ i=1 ci [ 1− (1− ri)xi ]ai subject to Rs = n∏ i=1 Ri = n∏ i=1 [ 1− (1− ri)xi ] ≥ RG, 0 < ri ≤ 1, i = 1, 2, ..., n. The positivity constraints ri > 0 will be satisfied at a minimizing point because of the inverse relationship between process costs and the variables ri, that is, as ri approaches zero, the corresponding cost term ci[1− (1− ri)xi ]ai approaches positive infinity. The diffeomorphism Ri = 1− (1− ri)xi changes the previous program into a geometric program. 3.2. Similar to transmission compressor design The problem of design a reliable system is similar to transmission compressor design. Indeed a classical problem in compressor design is finding the interstage pressures for an adiabatic reversible compression of an ideal gas (see, [3], pp. 426-432; [11], pp. 180-181). We seek to minimize the energy consumption of an (n+ 1)-stage system whose work is E(R1, . . . , Rn) = c [( R1 c1 )α + ( R2 R1 )α + . . .+ ( Rn Rn−1 )α + ( c2 Rn )α] , where: c1 = inlet reliability, c2=outlet reliability, α = adiabatic index (the surrounding do not influence the reliability). This free geometric program has zero degree of difficulty. The dual program is max ( c c−α1 δ1 )δ1 ( c δ2 )δ2 · · · ( c δn )δn ( c cα2 δn+1 )δn+1 , with the constraints n+1∑ i=1 δi = 1, δ1 = δ2, ..., δn = δn+1. It follows δ1 = ... = δn+1 = 1 n+1 and the searched minimum is (n + 1)c ( c2 c1 ) α n+1 . Since, at optimality, all dual variables are equal, the stages contribute equally to the minimizing energy policy, and all subsystem ratios must be equal. The minimum point is the solution of the system c ( R1 c1 )α = . . . = c ( c2 Rn )α = c ( c2 c1 ) α n+1 , S. A. Abed , C. Udriste, Ionel Tevy / Eur. J. Pure Appl. Math, 10 (4) (2017), 877-889 882 i.e., Ri = c i n+1 2 c 1− i n+1 1 , i = 1, ..., n+ 1. Remark 3. Let us point two interesting problems: max R E(R1, . . . , Rn) subject to n∏ i=1 Ri ≥ RG, 0 < Ri ≤ 1 and max R E(R1, . . . , Rn) subject to H(R) = − n∑ i=1 Ri log2Ri ≥ H0, 0 < Ri < 1. 3.3. Similar to statistics metric In statistics there exists a wide variety of metrics such as median, standard deviation, arithmetic mean, power mean, geometric mean and many others. In many important problems, we seek to minimize the geometric mean( n∏ i=1 Ri ) 1 n = n √ R1R2 · · ·Rn. subject to some restrictions. 3.4. Symmetric polynomials of reliabilities Let R = (R1, ..., Rn), with 0 ≤ Ri ≤ 1. We build the symmetric polynomials s1(R) =∑n i=1Ri, s2(R) = ∑ i0 1 I = min n>0 ( R E n−1 + r NE n ) . This is a posynomial geometric program with the solution n = √ N R r and Imax = E 2 √ N Rr . If n is supposed to be a natural number, then the geometric program should be solved in steps. Dictionary To pass to the reliability domain, we use E = eE , I = eI , r → − lnR, R→ − lnR1. It follows − ln I = ln ENn lnRN1 Rn 2 = logRN1 Rn 2ENn, i.e., I = exp ( − ln ENn lnRN1 Rn 2 ) . Dual example of reliability batteries with minimum intensity Consider a reliability subsystem with N components, with the same reliability r. The components are grouped many n in series, and the series in parallel. To the total system we attach a subsystem, consisting of a single element with reliability R, connected in series. The total reliability is Rs = [1− (1− rn)N/n]R. We use a sequence of arrows based on previous commutative diagrams, Rs = [1− (1− rn)N/n]R → − ln[1− (1− rn)N/n]− lnR → − ln 1 (1− rn)N/n − lnR → n N ln(1− rn)− lnR → n N ln 1 rn − lnR → −n 2 N ln r − lnR → n2 N r +R = nE I . The analog ”reliability emf” of the battery is E. To close the loop, we consider that the analog ”reliability intensity” is I = c(r;n)(1−Rs) = exp ( − ln ENn lnRN1 Rn 2 ) . The function n→ I(n) has minimum for n = √ N lnR1 lnR . Open problems (i) Clarify and extend the ideas in this Section. (ii) Extend the previous similarity to magnetic reluctance or general circuits. REFERENCES 888 5. Generated algebraic structures The commutative monoid ([0, 1], S) : aSb = ab is isomorphic to the commutative monoid ([0, 1], P ) : aPb = 1− (1− a)(1− b) via an isomorphism f . For example, f(x) = 1 − x (involution). The general isomorphism is of the form f(x) = h(1 − h−1(x)), where h : [0, 1]→ [0, 1] is an arbitrary bijection. The commutative monoid ([0,+∞], s) : AsB = A + B is isomorphic to the commu- tative monoid ([0,+∞], p) : ApB = 1 1 A + 1 B via an isomorphism g. As example g(y) = 1 y (involution). Generally, the isomorphism is of the form g(y) = `(1/`−1(y)), where ` : [0,+∞]→ [0,+∞] is an arbitrary bijection. The commutative monoid ([0, 1], S) is isomorphic to the commutative monoid ([0,+∞], s) by the isomorphism ϕ(x) = − lnx. Then ([0, 1], P ) is isomorphic to ([0,+∞], p) by the same function ϕ(x) = − lnx if and only if g(− ln f−1(x)) = − lnx. Open question Does exist a bijective and decreasing morphism ϕ : [0, 1]→ [0,+∞] such that ϕ(aSb) = ϕ(a)sϕ(b), i.e., ϕ(ab) = ϕ(a) + ϕ(b); ϕ(aPb) = ϕ(a)pϕ(b), i.e., ϕ(a+ b− ab) = ϕ(a)ϕ(b) ϕ(a)+ϕ(b) ? 6. Conclusions In Sections 2-3 were examined some reliability optimization problems: (1) optimiza- tion of series-parallel systems, (2) optimization of series-parallel systems, with additional assumptions, (3) designing series-parallel systems by similarities, (4) minimizing the ge- ometric mean with significative constraints. Section 4 rises and solves the problem of mathematical relationship between total resistance of an electrical circuit and total relia- bility of a reliability system. As consequence, there are introduced ”reliability batteries” with minimum intensity. The fundamental characteristic of our techniques is that similar techniques can be applied for simple and complex reliability systems. References [1] K. K. Aggarwal and J. S. Gupta. 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