Optimal Resource Allocation: With Practical Statistical Applications and Theory

CHAPTER 10

Optimal Redundancy with Several Limiting Factors

10.1 Method of “Weighing Costs”

A number of cases arise where one has to take into account several restrictions in solving the optimal redundancy problem. For example, various objects such as aircraft, satellites, and submarines have restrictions on cost and also on weight, volume, required electric power, and so on. (Apparently, the cost for most of these technical objects is an important factor, but, perhaps, less important than others mentioned.)

In these cases, one has to solve the optimization problem under several restrictions and to maximize the system reliability index under restrictions on all other factors.

Consider a system consisting of n redundant groups connected in series. For each additional redundant unit of the system, one has to spend some quantity of M various types of resources (for instance, cost, weight, or volume), say, C_j(X). There are constraints on each type of resource: . The optimization problem is formulated as:

(10.1)

where X = (x₁, x₂, … , x_n) is the vector of the system redundant units.

Let us assume that each C_j(X) is a linear function of the form

(10.2)

where c_ji is the resource of type ...