SUPPLEMENTARY RESULTS IN THE COMPARISON OF VARIOUS MAINTENANCE STRATEGIES FOR DETERIORATING SYSTEMS C. Teresa Lam and R. H. Yeh Department of Industrial and Operations Engineering The University of Michigan Ann Arbor, MI 48109-2117 Technical Report No. 92-33 June 1992

Supplementary Results in the Comparison of Various Maintenance Strategies for Deteriorating Systems C. Teresa Lam* and R. H. Yeh* The University of Michigan, Ann Arbor The purpose of this report is to provide detailed proofs for some of the results in the comparison of various maintenance strategies for deteriorating systems. The readers should also consult Technical Reports 92-22 and 92-31. 1. EQUIVALENCY OF OBJECTIVE FUNCTIONS Lemma 1.1. When Xg(0) > 0 for all 6 E As, finding a policy 6 e E A such that =- inf Y() =Y(0 ) (0) ^A^ ^W Y8/n\(1) 6EAe Xo(0) X66~(0) is equivalent to finding a g E JR+ and a policy 65 E As such that We(ge) inf [Y (0) - gX(0)] = Y:(0) - g (0) = 0. (2) -O -^ goX(o) - o. (2) Proof. Suppose that Equation (1) holds, then (0) inf Y (0) - for all X(0) - 6EA X for all 6. e X(0) >0e he X )- X~(0) When X (0) > 0, we have Y(0) - gX() > Y/.(0)-gX (0) = 0 for all 6 E A9, and the equality holds if 6 = 6. Therefore, Equation (2) holds. Conversely, if Equation (2) holds, then Y~(O) - gX~(O) > inf [Y~(0) - gX~(0)] = YA(0) - gSX (0) = 0 'Research Supported in part by a grant from the Research Partnership Program, Horace H. Rackham School of Graduate Studies, University of Michigan. 1

for all 6 E Ao. When X6(0) > 0, it follows that g6 = X(0) < for all 6 AO. Hence, 6-/ X6^ (0) -- x 0(0) Equation (1) holds. D Lemma 1.2. Wo(g) is a continuous and nonincreasing function of g. Proof. Let gl < g2 and 61,62 E AO such that Wo(gl) = inf [Y(0) - gliX (o)] = Y (O) - g x1 (0) < Ys(O) - gl X(0 ) 6EA2 We(92) = inf [Y8(0) - g2X(0)] = Y (0) - g2X~ (0) < Y6 (0) - g2X1 (0) Since X,(0) > 0 for all 6 E A9, the following inequalities hold. WO(gi) - W(g92) = [(o) - 7iX|1(0)] - [Y82(0) - 92X2(0)] > [Y6(o)- gix6(o)] - [y (o) - 92X (O)] = (92 - 91)X (0)> 0. We(g)- W(9g2) = [Y(0) - 9giX1(0)] - [Y62(0)- 92X2(0O)] < [y62 (o) - gyX(2()]- [Y(o) 92X (0)] = (92 - 91)Xs(O). Letting 1 g2 - g9 I — 0, result follows. LU 2. PROPERTIES OF DETERIORATING SYSTEMS Since the transitions of the deteriorating system follow a continuous time Markov process, the transition probabilities Pij(t), i,j E S and t E [0, oo) satisfies the following Kolmogorov's equations. Kolmogorov's forward equations: tP.ii(t) = -AiPii(t)= -Xie-xt dt -7 Pi t) = -Aj;j(t) + Oj_ I;,P_(t) tt ' d d- Fi(t) Tt n = E ajPij(t) 3=i for 0 < i < n, for 0 < i < j < n, for 0 < i < n. for 0 < i < j < n, Kolmogorov's backward equations: Pij(t) = -AiPij(t) + i Pi+l,j(t) dt d -Fi(t (t) = -iFi (t) - +( = -AiFi(t) + PiFi+l(t) + ai for 0 < i < n. 2

By solving Kolmogorov's equations, explicit formulas for the transition probabilities P23(t) can be derived easily when Ao, Al,. and A, are all distinct. 0 for j < i e- ~~~~~~~~~~for j = I =i (t d [ e-At fo <j<J U~t Ll-i k-i k#A1 n l- Pik(t) forji= n+1I By integrating P23 (t) over [0, t], explicit formulas for Qi (t) can also be obtained. 0 for j < i 1- ~~~~~~~~~~~~~forj=i t - > Qi(t) for j = n +1 k=i The expected time to fallure pi can be easily derived using the following recursive equation with /Ln+1 = 0. Furthermore, Pi3(t) and Qi3(t) have the following properties. Lemma 2. 1. For each t E [O, o), Pi -(t) is totally positive of order 2 (TP2) in i and j E S\ {n+ 1}, that is, > 0 for 0 < i <j i~ n and 0 < k < I < n. P~jk(t) P,1(t) Proof. Let Oii(t) = Pik(t)Pjl(t) - PJk(t)Pi1(t). It is obvious that Oii(t) = 0, OkAO) ~ 0, and Oi~)= Pik(t)P31(t) ~ 0 for all j > k. Using Kolmogorov's backward equations, we can show that dt Solving Equation (3), we have e-e(Ai+A-)t {j e(A-+A )u [i,(O,+i',(U) + /3,oij+i(u)] du+ +i( 3

Ys;(O) Y~(0) for all 6 E A6. When Xg(O) > 0, it follows that g; = (0) < l) for all 6 E A6. Hence. X6(0) - x0(0) Equation (1) holds. 1 Lemma 1.2. Wo(g) is a continuous and nonincreasing function of g. Proof. Let gl < g2 and 61, 62 E Ae such that Wo(gi) = inf [Y6/(O) - g1X(0)] = Y6() - g1X1 (0) < Wo(g2) = inf [Ys(O) - 2X(O)] = Y62(O) - 92X (0) < 6 EA9 - 2 Since XO(O) > 0 for all 6 E As, the following inequalities hold. > [Y (0) - 91 X1 (0)] - [YS (0) - g92Xs (0)] W(g1) - We(92) = [Y1 (O) - g1Xx (o)] - [Y6s(0) - 952X (0)] < [Y(o)-gX - x2()] - [Ys2(o)- 92X (o)] Y () - g1 X (0) Y61 (0) - 92X1 (0) = (92 - gl)X (0) > 0. = (92 - g1)Xs (0). Letting 1 92 - 9 I - 0, result follows. 2. PROPERTIES OF DETERIORATING SYSTEMS 1 Since the transitions of the deteriorating system follow a continuous time Markov process, the transition probabilities Pij(t), i,j E S and t E [0, oo) satisfies the following Kolmogorov's equations. Kolmogorov's forward equations: d dPii(t) = -AiPii(t) = -Aie- dt dt Pi t). dt n = E> jPi,(t) j=i for 0 < i < n, for 0 < i < j < n, for 0 < i < n. for 0 < i < j < n, Kolmogorov's backward equations: d t dP F(t) = -AiPij(t) + 3iPi+l(t) dt = -AiFi(t) + OiFi+l(t) + ai for 0 < i < n. 2

By induction, it is easy to show that 0- 1(t) ~ 0 for 0 < i < j ~ n and 0 K k <1 K n. E Lemma 2..For each i E S, Pjj-(t) is totally positive of order 2 (TP2) in j E S \ {n + 1} and t E [0 o) that is, Pij(U)Pij(V)> 0 for 0 < j < k K n and 0 K u < v. Pi k(U) Pik (V) Proof. Using Kolmogorov's backward equations and Lemma 2.1, it is easy to show that for all t E (,0).d Pik(t) =_ /3 [Pi+I,k(t)Pi)(t) - Pik(t)P,+1,3(t)] > o. dt P,, (t) [i(1 HencePk t is nondecreasing in t for all j < k. Result follows. P,3 (t) Lemma 2.3. For each t E [0, cc), Qj 3(t) is totally positive of order 2 (TP2) in i and jE S \ n + 1}, that is, Qik~) Qi~t)> 0 for 0 < i <i < n and 0 < k < 1 < n. Qjk(t) Q,1(t) Proof. Let ~pi,(t) = Qik(t)Ql(t) - Qjk(t)Qia(t). Obviously, (pi,(t) = 0 and ~p,3(O)= 0. Now, we prove ~pi (t) ~ 0 for three different cases. Case 1: For j > k, since Qjk(t) = 0, it follows that pi,(t) = Qik(t)Ql(t) > 0 Case 2: For 0 < i <j =k < I < n, from Kolmogorov's forward and backward equations, we can show that d Qji(t) = -AjQjj(t) and d Qj,(t) = -AjQj,(t) + O3,Qi,1, (t) for 0 < i < j K n. Taking the first derivative of ~pj (t) with respect to t, we have d 4 =tp.?( -(Ai + A,)pOi,(t) + o3i;oi1,j(t) +!33Q3+1,1(t)Qi,(t) - Qi1(t).(4 dt p,,(t) =e-(Ai+A)(tiu) [O3~pj+1,1(u) + i33Q3+1,,(u)Qi3(u) - Qjj(u)] du. Let i7,(t) = 33Qi+1,1(t)Qj,(t) - Qja(t). Since ~p - (t) = 0, if we can show that 7i(t) > 0 for all 0 < i < j -1, then it will hold by induction that ~p- > 0 for 0 < i < j. By differentiating 71i(t) with respect to t, we have d 4

property of TP2 functions, Equation (7) changes its sign at most once in i and t and the possible change is from negative to positive for any c. This result implies that the failure rate function hi(t) is nondecreasing in i and t. Since Fi(t) = 1 - e-o h,()du, F(t) is therefore nondecreasing in i. E Lemma 2.6. If ai is nondecreasing in i E S \ {n + 1}, then pi is nonincreasing in i. Proof. It is obvious that for i E S \ {n + 1}, pi - i+i = j [FP(t) - Fi+(t)] dt = [Fi+(t) - Fi(t)] dt. From Lemma 2.5, it is clear that pi is nonincreasing in i. O 3. PROPERTIES OF COST AND TIME STRUCTURES Given that the system starts in state i, the expected operating cost to failure Ai(oo) can be easily calculated by the following recursive equation providing that An+(1o) = 0. (Note that Ai(oo) = pi when ao = a = *.. = an = 1.) ai /i a~ n ak k-i Ai(oo) = + Ai+l(() o) ( A A Ai k=-i+lk A =i If the system satisfies the following assumptions, it can be shown that the optimal policies of various strategies have structural properties. (Al) 0< Ao< A1 <... <An (A2) 0 <ao < <...<an (A3) 0 < ro < r, <... < rn < rn+1 - q (A4) 0 <Co + M< C1 M<... <Cn+ +M < Cn+ ro+q rl +q r+l + q rn+l ao Al An Assumptions (Al) to (A5) above imply the following properties. 6

We know that ri + q < r+li from (A3). Result therefore follows. Co + Mi Property 3.6. Recall that bi(g) = a; -A K- (g)+Pi'i+l(g)+c [ [Kn+((g)- M] For g E [0, m + ] ro + q bi(g) is a nondecreasing function in i E S \ {n + 1}. Proof. Given any g E [0, m + ro + q] and i S \ {n, n + 1}, using Property 3.5, we have ro+ qJ bj+l(g) - bj(g) = {[aj+i - j+Kj+l(g)] - [a - AK(g)]} + 3j+IKj+2(g) - /3Kj+l(g) + (aj+l- aj) [Kn+l(g)- M] {[aj+ - Aj+,Kj+,()] - [aj - AjK,(g)]} + (3,j+1 - 3) Kj+,(g) + (aj+i - aj) Kj+(g) = {[aj+ - Aj+Kj+,()]- [ - (g) -} + (Aj+i - jA) Kj+(g). Furthermore, from assumptions (Al), (A3) and (A5), it is clear that aj - AjKj(g) is nondecreasing in j E S \ {n + 1}. Result follows. ' 4. PROPERTIES OF OPTIMAL POLICIES Sequential Inspection Strategy: Lemma 4.1. For each fixed g E [O,gmin], if D6(i) $ R, then G=(i, g, oo) - V'(i, g) < i [G'(i + 1,g, oo) - V*(i + 1, g)] for all 6g constructed in Step 2 of the algorithm and i E S \ {n + 1). Proof. If Dag(i) $ R, then V'(i,g) = G'(i,g,tf(g)) and G'(i,g,t) = 0. Using Koldt t=t:(g) mogorov's backward equations, we have G 1 tG'(i(,g, (t) = ({a - AiG-(i,g,t) + /iV'(i + 1,g) + aiK,+l(g) - g +i [1 - Pi+1,,i+(t)] [G(i +,g,t)- '(i + 1,g)]}. Since -G*(i g t) = 0 and V'(i + 1,g) < G'(i + 1,g, t'(g)), we have dt t=t, (g) ai - AiG"(i,g,t,(g)) + /iV*(i + 1,g) + aiKn+(g) - 9 < 0. (8) 8