Problems 583
(c)The objective function,f=(R 1 +R 2 )R 3 , is separable.
(d)A nonserial system can always be converted to an equivalent serial system by
regrouping the components.
(e)Both the input and the output variables are specified in a boundary value problem.
(f)The state transformation equations are same as the design equations.
(g)The principle of optimality and the concept of suboptimization are same.
(h)A final value problem can always be converted into an initial value problem.
Problems
9.1 Four types of machine tools are to be installed (purchased) in a production shop. The
costs of the various machine tools and the number of jobs that can be performed on each
are given below.
Machine tool type
Cost of machine
tool ($)
Number of jobs that can be
performed
1 3500 9
2 2500 4
3 2000 3
4 1000 2
If the total amount available is $10,000, determine the number of machine tools of various
types to be purchased to maximize the number of jobs performed.Note:The number of
machine tools purchased must be integers.
9.2 The routes of an airline, which connects 16 cities (A, B,... , P), are shown in Fig. 9.23.
Journey from one city to another is possible only along the lines (routes) shown, with
the associated costs indicated on the path segments. If a person wants to travel from city
Ato cityPwith minimum cost, without any backtracking, determine the optimal path
(route) using dynamic programming.
9.3 A system consists of three subsystems in series, with each subsystem consisting of several
components in parallel, as shown in Fig. 9.24. The weights and reliabilities of the various
components are given below:
Subsystem,i
Weight of each component,
wi(lb)
Reliability of each
component,ri
1 4 0.96
2 2 0.92
3 6 0.98
The reliability of subsystemiis given byRi= 1 −( 1 −ri)ni ,i= 1 , 2 , 3 , whereniis the
number of components connected in parallel in subsystemi, and the overall reliability of
the system is given byR 0 =R 1 R 2 R 3. It was decided to use at least one and not more
than three components in any subsystem. The system is to be transported into space by
a space shuttle. If the total payload is restricted to 20 lb, find the number of components
to be used in the three subsystems to maximize the overall reliability of the system.
