Apr 7, 1991 - Krishna Behari Misra ... Special math needed for explanations: Integer programming ..... "series-parallel'' system subject to any constraints.
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IEEE TRANSACTIONS ON RELIABILITY, VOL. 40, NO. 1 , 1991 APRIL
An Efficient Algorithm To Solve Integer-Programming Problems Arising In Sy stem-Reliability Design Krishna Behari Misra Indian Institute of Technology, Kharagpur
Usha Sharma
blems of any type in which the decision variables are restricted to integer values. Several illustrative examples substantiate these assertions.
Indian Institute of Technology, Kharagpur
Key Words - Mixed-redundancy, Reliability optimization, Integer programming,Mixed-integer programming, Parametric programming, Multi-criteria optimization, Zero-One programming Reader Aids Purpose: Widen state of the art Special math needed for explanations: Integer programming Special math needed to use results: None Results useful to: Reliability designers and operation-research specialists Abstract - A simple and efficient technique for solving integerprogramming problems that normally arise in system-reliability design, is introduced. The algorithm is based on functional evaluations and a limited search close to the boundary of resources. Our experience shows that it is fast to solve even a very large system problem. We believe that it can be effectively used with other general integer programming or Zero-One programming problems from the operations research area.
1 . INTRODUCTION In many reliability design problems, decision variables are constrained to integer values; redundancy allocation is one such problem. Several other problems (eg, spare parts allocation and repairmen allocation) also involve an integer programming formulation. The exact techniques which have been used for solving redundancy optimization problems, except those which are strictly based on some heuristic criteria, are computationally difficult and sometimes unwieldy since they aim at solving a general integer-programming problem. On the other hand, many techniques from the literature treat the decision variables as continuous, even though they must be integers and the optimal solution is obtained by rounding the solution to integers. The solution, obtained in this way, is not always optimal. From our literature survey, it is evident that a vast majority of the existing techniques are approximate, while the other techniques, which provide exact solutions, are computationally tedious, time-consuming, and uneconomical. Our MIP algorithm (so named by the editors: Misra Integer Programming) solves a very general type of integer programming problem and is simple, besides being amenable to computerizationand easy to formulate. It consists of a systematic search near the boundary of constraints and involves functional evaluations only. It can handle system-reliability design pro-
2. SURVEY OF EXISTING TECHNIQUES
The integer programming techniques, available in the literature to solve reliability optimization problems, can be classified into three broad categories: approximate, exact, and heuristic. Approximate techniques: The optimal values of the decision variables are generally non-integers and must be rounded to integers to yield the (so called) optimJ allocation of the decision variables. These techniques include Lagrange multiplier [2], geometric programming [4, 51, discrete maximum principle [3], linear programming [6], differential dynamic programming [7], sequential simplex search [2], and penalty function [2]. Exact techniques: These include dynamic programming [8, 91, branch and bound [2, 131, implicit search [lo] and cutting plane [1 11. See [ 1 , 31 for a survey of these techniques. Among these, dynamic programming is perhaps the most widely used and well-known method. The major disadvantage of dynamic programming is the curse of dimensionality. The volume of computation required for an optimal solution increases exponentially with the number of decision variables [12]. This can be reduced to some extent using Lagrange multipliers [8, 91. Dynamic programming is unsuitable for large systems or for problems with more than two constraints. Branch & bound techniques can solve fairly large nonlinear integer-programming problems in a reasonable time but most branch & bound algorithms are confined to linear constraints with an objective function that need not be linear. The implicit enumeration search technique of Geoffrion [lo] and the partial enumeration search technique of Lawler & Bell [ 141, like the branch & bound techniques, necessitate the conversion of integer variables into binary variables. Both techniques yield optimal solutions but require a monotonic objective function. Although the former requires separability of the objective function and constraints, no such assumption is necessary with the latter. While the Lawler & Bell approach [14] can tackle nonlinear constraints (which is its advantage over [lo]) the major limitation of [141 is that the search involves many binary variables, even for a small problem. However, these techniques do not seem suitable for the problems in which the variables are bounded above by large integers. The cutting plane technique [ l l ] for solving the linear integer-programming problem is an efficient tool for solving
001 8-9529191/0400-O08 1$0 l.OOO1991 IEEE
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IEEE TRANSACTIONS ON RELIABILITY, VOL. 40, NO. 1, 1991 APRIL
reliability optimization problems, its limitation is the dimensionality and the cost of achieving a solution. All exact methods mentioned above become computationally unwieldy for solving a large reliability optimization problem. Heuristic techniques: A heuristic is an intutive procedure. None of these methods establishes the optimality of the final soutions. Several papers [ 18-21] suggest many heuristics. Very frequently, heuristics lead to solutions which are near optimal (if not truely optimal) in a reasonably short time.
n
gi(r,x)
j= 1
mps,
bi Si
The MIP technique [22] has the following advantages over existing techniques: It does not require conversion of the original decision variables into binary variables - unlike Lawler & Bell [14] or Geoffrion [lo]. It applies to a wide variety of problems, with arbitrary objective and constraint functions. The objective function need not be separable. The system-reliabilityfunction need not be partially differentiable with respect to the component reliabilities. Thus, the algorithm is independent of the nature of the objective function or constraints. It can solve integer programming as well as zero-one programming problems, effectively and with ease.
4. THE PROBLEM 4.1 Notation
X
‘i
PER
system reliability optimal value of the variable 4 percentage effort ratio (in obtaining the optimal solution): ratio of points searched to total search points of the region
Other, standard notation is given in “Information for Readers 8~Authors” at the rear Of each 4.2 Problem Statement
The problem must be put in the form: Minimize f (x)
(1) =
such that
‘y2>...
(2)
J~
Very often, in reliability design problems, the xJ are restricted to be positive integers.
5. MOTIVATION FOR THE MIP PROCEDURE
Amongst the search techniques in the literature, the Lawler
number of subsystems (stages) of a system redundant units corresponding to subsystemj, 1 rjr n; a non-negative integer lower, upper limit of decision variable xJ, 1 s j r n ; both are non-negative integers (x1,x2,. ..,xn) - redundancy vector component reliability corresponding to subsystem j , lrjln ( r1,r2,...,rn) - component reliability vector objective function; non-decreasing in xJ objective function using augmented decision vector,
(r,x) number of constraints constraint i ; non-decreasing in xJ constraint i using augmented decision vector, ( r , x ) constraint coefficient function corresponding to constraint i and subsystemj ( a , is constant if constraints are linear w.r.t. x) a function representing nonlinearity of the constraint; h,(xJ) = xJ, iff constraints are linear rl
g,(x)
for nonlinear constraint i maximum permissible slack for constraint i ; mpsi < min( {aij}) , for linear constraint i J budget for constraint i slack for constraint i; si = bi - g i ( x ) or si = bi gi(rtX)
R,
E* 3. ADVANTAGES OF MIP ALGORITHM
aij(rj) . hij (xi)
E
=
a, xJ; for linear constraint i ; J=1
& Bell algorithm [14] is distinctly superior to other existing
search methods for exactly solving integer-programming problems. However, as mentioned earlier, it has disadvantages the foremost being the sharp increase in binary variables due to the integer variables being transformed to binary variables from the knowledge of upper bounds of the decision variables. It is worthwhile [22] to provide an idea of dimensionality difficulty and computational effort involved in the Lawler & Bell algorithm; [15, 161 discuss @is subject in detail. 5.1 Example
We are interested in improving the reliability of a l-outof-5:F system employing active redundancy of components. The system data are given in table 1 . The components in each subsystem are i.i.d. TABLE 1 Data for the Example 1
2
3
4
5
component cost ( c j )
2
component reliability ( r j )
0.700
3 0.850
0.800
3 0.800
0.900
Subsystem j
2
1
MISMSIWRMA: EFFICIENT ALGORITHM TO SOLVE INTEGER-PROGRAMMING PROBLEMS ARISING IN SYSTEM-RELIABILITY DESIGN
The problem is to determine an optimum allocation of redundancy to each subsystem such that system reliability is maximized.
n 5
Maximize R, =
1 - ( 1 - rj)’j
j= 1
5
cj xj
such that
I20.
j= 1
The lower and upper limits of each decision variable are in table 2. TABLE 2 Bounds of the Decision Variables j
1
2
x i ” 5 X
j
3 5
4
l
l
l
4
5
4
10
l
1
The region bound by the upper and lower limits of xj has 4000 solution points and only one of these points (if the solution is unique) is the optimum. In the Lawler & Bell algorithm [14], the limits xj and xi”are used [15, 161 to generate binary variables, xjp,for subsystem j : Pj’
2p-’ xjp,
(xi” - xj) 5 p=l
Notation pi.
minimum value of p for which the inequality (5) is satisfied for subsystem j .
There are 14 binary variables to generate 214 = 16384 solution points. Out of these solution points, through the Lawler & Bell algorithm, we would obtain the optimal solution by testing 635 solution points [22] and get the result:
2 = (01 1 1 1 0 1 1 0 1 1 0 1 l} with equivalent decision variables, corresponding to (3) and (4) as x* = {2,1,2,2,3}, with R,* = 0.69545.
PER = 635116384
=
3.9%.
The main disadvantage of Lawler & Bell algorithm (otherwise versatile) is the tremendous amount of computation due to conversion of integer variables to binary variables. However, table 2 shows that there are
83
integer points in the search region as compared to 16384 in the binary search. This is our main motivation, for developing a search procedure in integer frame of variables. Fortunately, not all of the 4000 solution points are feasible; only 157 of them lie in the feasible region defined by (4). Again, out of these 157 feasible solutions, many lie far away from the boundary within the feasibility region and are of no interest since we can always pick up a point with better objective function value close to the boundary since all functions including the objective function are non-decreasing functions of the decision variables. 5.2 General Discussion
It would save time if we search the feasible region close to the boundary (from within) only. To achieve this, we always allocate a maximum value to x l , say, x ~ , which ~ ~ does ~ , not violate the constraints, while we retain the previous allocation at other subsystems. In this way, for the example above, we skip several vectors from amongst the 157 feasible vectors. If any xk reaches its maximum, x;, then we initialize all xj to x:, for j < IC, j z 1 , and increment X k + l by one. However, for j = 1 , we still have to compute xl,mxwhich does not violate the constraints. In this way we can scan the entire feasibility region while skipping many solution vectors. Due to the structure of the constraints, it is possible that even after setting x1 = xl,max, the slacks for some constraints are large enough that we can increment some xk, 2 Ik 5 n without violating any of the constraints. If such a possibility exists, due to nondecreasing nature of the objective function, the objective function at the new point is better than the point with x1 = x ~ , For ~ the ~ ~ example . in section 5.1 there are several feasible solution points during the search which have a system cost of 19 units. For example, a feasible point such as (3,1,1,2,2)with c, = 19 and RS = 0.58952 would also be obtained during the search but this point is covered by another point (3,1,1,2,2),which has a R, = 0.59488 and is generated following the above procedure. The latter is definitely superior to the former. Similarly, another feasible point (2,1,1,2,4)with c, = 19 and R, = 0.55686 is also covered by another point (2,1,1,2,5) with c, = 20 and R, = 0.55691. Therefore, in order to avoid such a situation, it is necessary to: a) compare ~has ,been~determined) ~ ~ with a preassigned slacks (after x mpsi for each constraint type, and b) ensure that slack i does not exceed mps, during the search. Each mpsi, i = 1,2,...,m can be assigned a value less than the minimum of the incremental costs of the components. This eliminates many feasible points near the boundary which otherwise might be included in the list of feasible solutions [22].
6. THE MIP ALGORITHM
The main steps are as follows [22].
1 . Compute x/ and xj” of all the decision variables. These bounds determine the feasible region of search. The lower bounds are generally known from the system description, and
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IEEE TRANSACTIONS ON RELIABIZSTY, VOL. 40,NO. 1, 1991 APRn
upper bounds are determined from the constraints. The latter can be done by allocating the whole of resource i to xj and determine xj,- which can be allocated while keeping all other variables at minimum level (lower bound). This is repeated for all i, i = 1,2,...m; xi” = m;ln({xj,mx}).
cremental costs computed for the whole range of x1 and store them in memory rather than evaluate it, every time xl,max is desired.
The search begins at a corner of the polyhedron, eg, (xy, x i ... ,xi) and finishes at another comer, eg, (xi,xi ...&). Both corners are always feasible. Set f ( x * ) = 03 (for minimization) or -03 (for maximization). Set t = 2, and v = 0.
Notation cost coefficient of constraint i corresponding to x1
cql
+
Set x = (xy,xL... J:), and x* = x. If this point is within the slack band < bi - mpsi, bi > for i = 1,2,...,m, then go to step 8. [See note 1 at the end of this section.] 2. Set x2 = x2 1. If x2 Ix f , go to step 4. 3. Keep all other variables, (j= 2,3,. .. , n ) at the current level. Determine the xl,max, which does not violate any of the constraints; if xl,max# 0 then go to step 7. [See note 2 at the end of this section.] 1. If v > n - 2, STOP, and print out 4. Set v = v the optimal result. v and xk = xk 1. If x k > x i , return 5. Set k = t to step 4. 6. Set xi = xj’, f o r j = 2,3 ,...,k - 1.
+
+ +
.
7. Calculate slacks, si; i = 1,2,...,m. If x does not lie within the band < bi - mpsi, bi> for i = 1,2,... ,m, ie, si 5 mpsi, for i = 1,2,...,m, then return to step 2. 8. Evaluate the objective functionf(x) at x. Comparef(x) with current f ( x * ) and store x in x* andf(x) inf(x*) if the objective functionf(x) is better thanf(x*). Return to step 2. Note 1. In order to see if x lies within the tolerable slacks specified for the constraints, the current test point must be close enough to the boundary. Define maximum permissible slacks for linear constraint i as a quantity slightly less than the minimum of the “incremental costs” among the n subsystems. For linear constraints, mpsi = min{cii} - E . These I slacks are defined on the inner side of the feasible region since the constraint inequalities are of the “less than or equal to” type. This ensures that only those feasible points, which are close to the boundary of the constraints, are considered for functional evaluations. Notation
cii
Most of these problems have been taken from published papers. 7.1 Problem 1. Maximize the reliability of “series-parallel’’ systems subject to linear constraints.
cost coefficients of constraint i positive number representing tolerance permitted on the constraint.
n n
Maximize, R, =
(1
- (1
- rj)”j)
(6)
j= 1
subject to the linear cost-constraints
+
Set v = 0. Return to step 3.
E
7. EXAMPLE PROBLEMS
n
c i i x j Ibi, i = 1,2,...,m.
gi =
(7)
j=l
Example 1. Same as the example of section 5.1. Only 48 solution points were generated; all are feasible and have a system cost of 20 units since mpsl = 0. The optimal solution is (2,1,2,2,3) with system cost = 20 units and R,* = 0.69545. The statistics of obtaining this result are [22]: Total points in the Region CR, ( n T ) Total feasible points in the Region CR Search point visited by the algorithm Number of points where functional evaluations were done (nf) Number of points for which objective function has been compared
4000 157 83 (2.1 %) 48 (1.20%) 5
The figures in parentheses represent the effort as a fraction of the points in the Region; when compared with those re- , quired by the Lawler & Bell algorithm, they show the efficiency of the MIP algorithm. Another advantage is that since the dimensionality of the problem does not increase rapidly, the algorithm can solve large problems too. Example 2.
Table 3 lists results for a 4-stage system with various cost vectors and budgets. The component reliability vector r = Note 2. For linear constraints, (0.80, 0.70, 0.75, 0.85). Table 3 highlights the influence of the xl,max= min {xl:xl = [bi - C g i ( x j ; j = 2,3,...,~ ) ] / c c ~ , ~ cost vector and/or budget on the search effort. I Table 3 shows an appreciable reduction in the PER with If the constraints are nonlinear, xl,- is obtained by incremen- the increase in the size of the search region. For example, when ting x1 successively by 1 until some constraint is violated. It the total search points ( n T ) increase from 48 to 1080 to 9072 may be computationally advantageous to keep nonlinear in- the PER reduced from 10.4% to 2.9% to 1.1%.
MISFWSHARMA: EFFICIENT ALGORITHM TO SOLVE INTEGER-PROGRAMMING PROBLEMS ARISING IN SYSTEM-REI-IABILITY DESIGN
85
7.3 Problem 3. Maximize the reliability of "series-parallel'' systems subject to nonlinear constraints.
TABLE 3 Influence of Cost vector and Budget on Search Effort
Example. A 1-out-of-5:F system [2] with nonlinear constraints. (4,2,3,5) ; 20 (4,2,3,5) ; 30 (1,2,3,4) ; 30
(1,2,2,1) ; (2,3,2,2) ; (5,4,3,2) ;
19 30 30
0.58012497 0.85599673 0.95412707
48 1080 9072
5 31 108
10.41 2.87 1.11
n 5
Maximize R, =
(1 - ( 1 - rj)'j)
(11)
j= 1
5
subject to gl
Maximize R,
=
5
g2
=
cj(xj
+ exp(xj/4))
I C,
j= 1 5
(1 - ( 1 -0.6)"1) (1 - (1-0.9)'2)
( 1 - (1 -0.55)'3)(
(12)
j=l
7.2 Problem 2. Parametric maximization of reliability of "series-parallel'' system subject to any constraints. Example. A 1-out-of-4:F system with linear constraints.
P ~ ( x , )I ~P
3
g3 =
1 - ( 1 -0.75)'4)
wj xj exp(xj/4) )
I W,
j= 1
(8)
Table 5 shows the system data. This problem has been solved by many authors [2]; the symbols have the same meaning as in Tillman et a1 [2].
subject to
+ + + + 9 . 5 ~ + ~5 . 5 ~+ ~3 . 8 ~+ ~4 . 0 ~I~67.8 ~ ~51.8 6 . 2 ~ ~3 . 8 ~ ~6 . 5 ~ ~5 . 3 I
10(01)
(9)
15(02)
(10)
All xi are positive integers; - 1 I0, I 1. Table 4 provides the results: optimal allocation, optimal system reliability, total number of search points ( nT), total number of functional evaluations ( nf), and for various values of 6 used in [23]. This example demonstrates the capability of generating all solution points economically, and should be compared with the effort involved in [23].
TABLE 5 System Data for Example 3 j
Pj
'1
0.80 0.85 0.90 0.65 0.75
1 2 3 4 5
1 2 3 4 2
P
cj
110
7 7 5 9 4
c
wj
w
175
7 8 8 6 9
200
The results, for various values of allocated resource, are listed in table 6.
TABLE 4 Results of Problem 2, Example
TABLE 6 Results for Problem 3, Example 0.0 0.07 0.08 0.23 0.34 0.37' 0.46 0.59 0.61 0.694 0.97 0.973 1.0
51.8 52.8 52.6 54.1 55.2 55.5 56.4 57.7 57.9 58.74 61.5 61.53 61.8
67.8 66.75 66.60 64.35 62.70 62.25 60.90 58.95 58.65 57.39 53.25 53.20 52.8
50.1 52.5 52.5 52.5 55.1 55.4 56.3 56.3 57.8 56.6 56.6 58.1 58.1
49.4 53.4 53.4 53.4 59.1 53.4 58.9 58.9 57.4 53.2 53.26 51.7 51.7
0.70858 0.71778 0.71778 0.71778 0.72745 0,74401' 0.78956 0.78956 0.75367 0.74766 0.74766 0.7 1367 0.7 1367
1200 1200 1200 1400 1680 1680* 1470 1176 1176 1176 1344 1344 1344
43 45 45 49 51 50' 52 47 48 47 45 42 44
3.60 3.8 3.8 3.5 3.4 2.97' 3.5 4.0 4.1 4.0 3.3 3.1 3.3
Table 4 shows that the PER was minimum at 2.97 % when the total number of search points was maximum at 1680.
100 114 116 116 90
175 185 190 145 195
200 212 218 236 256
83.0 83.0 93.0 92.0 90.0
146.12 146.12 156.40 142.25 157.33
192.48 192.48 216.91 211.81 224.14
(3,2,2,3,3) (3,2,2,3,3) (3,3,2,3,3) (2,2,2,3,4) (4,2,2,3,3)
0.9044 0.9044 0.9221 0.8857 0.9103
3125 3125 3125 3750 3600
173 202 211 230 184
5.5 6.5 6.8 6.1 5.1
Table 6 shows that the PER for this problem with various input data, remains about the same. Tillman et a1 [2] solved this problem with a dynamic programming procedure using the concepts of dominating sequences and sequential unconstrained minimization technique. Their effort was much greater than required by the MIP algorithm.
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7.4 Problem 4. Maximize availability of a “series-parallel” maintained system, subject to linear constraints involving redundancy, spares, and repair facility as decision variables.
Table 8 provides optimal allocation and system availability for various values of allocated budget. Table 8 shows that when the allocated resource of each type is increased by 3 units and5 units thePERreduces from0.38% to0.14% to 0.013%. We believe that no other search procedure provide results so economically as the MIP algorithm.
n n
Maximize ASS =
Aj”
j= 1
Subject to the constraints
7.5 Problem 5. Maximize the reliability of non-series parallel systems, subject to any constraints.
n
gj =
gi{(xj
- k j ) , uj, pi}
Ibi, i=1,2
,...,m
(16)
j=i
Maximize R, = f ( r , x ) Example 1. A bridge network with 5 elements.
Notation
+
+
1 - p(uj (Xi - k j ) l),j probability of being in state i for subsystem j [24] number of spares for subsystemj number of repairmen for subsystemj
Aj” pij uj pj
The Aj” in (15) is quite involved; see [24, (5)-(7)]. We are not providing the complete mathematical formulation of this problem, since it is quite lengthy and is available in [24] with full details. Example. Maximize the steady state availability of a system consisting of three subsystems, subject to a single constraint. The failure and repair rates of these subsystems are given in table 7. The constraints are given by
+
3 . 0 ~ 5~ . 0 +6.0x3 ~ ~ +4.5sl
Example 2. The ARPA network of 5 nodes and 7 links.
subject to the following linear constraints
+ 6 . 5 +7.5s3 ~ ~ +2.Or, +2.5r2
n
gi
+3.0r3
Ibl
+
(17)
+
+
+
+
2. 5x1 4.5X2 5. Ox3 5.Osl-k 9.0~2 9. 5s3 1.5r1+ 2. Or2
=
ci xj
Ibj,
i = 1,2,...,m;
j= 1
all xi are positive integers. Cost Coeficients The cost coefficients ( cii) for examples 1 & 2 are given in table 9.
TABLE 7 Input Data for Problem 4, Example
Solutions Subsystem j
Failure rate
Repair rate
4
b
Number of essential component, kj
1 2 3
0.01285 0.01500 0.02500
800 8500 900
2 1 2
Table 9 lists the optimal solutions, and shows that the PER reduces drastically as the region of search becomes large. The last row of table 9 shows that even for a problem involving the total number of search points as large as 1161600, the PER is only 0.07 % .
TABLE 8 Results of Problem 4, Example Resource allocated
b,
6,
Redundancy
Spares
Repair facility
Optimal system availability
40.0 43.0 45.0
45.0 48.0
(l,l,l) (2,0,1) (5,0,2)
(l,l,l) (l,l,l) (0,1,0)
(1 1,1) (2,192) (3,1,1)
0.95798 0.99543 0.99911
50
Optimal allocations
3
Resource Consumed
PER g,
g2
nT
“f
(%)
40.0 43.0 45.0
41.5 43.5 40.5
46656 25920 486000
180 379 634
0.38 0.14 0.013
MISRA/SHARMA: EFFICIENT ALGOmHM TO SOLVE INTEGER-PROGRAMMING PROBLEMS ARISING IN SYSTEM-RELIABILTY DESIGN
TABLE 9 Input Data and Results for Problem 5, Example Input data
Network
3
1
x'; R: g, (top line) nn; 4; PER (%) (bottom line)
reliability vector cost vetor
bl
&
0
Results
20
(0.7,0.85,0.75,0.8,0.9); (1.4,2.5,1.5,2.5,3)
(4,2,2,1,1); 3136; 41; 1.30
0.9947; 19.5
20
(0.8,0.8,0.8,0.8,0.8); (2,3,2,3,1)
(2,3,2,3,1); 4OOO; 48; 1.2
0.9911; 20
20
( 2 3 2 ,2,1) (0.9,0.85,0.75,0.8,0.7);
(2,3,1,1,2); 4OOO; 48; 1.2
0.9966; 20
25
(0.7,0.85,0.75,0.8,0.9); ( 2 3 2 ,3,1)
(4,3,2,1,1); 0.9986; 25 24000; 161; 0.67
44
(0.7,0.85,0.75,0.8,0.9); (6,4,5,6,3)
(1,2,3,2,1); 3360; 49; 1.45
45
(0.7,0.9,0.8,0.65,0.7,0.85,0.85); (4,5,4,3,3,4,5)
(1,1,2,1,1,3,2); 0.9967; 45 72000; 182; 0.25
45
(0.85,0.8,0.8,0.65,0.7,0.85,0.9); (4,5,2,2,3,3,5)
(1,1,2,1,1,4,2); 0.9997; 45 1161600; 815; 0.07
.
0
q-p.
0.9932; 44
0
6
7
0
7.6 Problem 6. Maximize the global availability & reliability of a communication network subject to a linear cost constraint
+k(l)A(2)A(3)A(5)A(6)
+ k(l)A(2)A(3)A(4)k(S)A(6)
[22]: Maximize A, = f(A1,A2,...,A,) = f ( x l , x z,...,x,)
(21)
since, A, would be a function of AJ provided xJ = 1. Example 1 . For the ARPA network with 5 nodes and 7 links -
+A( l)A(2)A(3)A(4)2(5)k(6)&7)
A, = A( l)A(2)A(5)A(6)
+ A( l)A(2)k(4)k(5)A(6)A(7) +A( l)A(2)A(4)k(6)A(7) + A( l)A(2)&4)A(S)k(6)A(7) +A( l)k(2)A(3)A(5)A(6) + A( l)k(2)A(3)A(4)k(5)A(6) +A( l)A(2)A(4)@)A(6)
+A( l)k(2)A(3)k(4)k(5)A(6)A(7)
+k(1)A(2)k(3)A(4)A(5)k(6)A(7) subject to n
+A( l)k(2)A(3)A(5>R(6)A(7) +A( l)k(2>A(3)A(4)/1(5)k(6)A(7)
6 ( x ) 1 nn - 1
1
+A( l)R(2)R(3)A(4)A(6)A(7) n
cflj
+A( l)k(2)k(3)A(4)A(5)k(6)A(7)
I
c,
j= 1
+A( l)A(2)k(3)A(4)A(s)A(6)k ( 7 )
xj = 0 or 1, for a l l j = 1,2 ,...,n
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88
subject to n
AO’)
1 - AO’) yjlyj2...yjnn a string of binary variables corresponding to link j . If link j connects nodes u & v then yj, and yjy are 1 and yil = 0 for all 1 = 1,2,...,nn, 1 # u # v. number- of non-zero xi in x, 6( x ) number of nodes in the network. nn Numerical Example. Determine the optimal configuration of a network with nn = 5 (computer centres) and n = 7 links, and t = 500 hours (mission time). For complete mathematical formulation of the problem, see [22]. The input data and results are in tables 10;ll.
Xj(Yjl
yj2 ...ynn) = 11... 1
j= 1
nn -
1
(continuity constraints) (25)
xi = 0 or 1 for a l l j = 1,2, ...,n
6
cryl 5 c,,
=
Notation
aj exp
RO’) ndcj
Figure 1 . ARPA Network with n = 7 and nn=5
A,,
TABLE 10 ci for Figure 1 3
4
5
2.1
1.2
1.0
2.5
7.1 3.4
2.1 4.5
5
3
2
4
1
AJ(IO-3)
4 1 - RO’)
j = 1,2,...,n
a vector, representing the multiple discrete choices for link j number of discrete choices of component reliability for element j
6
7
2.1
1.2
7.1
3.7
4.0
5.0
6
4
2
0 @@@ 3
7.7 Problem 7. (Multiple choice, Zero-One Programming)
0
Maximize the global availability of the system
A, = f ( x , , x2, ...,x n )
1,
Illustration. Determine the optimal configuration of a communication network with 4 nodes (computer centres) and 6 links (communication channels).
2
j
lo-^)
pi,
[
Figure 2. A Network with n = 6 and n n = 4
TABLE 11 Optimal Configuration for Problem 6, Example Network
Optimal configuration
Availability
units
MISWSHARMA: EFFICIENT ALGORITHM TO SOLVE INTEGER-PROGRAMMING PROBLEMS ARISING IN SYSTEM-RELIABILITY DESIGN
R2 (x2) = 1 - ( 1 - 0.81)'2 (parallel redundancy)
TABLE 12 Input Data for Problem 7, Illustration
i ffj
1 4.4
2
3
4
5
6
0.65
0.45
1.4
2.4
2.5
0.002
0.25
0.016
ndcj
4
4
3
R(j)
0.88 0.92 0.8 0.99
0.7 0.75 0.8 0.85
0.9 0.95 0.99
0.02
0.03
3
2
4
0.8 0.85 0.9
0.95 0.98
0.85 0.90 0.92 0.95
0.12
89
2 6)
(0.77) ( 1 - 0.77) x 3 - i (k-out-n redundancy)
=
i=2
(28)
and minimize, c,(x)
=
4 exp
0.02
subject to
Optimal configuration: 65.0 - g2(x) 2 0 230.0 - g 3 ( ~ )2 0 g 4 ( ~ )- 0.9 2 0 where, gl(x)
=
4 exp
0.02
Figure 3. Optimal Configuration for Illustration
g3(x) = 8x2ex2/4 Optimal allocation: (4,2,2,0,1,3)
n
+ 6(~~-l)e('3-~)/~
3
g4(x)
E
Rj(xj)
j= 1
Optimal system reliability and cost: 0.993703; 14.98
The minimum reliability of each subsystem is 0.95. The optimal allocation (4,2,2,0,1,3) with the optimal configuration as shown in figure 3 which consists of links (1,2,3,5,6) has link reliability
R j ( x j ) - 0.95 2 0 f o r j = 1,2,3
(34)
The solution is:
r = (0.99, 0.75, 0.95, 0.0, 0.95, 0.92). 7.8 Problem 8. Mixed Redundancy System and Multicriteria Optimization [171
n 3
Maximize, R , ( x )
Rj(xj),
j= 1
1::::1,
Optimal redundancy allocation x* = (3,3,6) Objective function values are:
R s ( x * ) = 0.97024 C,(x*) = 37.88 units
The resources consumed and the slacks spared for each resource are:
0.88
where, R l ( x l ) =
for x1 = 1,2,3,4
1 0.99 1
(multiple choice component reliability)
gl(x*) = g2(x*) = g3(x*) = g4(x*) =
37.88; s1 = 7.12 units 64.26; s2 = 0.74 units 155.52; s3 = 74.48 units 0.97024; s4 = 0.07024, respectively.
Optimal subsystem reliabilities are: (0.98, 0.993141, 0.9968783)
90
IEEE TRANSACTIONS ON RELIABILITY,VOL. 40, NO. 1, 1991 APRIL
The optimal allocation, optimal system reliability, resources consumed and the slacks left on constraints are the same as in [17].
The MIP algorithm was used to determine x corresponding to a randomly generated r in order to solve the mixed integer programming problem. This problem was also solved by Sakawa [25] using a Surrogate Worth Tradeoff method. Our results are better than those obtained by Sd~awa.
7.9 Problem 9. Multiobjective and Mixed Integer Programming [251
4
Minimize,
(36)
Cj Xj
C, E
Optimal allocation ( x , r )
= (10,8,9,10,0.573,0.587,
Optimal system reliability Optimal system cost
= 370.94
= 0.99767
j= 1
Resources consumed and slacks left on cost, weight and volume are listed in table 14.
Subject to a cost constraint, 4
400.0 -
r 0
cjxj
TABLE 14 Results of Problem 9
(37)
j= 1 Cj
=
aj
exp[Pj/ ( 1 -r,)].
The weight and volume constraints are:
Type
Resources consumed
Slacks
cost Weight Volume
370.95 68.90 79.79
29.05 6.10 0.21
4
75.0 -
w j ~2 j
0
j= 1
ACKNOWLEDGMENT 4
E
80.0 -
vjxj
2 0,
(39)
j= 1
wj and
V,
are of the form
CY,
rJp’.
Table 13 gives the aj and
n
Pj for each subsystems
We thank the Reliability Engineering Centre, Indian Institute of Technology, for providing necessary facilities for carrying out the present research work. We also appreciate the assistance rendered by the editors, Dr. Evans and Dr. Pecht for bringing the origirial manuscript to the present improved form.
4
- 0.9 2 o
(l-(l-rj)x.)
REFERENCES
j=l
( R , 1 0.9)
R~ = (1- ( 1 -rj)’j) - 0.95 2 0, for j=1,2,3,4
(41)
Also, 0.4 5 rj
(42)
5
0.99, forj=1,2,3,4
Both x, (integer) and rj (real) are the decision variables. TABLE 13 cyi
and pi for Constraints Subsystem
1 2 3
5,0
1.0
0.3 2.0
3.5 4.0
0.55 2,0
2.0 8.0
4.0
2.0
8.0
2.0
6.0
0.4 2.0 2.0
5.0 7.0
0.65 2.0
10.0
2.0
[l] K.B. Misra, “On optimal reliability design: A review”, WAC, 6th World Conference, Boston, Mass., USA, pp 3.4.1-3.4.10, 1975 Aug. [2] F. A. Tillman, C. L. Hwang, W. Kuo, Optimizationof System Reliability, 1980; Marcel [3] K. B. Misra, “On optimal reliability design: A review”, System Science, V O ~12, 1986, pp 5-30. [4] A. J. Federowicz, M. Mazumdar, “Use of geometric programming to maximize reliability achieved by redundancy”, OperationsResearch, vol 19, 1968 Sept.-Oct., _pp _ 948-954. [5] K. B. Misra, J. D. Sharma, “A new geometric programming formulation for a reliability problem”, Int’l J. control, vol 18, 1973 Sep, pp 497-503. [6] R. Gomory, An Algorithm for Integer Solutions to Linear Programs, Princeton IBM Mathematical Research Report, 1958. [7] D. M. Murray, S. J. Yakowitz, “Differential dynamic programming and Newton’s method for discrete optimal control problems”, J. Optimization Theory and Applications, vol 43, 1984, pp 395-414. [8] R. E. Bellman, E. Dreyfus, “Dynamic programming and reliability of multicomponent devices”, Operations Research, vol6, 1958 Mar-Apr,
pp 200-206. [9] K.B. Misra, “Dynamic programming formulationof redundancy allocation problem”, Int ’1 J. Mathematical Education in Science and Technology, (UK), V O ~2, 1971, pp 207-215.
MISRA/SHARMA: EFFICIENT ALGORITHM TO SOLVE INTEGER-PROGRAMMING PROBLEMS ARISING IN SYSTEM-RELIABILITY DESIGN
[lo] A. M. Geoffrion, “Integer programming by implicit enumeration and Bala’s method”, Soc. Industrial and Applied Mathematics Review, vol 9, 1967, pp 178-190. [ l l ] J. Kelley, “The cutting plane method for solving convex programs”, J . Soc. Industrial and Applied Mathematics, vol 8, 1960, pp 708-712. [12] D. E. Fyffee, W. W. Hines, N. K. Lee, “System reliability allocation and a computational algorithm”, IEEE Trans. Reliability, vol R-17, 1968 Jun, pp 64-69. [13] K. B. Misra, J. D. Sharma, “Reliability optimization of a system by zeroone programming”, Microelectronics and Reliability, vol 12, 1973 Jun, pp 229-233. [14] E. L. Lawler, M. D. Bell, “A method for solving discrete optimization problems”, Operations Research, vol 14, 1966 Nov-Dec, pp 1098-11 12. [15] K. B. Misra, “A method of solving redundancy optimization problems”, IEEE Trans. Reliability, vol R-20, 1971 Aug. [16] K. B. Misra, “Optimum reliability design of a system containing mixed redundancies”, IEEE Trans. Power Apparatus and Systems, vol PAS-94, 1975 May-Jun, pp 983-993. [17] Y. Nakagawa, “Studies on optimal design of high reliable system: Single and multiple objective nonlinear integer programming”, PhD &sis, Kyoto University, Japan, 1978 Dec. [18] P. M. Ghare, R. E. Taylor, “Optimal redundancy for reliability in series system”, ORSA, vol 17, 1969 Sep, pp 838-847. [19] J. Sharma, K. Venkateswaran, “A direct method for “ i z i n g the system reliability”, IEEE Trans. Reliability, vol R-70, 1971 Nov, pp 256-259. [20] K . B. Misra, “A simple approach for constrained redundancy optimization problems”, IEEE Trans. Reliability, vol R-21, 1972 Feb, pp 30-34. [21] K. K. Aggarwal, J. S. Gupta, K. B. Misra, “A new heuristic criterion for solving a redundancy optimization problem”, IEEE Trans. Reliability, vol R-24, 1975 Apr, pp 86-87. [22] Krishna B. Misra, “An algorithm to solve integer programming problems: An efficient tool for reliability design”, Microelectronics and Reliability (to appear). [23] M. S. Chern, R. H. Jan, “Parametric programming applied to reliability optimization problems”, IEEE Trans. Reliability, vol R-34, 1985 Jun, pp 165-170. [24] U. Sharma, K. B. Misra, “Optimal availability design of a maintained system”, Reliability Engineering and System Safety, vol 20, 1988, pp 146-159.
MANUSCRIPTS RECEIVED MANUSCRIPTS RECEIVED “Estimation & testing problem in an incomplete-inspection model”, M . S. Srivastava 0 Dept. of Statistics 0 University of Toronto 0 100 St. George Street 0 Toronto, Ontario M5S 1Al 0 CANADA. (TR90-215) “Conservative Bayes experimental design”, Dr. Ronald V. Canfield, Professor 0 Dept. of Mathematics 0 Utah State University 0 Logan, Utah 84322-3900 0 USA. (TR90-216) “Comment on: Calculation of the Poisson cumulative distribution function”, Dr. N. K. Srinivasan 0 Centre Aeronautical Systems Studies&Analyses 0 Defence Research & Development Organisation 0 C V Raman Nagar 0
Bangalore-560
093 0 INDIA. (TR90-217)
“An iterative algorithm for computing the reliability of a k-out-of-n system”, Dr. Janusz Biernat 0 ul. Gwarecka 13 m.2 0 54-143 Wroclaw 0 POLAND. (TR90-218) “MTBF & passage-times for a k-out-of-n:G repairable system with possibly state-dependent failures”, John Newton 0 Dept. Mathematical & Computational Sciences 0 North Haugh 0 University of St. Andrews 0 St. Andrews KY16 9SS 0 GREAT BRITAIN. (TR90-219) ‘‘Software-reliabilitymeasurement based on stochastic differential equations”, Dr. Shigeru Yamada, Associate Professor 0 Dept. of Industrial & System Engineering 0 Faculty of Engineering 0 Hiroshima University 0 HigashiHiroshima 724 0 JAPAN. (TR90-220)
91
[25] M. Sakawa, “Multi-objective reliability and redundancy optimization of a series parallel system by the surrogate worth trade off method”, Microelectronics and Reliability, vol 17, 1978, pp 465-467.
AUTHORS Prof. K. B. Misra, PhD; Reliability Engineering Centre; Indian Institute of Technology; Kharagpur - 721 302 W.B., INDIA. K. B. Misra was born on 1943 Jan 23 and received his BE, ME, and PhD degree in 1963, 1966, and 1970. He has 27 years of teaching, research, and professional experience, and has taught at the University of Roorkee and IIT-Kharagpur besides working at GRS (Garching), Technical University (Munich), Aachen University, and Kernforschungszentrum Karlsruhe. He has published over 120 technical papers and guided 15 PhD students. Prof. Misra is on the editorial board of several internationaljournals. He has visited several European countries to deliver invited lectures. He began the first master’s degree course in Reliability Engineering in India in 1982 and founded the Reliability Engineering Centre at IIT-Kharagpur in 1983. He is Fellow of several professional societies and recipient of a few awards and prizes, including the first Lal C. Verman award in 1983 for his pioneering work in reliability engineering. At present he is the coordinator of Ministry of HRD project on Reliability Engineering and is Dean, Institute Planning and Development at IIT-Kharagpur. Ms. Usha Sharma, PhD; Reliability Engineering Centre; Indian Institute of Technology; Kharagpur - 721 302, W.B.; INDIA, Usha Sharma was born on 1962 Jun 16 and received her BSc, MSc in Mathematics and PhD in Reliability from IIT Kharagpur in 1982, 1984, and 1990. She has published 10 technical papers in various reputed international journals. Her research interest includes reliability evaluation and optimization of complex systems, fuzzy-set theory and its application to reliability and analysis of communication networks. Manuscript TR87-102 received 1987 July 24; revised 1989 March 2; revised 1990 April 30. IEEE Log Number 37395
4TRb
MANUSCRIPT RECEIVED MANUSCRIPTS RECEIVED “Estimation for a life model of transformer-insulation under combined electrical & thermal stress”, Dr. S. B. Pandey 0 Dept. of Electrical & Computer Engineering 0 New Jersey Institute of Technology 0 University Heights 0 Newark, New Jersey 07102 0 USA. (TR90-222) “Robustness of sequential Weibull life-test plans”, Dr. K. K. Sharma 0 Dept. of Statistics 0 Institute of Advanced Studies 0 Meerut University 0 Meerut (U.P.) - 250 005 0 INDIA. (TR90-223) “Comment on: Time-to-fdure & availability of parallel systems with repair”, Dr. Brahim Ksir 0 Institut de Mathematiques 0 Universite de Constantine 0 25000 Constantine 0 ALGERIA. (TR90-224) ‘‘A neural-network approach to evaluating electrical-distribution-system reliability”, Dr. Jiann-Liang Chen 0 4F 20 Lane 148 0 Yuan Shao Road 0 Jong Her, Taipei 0 TAIWAN-R.O. CHINA. (TR90-225) “Representation of reliability-decay processes by time-dependent autoregressive models”, Dr. N. Singh 0 Dept. of Mathematics 0 Monash University 0 Clayton, Victoria 3 168 0 AUSTRALIA. (TR90-226) “A new reliability measure for computer networks”, Dr. Victor 0. K. Li 0 Dept. of Electrical Engineering 0 University of Southern California 0 University Park 0 Los Angeles, California 90089-0272 0 USA. (TR90-227)
