One of the first attempts to solve the optimal redundancy problem was based on the classical Lagrange Multiplier Method. This method was invented and developed by great French mathematician Lagrange (see Box). This method allows one to get the extreme value of the function under some specified constraint on another involved function in the form of equality. The Lagrange Multiplier Method is applicable if both functions (optimizing and constraining) are monotone and differentiable.

Joseph-Louis Lagrange (1736–1813)

Lagrange made outstanding contributions to all fields of analysis, to number theory, and to classical and celestial mechanics. He was one of the creators of the calculus of variations, and also introduced the method of Lagrange multipliers where possible constraints were taken into account. He invented the method of solving differential equations known as variation of parameters and applied differential calculus to the theory of probabilities. Lagrange studied the three-body problem for the earth, sun, and moon and the movement of Jupiter's satellites. Above all, he contributed to mechanics, having transformed Newtonian mechanics into a new branch of analysis, Lagrangian mechanics.

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Strictly speaking, this method is not appropriate for optimal redundancy problem solving because the system reliability and cost are described by functions of discrete arguments x_i (numbers of redundant units) and the restrictions on accessible resources (or on required values of reliability) are fixed in the form of inequalities.

Nevertheless, this method is interesting in general and also gives us some useful hints for the appropriate solution of some practical problems of a discrete nature.

Let us begin with the direct optimal redundancy problem (Fig. 3.1 below). To solve this problem, we construct the Lagrange function, :

(3.1)

where C(X) and R(X) are the cost of the system redundant units and the system reliability, respectively, if there are X redundant units of all types,

Figure 3.1 Procedure of solving the direct problem of optimal redundancy.

The goal is to minimize C(X) taking into account constraints in the form of R(X^opt) = R⁰. Thus, the system of equations to be solved is

(3.2)

The values to be found are: , i = 1, 2, …, n, and Λ.

If both function C(X) and R(X) are separable¹ and differentiable, the first n equations of (3.2) can be rewritten as

(3.3)

On a physical level, Equation (3.3) means that for separable functions and C(X), the optimal solution corresponds to equality of relative increments of reliability of each redundant group for an equal and infinitesimally small resources investment.

(3.4)

In the general case, Equation (3.3) yields no closed form solution but it is possible to suggest an algorithm for numerical calculation:

1. At some arbitrary point calculate the derivative for some fixed redundant group, say, the first one:

(3.5)

Of course, one would like to choose a value of close to an expected optimal solution . For example, if you consider spare parts for equipment to operate failure-free during time t and know that the unit mean time to failure (MTTF) is T, this value should be a little larger than t/T. In other words, this choice should be done based on engineering experience and intuition.

2. For the remaining redundant groups, calculate derivatives until the following condition is satisfied:

(3.6)

3. Calculate value:

(3.7)

4. Compare R⁽¹⁾ with R⁰. If R⁽¹⁾ > R⁰, choose ; if R⁽¹⁾ < R⁰, choose . After choosing a new value of , return to step 1 of the algorithm.
The stopping rule: X^(N) is accepted as the solution if the following condition holds:

(3.8)

where ε is some specified admissible discrepancy in the final value of the objective function R(X).

Solution of the inverse optimal redundancy problem (Fig. 3.2) is analogous. The Lagrange function is:

(3.9)

and the equations are:

(3.10)

Figure 3.2 Procedure of solving the inverse problem of optimal redundancy.

The optimal solution has to be found with respect to goal function C(X). The stopping rule in this case is

(3.11)

where ε* is some specified admissible discrepancy in the final value of the goal function C(X).

Unfortunately, there is only one case where the direct optimal redundancy problem can be solved in a closed form. This is the case of a highly reliable system with active redundancy where:

(3.12)

or

(3.13)

that is, instead of maximizing R(X), one can minimize Q(X) and find the needed optimal solution.

Taking into account that both objective functions are separable, Equation (3.10) can be written as:

(3.14)

From equation

(3.15)

we can finally write

(3.16)

Now returning to the last equation in Equation (3.10), we have

(3.17)

and, consequently, the Lagrange multiplier has the form

(3.18)

From Equation (3.16), one has

(3.19)

Finally, after substitution of Equation (3.17) into Equation (3.19), one gets

(3.20)

Hardly anybody would be happy to deal with such a clumsy formula!

The solutions obtained by the Lagrange Multiplier Method are usually continued due to requirements to the objective functions. Immediate questions arise: Is it possible to use an integer extrapolation for each non-integer x_i? If this extrapolation is possible, is the obtained solution optimal?

Unfortunately, even if one tries to “correct” non-integer solutions by substituting lower and upper integer limits , this very rarely leads to an optimal solution! Moreover, enumerating all 2ⁿ possible “corrections” can itself be a problem. We demonstrate this statement on a simple example.

Chronological Bibliography

Everett, H. 1963. “Generalized Lagrange multiplier method for solving problems of optimum allocation of resources.” Operations Research, no. 3.

Greenberg, H. J. 1970. “An application of a Lagrangian penalty function to obtain optimal redundancy.” Technometrics, no. 3.

Hwang, C. L., Tillman, F. A., and Kuo, W. 1979. “Reliability optimization by generalized Lagrangian-function and reduced-gradient methods.” IEEE Transactions on Reliability, no. 28.

Kuo, W., Lin, H. H., Xu, Z., and Zhang, W. 1987. “Reliability optimization with the Lagrange-multiplier and branch-and-bound technique.” IEEE Transactions on Reliability, no. 5.