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 IE I may be determined for the given value of U. Then one need only 1° specify whether r≤rg or r≥ r1 to conclude that the ASAP policy or ALAP policy, respectively, is optimal. of and Nt One run was made with Mt = N. = 10 for all t, with the other data re't maining the same. With r = 0, the ASAP policy resulted; in this case d11 = d21 d12 = d22 = 10, d, = d. = 0 (t=3,4,5). Unfortunately, lack of 1t 2t time prevented further study of this case. Intuitively, one might expect a greater "critical" range of r since the larger values of = and Nt to a greater difference between the ASAP and the ALAP policies. give rise 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 (x,d) 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 The full derivations and explanations of these formulas are given variables. in [7]. 1t' Notations Before presenting the bounds, certain notational conventions should be explained. Notations of the form (X1) and (x1t) will denote the greatest lower bound (GLB) and least upper bound (LUB) of X1t, respectively. of the form (x3t; X1t' x2+) will denote the GLB of X3+, given specific values of X1t and x2t' namely Xit x. = X. and x *2t = X2t. As in section 3, it is understood that whenever the lower limit of summation exceeds the upper limit, 2t the summation is taken to be zero. A With these notational conventions in mind, table 4 lists all of the formulas needed to calculate the ranges of the vari ables used in the dynamic programming model. 't It should be emphasized that the DP model has considerably greater generality 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., M. and Nt must be consistent with Dt) If, for example, an urban fire department sees fit to reduce its fleet size because of overkill capacity or perhaps because of declining demand, and the values of Mt 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. = a constant for all 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., Mt = constant, or Dt 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 Dt and R. For instance, the optimal 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 varying, or using a different (perhaps linear) depreciation policy, would be desirable. 6. REFERENCES [1] McCall, J. J., Maintenance Policies for Stochastically Failing Equipment: A Survey, Management Science, Vol. 11, No. 5, pp. 493-524 (1965). [2] 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 Zelen, M. (ed.), Statistical Theory of Reliability (The University of Dreyfus, S., A Generalized Equipment Replacement Study, JSIAM, Vol. 8, Balcolm, R. D., A Systems Analysis for the District of Columbia Fire Ku, R. and Saunders, P. B., Sequencing the Purchase and Retirement of Ackoff, R. L., Progress in Operations Research (John Wiley & Sons, New Britten, A. A., Decision Making in Vehicle Management, Report No. S.15, 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 stateof-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 interested in studying methods of general fire cause determination, since most existing literature is arson oriented, and (b) that standardized 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-ofelimination investigation sequence known as HEP (Hexagonal Elimination 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 investigation sequence in their initial, on-scene, routine fire investigations. The prototype training packages were field tested in six fire departments of varying sizes and types. Key words: Field investigation; fire information; fire investigations; 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 pro tection. 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. Fire research requires an input of valid qualitative and quantitative data on fire behavior. This work was done under a contract with the National Bureau of Standards. |