It is clear that if a value of r is specified, then the DP model may be run to determine the optimal policy. If r cannot be specified, then the values

of r IE and r1 may be determined for the given value of U. Then one need only specify whether r≤rg or r≥r to conclude that the ASAP policy or ALAP policy, respectively, is optimal.

a21 d12 d22 = 10, d. = d.

time prevented further study of this case. Intuitively, one might expect a greater "critical" range of r since the larger values of M and Nt give rise to a greater difference between the ASAP and the ALAP policies.

4. BOUNDS ON DP VARIABLES

Dynamic programming models are frequently subject to criticism because although the formulation can be accomplished, the numbers of states and decisions to be considered make the model computationally unfeasible. Fortunately, for the model described in this paper, there are relationships among the variables which make it possible to examine a limited number of states and decisions for which the stage return It (xt,dt) is calculated. (xt,dt) is calculated. Although technical in nature, this aspect of the problem is of great importance to computational feasibility in the sense that computer storage requirements and running time depend on the number of states and decisions that the algorithm must consider.

This section will simply present the set of formulas to be used in calculating upper and lower bounds on both the state variables and the decision variables. The full derivations and explanations of these formulas are given in [7].

Before presenting the bounds, certain notational conventions should be explained. Notations of the form (X1t) and u(x1t) will denote the greatest 1t' lower bound (GLB) and least upper bound (LUB) of X1t' respectively. Notations of the form (×3+ X1t' X2t) will denote the GLB of x3t, given specific values of X1t and X2t' namely X1t

A

= X.
1t

= and x

x2t

As in section 3, it is understood that whenever the lower limit of a summation exceeds the upper limit, the summation is taken to be zero. With these notational conventions in mind, table 4 lists all of the formulas needed to calculate the ranges of the variables used in the dynamic programming model.

(dat it' *2t it' = max [0,λ (×2,t+1; *1t - â1t) - ×2t)

It should be emphasized that the DP model has considerably greater generlity than was indicated in the limited application to Washington, D.C. The only model constraint on the data is that they be self-consistent (e.g., Mt and t must be consistent with D+). If, for example, an urban fire department sees Eit to reduce its fleet size because of overkill capacity or perhaps because of declining demand, and the values of M and Nt fluctuate because of a fluctuating budget, then a greater portion of the model's generality could be exploited. The interactions among the variables and parameters of the model which are evident in section 4 should support this contention.

=

On the other hand, time limitations prevented any attempts to examine the model with particular relationships among the parameters. It seems reasonable that certain conditions, e.g., M = Nt = N = constant, or Dt a constant for all t, could lead perhaps to closed-form optimal solutions, or at least might simplify the necessary DP calculations. Further research along these lines is recommended. In addition to these basic issues, there is a need for further sensitivity tests, with respect to the discount rate and the value of U1, for other values of the parameters Mt, Nt D+ and R. For instance, the optimal 't values of the objective f1 (m, 0, 0) could be compared for different values of R (in some reasonable range of maximum ages), leading to an "optimal" value of R (i.e., one which minimizes f1 (m, 0, 0)). Finally, runs with depreciation rate p varying, or using a different (perhaps linear) depreciation policy, would be desirable.

t'

[1]

[2]

McCall, J. J., Maintenance Policies for Stochastically Failing Equipment:
A Survey, Management Science, Vol. 11, No. 5, pp. 493-524 (1965).

Barlow, R. E. and Proschan, F., Planned Replacement, Studies in Applied
Probability and Management Science, K. J. Arrow et al (eds.) pp. 63-87
(Stanford University Press, Stanford, Calif., 1962).

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

Radner, R. and Jorgenson, D. W., Optimal Replacement and Inspection of
Stochastically Failing Equipment, Studies in Applied Probability and
Management Science, K. J. Arrow et al (eds.) pp. 184-206 (Stanford
University Press, Stanford, Calif., 1962).

Zelen, M. (ed.), Statistical Theory of Reliability (The University of
Wisconsin Press, Madison, Wis., 1963).

Dreyfus, S., A Generalized Equipment Replacement Study, JSIAM, Vol. 8,
No. 3, pp. 425-435 (1960).

Balcolm, R. D., A Systems Analysis for the District of Columbia Fire
Department Dealing with the Replacement of Wagon Pumpers, report to the
Washington, D.C. Fire Department (1970).

Ku, R. and Saunders, P. B., Sequencing the Purchase and Retirement of
Fire Engines, Nat. Bur. Stand. (U.S.), NBSIR 74-513 (to be published).
Nemhauser, G. L., Introduction to Dynamic Programming (John Wiley & Sons,
New York, 1967).

Ackoff, R. L., Progress in Operations Research (John Wiley & Sons, New
York, 1961).

Britten, A. A., Decision Making in Vehicle Management, Report No. S.15,
Local Government Operational Research Unit, Reading, England (1971).
Bellmore, M., Howard, G. and Nemhauser, G. L., Dynamic Programming
Computer Model 4, The Johns Hopkins University, Baltimore, Md. (1966).

NATIONAL BUREAU OF STANDARDS SPECIAL PUBLICATION 411, Fire Safety Research, Proceedings of a Symposium Held at NBS, Gaithersburg, Md., August 22, 1973, (Issued November 1974)

F. James Kauffman and Martin E. Grimes

National Fire Protection Association, Boston, Massachusetts

The Research Phase of FIFI defined and evaluated those state-
of-the-art investigation techniques used at the scene of fires to
report on the cause and circumstances of those fires. The two most
significant conclusions reached during the research phase were
(a) that very little literature is available to an officer inter-
ested in studying methods of general fire cause determination, since
most existing literature is arson oriented, and (b) that standard-
ized training programs in systematic investigative practices do not
exist. The Development Phase of the project isolated the most valid
of the investigative techniques evaluated in the research phase, and
synthesized them into a new, simplified, logical, process-of-
elimination investigation sequence known as HEP (Hexagonal Elimina-
tion Process) based on the NFPA Standard 901, Coding System For Fire
Reporting. Two prototype training packages were developed; designed
to aid firefighters in carrying out the systematic HEP investi-
gation sequence in their initial, on-scene, routine fire investi-
gations. The prototype training packages were field tested in six
fire departments of varying sizes and types.

Key words: Field investigation; fire information; fire investi-
gations; fire training; programmed instruction.

The National Fire Protection Association (NFPA) has maintained and published statistics and information on fire and its effects for many years. The information received is used by NFPA's technical committees in the preparation and evaluation of codes and standards. It is made available to any organization engaged in fire research. However, it has been quite apparent that much of the information on fire received by NFPA from various sources is deficient due to lack of adequate field investigation and that the reports received frequently do not contain material pertinent to research needs.

Recognizing this, NFPA employs investigators and utilizes its specialists to make field investigations of special incidents. This must, of necessity, be limited. If a higher standard of reporting from primary sources were available, its usefulness would be reflected, not only in NFPA's operations, but also in providing a sound basis of data for all research workers in fire protection.

In order to make progress towards a uniform national reporting system and with the obvious need for basic information for research purposes, a greater emphasis on sound information gathering in the field is essential.

2. PRIME DEFICIENCY

Fire research requires an input of valid qualitative and quantitative data on fire behavior.

1This work was done under a contract with the National Bureau of Standards.