Simulating money allocation problems on an analog computer

38

Annales de l'Association internationale pour le Calcul analogique

N ° I - - /anvier 2962

SIMULATING MONEY ALLOCATION PROBLEMS ON AN ANALOG COMPUTER by L u c i e n IV. N E U S T A D T .

*

SUMMARY A method of simulating a management problem on an analog c()mp.uter is described. The problem consists of finding an optimal way of alloting money to a number of projects. Both the mathematical model ,and the method of simulation - - including the representation or input and output - - are described.

Introduction. One of the many problems which occurs in operaeions research is that of the proper allo'cation of resources. Typically the problem is as follows: A fixed sum of money is available. There exist n projects among which the money is to be distributed. For each project the return depends on the sum of money allotted to it. How should the money be distributed so as to obtain an over-all optimum return ? In this paper a simple mathematical model is constructed, and a method for simulating this model on an analog computer is described. The advantages of using an analog computer are shown by the ease with which parameters may be introduced into the computer, and the convenient form in which the output may be displayed. Description o] the Problem. A typical problem facing a business or government executive is the following: A sum of M dollars has been allotted in a particular budget, These M dollars are to be distributed among n projets. For each project there is a (more or less wall-known) function which relates the time T, to complete the i-th project to the money M, allotted to it. Let us consider that an optimal allocation is one that minimizes ~ TL. In a certain way this represents equal priority. For once this optimum allocation has been made, taking money from one project, and giving it to, another, will result in a loss in time in the first system at least as great as the gain in time in the second. From a mathematical viewpoint, if the functions T|(Mt) are differentiable, an optimal allocation implies finding values of the M, such that all the derivatives dTi/dMi are equal. Various modifications may be-made to the above formulation. For example, the projects may be weighted,. so that the function to be minimized is £ at T t ,

where the a~ are positive weights, which may even depend on T i . Constraints may also be introduced, for example, that the time T1 must not exceed a fixed amount T1. A detailed description of the problem is given by Salveson [2]. For a discussion of the use of analog computers in operations research--including an excellent bibliography--see reference [1], Chapter 10.

Analysis. First consider the problem of optimizing 'the allocation of money to n projects having equal priority with the total amount of money, M, fixed. Assume that the time-cost functions are hyperbolas 4~, i.e., that M, = A,/(T, - - Ct) + B,

(1)

where Ml is the money available for the i-th project, Tt is the time to complete it, and A t , Bt and Ct are parameters. It was shown above that the optimal allocation of money can be found b y determining a constant r such that = r and

(aM'-aT')T,/=

MdTi* ) = M.

(2)

The M dollars are best distributed by allotting the i-th project Ml ~ = Mt(Tt "~) dollars. From (1) dMt/dTt

=

_

a l l ( T , - - Ci)- =

- - B,)2/A,

__

or

Ml = V - - A I T, =

(dMl/dTi) q- B l ,

and

V'--&/(aM,/dT,) + Ci.

(3) (4)

If dMl/dTl = r, then = M, =

I

* Space Technology Laboratories, Inc., Los Angeles, California. Manuscrit requ le 10 avril 1961. ** The author wishes to express his thanks to Melvin E. Salveson for his suggestion and form~ation of the problem; to the Reeves Instrument Corporation for lending the an~og computing equipment; arid to Paul Wilbur, Edwin Boy0.n (now with the Thompson Ramo-Wooldridge Corpora~or0, and Major General Austin Davis of the U.S. Air Force Air Materiel Command for their help and cooperation.

i ----- 1 ..... n

i

V'

# £ V A--] + £ B t , t

i

Thus, if M is given, r is given by

:~ V &

J

* This is an excellent approximation in many observed cases,

L.W. Neustadt : Simulating money allocation problems Method of Computation. It is clear that r is a monotonic function of M. Conversely M is a monotonic function of r. This property was used in the simulation. The parameter r was made an independent variable, and the M, as well as the T, were computed as a function of r from formulas (3) and (4), with r = d M , / d T , . The computation was stopped when £ M1 was equal to M. At this point the variables M, and T l w e r e read out. If the problem is modified so that each project is to have its own priority, then equation (2) is modified to

dMl/dTl = m r where at is the index of priority of the i-th project. The computation is done as before, with expression (5) substituted into relations (3) and (4). If certain times are to be fixed, e.g., T, --- T~, then M, must be obtained from (1) with T, = T1 and i = 1. All other Ml are computed as a function of r as above. The resultant solution gives the optimal allocation of M - - M~ to the projects 2 . . . . . n. One of the principal advantages of an analog computer is the simplicity of the input and output, particularly in the <~feel >>that the operator of the computer is given. The principal outputs are the completion times Ti for each project. These were presented in the form of a Gantt chart, which is a bar graph of the Tl (See Figure 1). The principal inputs are M (or T), and the timecost functions. The total money M was adjustable on a <> and was read on a digital voltmeter. The time-cost hyperbolas were shown on an osdlloscope. The hyperbola was presented by representing equation (1) parametrically: Mi = ~/'Ai e t + Bl TI = ~v/'AI e -t -{- CI where t is computer time. These functions were generated (over a finite t interval) on a repetitive computer, and the outputs were placed on the vertical and horizontal deflections of the scope. The operator could change the parameters A , , B i and Cl and visually see the effect on the curve. Evidently the time-cost curves were presented and adjusted before the actual computation was begun.

Equipment Requirements. The' total amount of computing equipment used for the simulation was as follows: ,, , 7 . , [ .: i .,,,:'~.:..i. 9~:..,i; '.!'if?::"• iq

WE*PoNS~SVEM ' WEAPON SYSTEM 2

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" ". ': ;

WEAPON SYSTEM 5

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- . I.'

WEAPON SYSTEM A I, ',":

[: ;:"

.'

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, . " . , ; '" , | " " '.

',;,

':.i, ,7.,';,

, (,

~'

":.i':i,-:. ~ ':i'[)[[':]

:

0

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Results. A pilot problem was demonstrated to the U.S. Air Force Air Materiel Command simulating six projects (weapon systems) with m = 2. Two Reeves Electronic Analog Computers (REAC's) were used. In the demonstration a Gantt chart would be plotted for a predetermined sum of money. A new chart would then be plotted for a smaller sum of money, showing the stretch outs on each of the programs. Then a constraint would be introduced on one of the systems and a new Gantt chart drawn, to show how redudng the time of this system would result in delays in the remaining systems. Finally the time cost curves for each system would be shown on the oscilloscope, and would be changed at random, after which any o.'t: the above steps could be repeated. The feasibility of using an analog computer for such simulations was thus well demonstrated. However, a more sophisticated (and realistic) model will undoubtedly be necessary--as wall as a larger and more complex analog computer--if realistic problems are to be simulated. REFERENCES

,.

I

3 TIME UNITS

Fig. 1, - -

(1) One integrator to generate ~ / - - r.' For simplicity the parametric function V - - r was set equal to t - - t o where t o was some sufficiently small positive quantity, and t was computer time. (2) A division circuit, typically requiring one amplifier and one servo multiplier to generate 1 { # - - r [ (3) One summing amplifier for each T l , and one for each M1. (4) One summing amplifier for M - - ~ M l . (5) For each option to fix a T l , one division drcuit to compute the corresponding M,. Typically this circuit includes one servo multiplier and two amplifiers. (6) To generate the exponential functions for the time-cost curve display, two integrating amplifiers in a repetitive mode, as well as four summing amplifiers. An oscilloscope is required for the display. (7) A stepping switch arrangement to generate the Gantt chart. This also required one integrating amplifier and one high gain amplifier. The plot was drawn on an X-Y plotter. In summary the amount of computing equipment required is : 11 -{- 2 n + 2 m amplifiers 1 + 2 m servomuItipliers 1 oscilloscope 1 X-Y plotter. Here n is the number of projects, and m is the maximum number of TI which may be fixed simultaneously. In addition a number of scale factor potentiometers, approximately two per amplifier, are necessary,

([d

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WEAPO,~ SYSTEMS~ WEAPON ~YSTEM 6

"

39

A Typical Gantt Chart,

4

[1] A.S. Jackson: Analog Computatio.n, McGraw-Hill, New York, 1960. [2] M.E. Salveson : <~Principles of Dynamic Weapons System Programming ~>,Naval ResearchLogistics Quarterly, futu,e issue,