Process which aids to the laying out of locations of a limited number of 100, personnel and equipments in functional organization

Abstract

A method which is of aid in the laying out of locations of personnel and equipments in functional organizations where the personnel and equipments do not exceed numeral 100 and can be represented as density units in rectangular or square lattices. The relationship among objects in a particular space can be accurately determined to minimize crowding. The method has utility in laying out military attack center items of equipments being operating upon by center personnel for purposes of enhancing man-machine interfaces throughout the attack center. The method of the invention may be used alone or in combination with other techniques such as multidimensional scaling (MDS) to improve the accuracy of such other techniques.

Claims

What is claimed is:

1. A method which aids in laying out spatial interrelationships among a plurality of objects including a predetermined number of items of equipment being operated upon by a predetermined number of people within a predetermined physical area, where said plurality of objects constitute a unitary functional organization, where said objects do not exceed 100 in number, and where functioning of said organization can be hindered by crowding, the method comprising the steps of:

(a) defining an initial configuration of said objects within said predetermined physical area;

(b) acting out a first scenario involving said objects within said predetermined physical area and recording movements of said predetermined number of people within said physical area performing at least one task, said recording step comprising photographing said movements of said predetermined number of people using time lapse photography;

(c) introducing information about said recorded movements of said predetermined number of people and said initial configuration into a computer;

(d) generating information on all possible locations of said predetermined number of people and said equipment in said predetermined physical area by constructing a finite number of experimental configurations of all said objects within said predetermined physical area with said computer;

(e) determining a first relationship between (I) the relative positioning of the objects within the predetermined physical area and (II)(i) a first distance among all possible pairs of the objects, (II)(ii) the number of objects, and (II) (iii) a magnitude of the predetermined physical area for said first scenario using said information from said time lapse photography and for all of said finite number of experimental configurations constructed by said computer;

(f) determining a minimum density measure among said predetermined number of people within said predetermined physical area;

(g) determining a maximum density measure among said predetermined number of people within said predetermined physical area; and

(h) selecting a set of relationships of said objects in said predetermined physical area wherein said first relationship for said selected set is at least equal to said minimum density measure and no more than said maximum density measure.

2. The method of claim 1 further comprising:

(i) adjusting the number of objects within said predetermined physical area;

(j) acting out a second scenario involving said adjusted number of objects and recording movements of people within said predetermined physical area using said time lapse photography; and

(k) repeating steps (c) through (g) when said method with said first scenario does not yield a first relationship which falls between said minimum density measure and said maximum density measure.

3. A method as set forth in claim 1

(l) wherein the relationship stated in step (e) varies (h) directly with the number of objects for a constant predetermined area; (ii) indirectly with the predetermined physical area for a constant number of objects; and (iii) indirectly with said distance among pairs for a constant number of objects and for a constant predetermined physical area.

4. A method as set forth in claim 1

(m) wherein the relationships stated in step (e) are defined as follows: ##EQU37## (n) where d.sub.act is the average Euclidean distance among all possible pair of said objects, A is the predetermined physical area, and n is the number of the objects within the predetermined physical area.

5. The method of claim 1

(o) wherein the predetermined physical area represents an area within a submarine available for the location of a control center, said control center constituting a unitary functional organization for controlling at least one aspect of the submarine's operation;

(p) wherein said predetermined number of people are crew members for said submarine and said predetermined number of items of equipment are located in said control center, each of said crew members having functional assignments in said control center in association with the operation of said items of equipment; and

(g) wherein said selecting step comprises selecting a set of relationships which yields a lay out for the control center having the items of equipment and the crew members positioned so that the crew members can carry out respective assigned functions in connection with said items of equipment with the least overall hinderance due to crowding within the control center.

Description

CROSS-REFERENCE TO RELATED APPLICATIONS

This patent application is co-pending with two related patent applications, respectively, Ser. No. 07/754,779, filed Aug. 30, 1991 entitled "Process Which Aids in the Laying Out of Locations of Personnel and Equipments in Functional Organizations, Including Features of Special Applicability Where a Capability for Handling an Unlimited Number of Personnel/Equipment Data Items is Desired" and Ser. No. 07/756,264, filed Aug. 30, 1991 "Process Which Aids in the Laying Out of Locations of Personnel/Equipments in Functional Organizations, Including a Feature of Use of Abbreviated Calculation Routines", both by the same inventor and filed on the same date as this patent application.

BACKGROUND OF THE INVENTION

(1) Field of the Invention

This invention provides a more accurate and flexible method for producing spatial layouts of objects (including equipments, personnel, or combinations of equipments and personnel) under circumstances in which it is important to consider discrete spatial density in any rectangular or square lattice containing 100 or less density points. Thus, the relationship among objects in a particular space can be accurately determined to minimize crowding.

(2) Description of the Prior Art

The conventional formula to measure or model two-dimensional discrete spatial density; i.e., population density or physical crowding is defined as the average number of objects (n) per unit area of space (A):

D=n/A (1)

This definition has severe shortcomings since actual spatial orientation within a specified area is disregarded. As an example of this shortcoming, refer to FIG. 1 which displays three different configurations of objects or density points. In each case, the "perceived density" of the four points is obviously different. Since the number of points and area are identical in each depiction, there is a constant value of 0.25 for Population density. FIG. 2 depicts geometrically the population demographer's model of population density shown for the distributions in FIG. 1. FIG. 2 shows that each point occupies four space units (such as feet); hence, population density or physical crowdedness (D=n/A) equals one object per four square feet. FIG. 2 represents the model for each depiction of FIG. 1. However, large differences in perceived physical crowding clearly exist among the three configurations shown in FIG. 1.

A formula was then derived by the inventor to capture the differences shown in FIG. 1 more accurately by taking the actual spatial orientation of objects into account. See O'Brien, F., "A Crowding Index for Finite Populations", Perceptual and Motor Skills, February 1990, 70, pp. 3-11 which is incorporated into this disclosure in its entirety by reference.

This formula, referred to as the Population Density Index (PDI), is as follows: ##EQU1## where n=number of objects

A=the geometric area, and

d=average Euclidean distance among all possible pairs of n objects.

Basically, the above proposed formula is a generalization of the bivariate Euclidean distance formula. The derivation of the proposed formula is patterned on the well known square-root law used in the physical sciences.

Assume two objects are plotted on an X, Y Cartesian coordinate system with a fixed origin O. The mathematical distance between the two objects is measurable by simple analytic geometry using the Pythagorean distance formula:

d.sub.12 =[(X.sub.1 -X.sub.2).sup.2 +(Y.sub.1 -Y.sub.2).sup.2 ].sup.1/2( 3)

where (X.sub.1,Y.sub.1); (X.sub.2,Y.sub.2) represent each object's coordinates.

If, now, we conceive of n objects, each given coordinates within the same geometric plane such as a room, it is possible to generalize the above formula to obtain an average Euclidean distance among the n objects. The average Euclidean distance of n points, considered pairwise, is given by: ##EQU2## where d.sub.ij is the Euclidean distance between any two objects. Note that for n=2 objects, d;d.sub.12 are equivalent.

The last step in deriving a density index is to scale d to adjust for a given number of objects residing within a specific area. A proposed general formula based on the square-root inverse law for distances incorporating size of area and the number of objects is: ##EQU3## where A=the geometric area in which objects reside, and

n=the number of objects within one area.

Dimensional analysis, as well as empirical Monte Carlo simulation investigations, of .DELTA. shows that the units are: ##EQU4## Essentially, .DELTA. is the average pairwise Euclidean distance among n objects scaled for a given unit area. As will become evident in the following numerical example, .DELTA. is inversely related to the average geometric distances among n points. Calculating the reciprocal of .DELTA., 1/.DELTA. will make the relationship monotonically increasing, that is, the more densely packed the objects, the higher the value of the index. This reciprocal of .DELTA., or 1/.DELTA., is arbitrarily referred to as the population density index, or PDI. The units for PDI are .sqroot.n/A.

A computational example is provided with the aid of FIG. 3. For four points, there are 4.times.3/2=6 pairwise distances to calculate. The coordinate points for the 4 units are (1,1), (1,3), (2,4) and (3,2). The area shown is 16 units. Applying .DELTA., ##EQU5## Calculating the reciprocal of .DELTA. and multiplying by 10 to give integer results, PDI=2.3.

The .DELTA. index appears to be valid even when areas differ by a large amount. To demonstrate this, consider FIGS. 4A and 4B. The average Euclidean distances are identical (1.6) in each situation depicted. The smaller value of .DELTA. in FIG. 4A (3.7) is in accord with the basic interpretation of .DELTA., that is, the smaller the value of .DELTA., the more densely packed are the points relative to the allowed area. The results also correspond to the intuitive notion of density.

The proposed crowding index, .DELTA. or PDI, should be interpreted as a relative measure much like a standard deviation in statistics. The theoretical mathematical minimum value of .DELTA. or PDI is always 0, a condition realizable with dimensionless points but not realizable with solid objects such as people.

The maximum value depends on the number of objects and the geometric area. Beyond three or four objects, it becomes difficult and perhaps meaningless to attempt calculating a precise maximum value of .DELTA. or PDI. For these reasons, hypothetical minimum and maximum bounds of the PDI formula are derived below and presented as an integral component of this disclosure. Three additional properties derived from the square-root law for average distances of .DELTA. or PDI appear to be critical to the usefulness and interpretability of the index: 1) for constant area, PDI varies directly with the number of objects; 2) for a constant number of objects, PDI varies indirectly with area; and 3) for a constant number of objects and constant area, PDI varies indirectly with distance. Small sample Monte Carlo simulations performed by the inventor have supported these square-root properties for the PDI formula. The values of PDI computed from randomly selected uniform distributions correlated 0.96 with the conventional formula for population density (n/A).

In addition, the PDI formula can be evaluated on three key scientific criteria. First, the model is very simple. It connects population density to three key variables--distance, number of points and area--through an equation that can be readily calculated. Second, the formula is justified by mathematical analysis. The inverse square-root properties of the index stated as conjectures are very reasonable and provide a context for prediction and explanation of observed results. Monte Carlo simulations support each conjecture, thereby providing preliminary justification until large scale simulations can be conducted. Third, the formula has been tested and verified by empirical research. The use of the formula in hypothetical military settings has produced results that were readily interpretable and which correlated with qualitative estimates of crowding made by independent expert observers.

The population density formula attempts to express differences such as those shown in FIG. 1 more accurately than the conventional population density formula. Since the index can vary widely, as indicated in FIG. 1, it was necessary to develop a new model to predict minimum and maximum bounds of the population density index values. It was in this light that the present invention was conceived and has now been reduced to practice.

SUMMARY OF THE INVENTION

The present invention concerns this newly developed model which provides a more accurate interpretation of the population density index values by predicting the minimum and maximum bounds of such population density index values. This model is designed for small sample research in the area of population density analysis. The population density index model includes a measure of interperson distance and is scaled for a geometric area enabling measurement of population density with greater precision and flexibility than the conventional population density model. The population density index model of the invention is designed for small-scale projects involving as few as two and as many as one hundred density points.

A review of the scientific literature and discussion with experts in the field of micro-population demography, resulted in the decision to provide the model for up to only 100 density points. This quantity is likely to be useful in a great majority of the tasks of researchers and other scientists or engineers involving concentrations and activities of small populations of objects, such as control rooms for various functional operations (including attack centers on warships), prisons, school classrooms and other micro-populations. It is to be understood that the term "Populations of objects" includes equipments, personnel, and combinations of equipments and personnel.

The advantage of the improved population density model is that it measures population density for 100 or less points with greater precision and flexibility, thereby enhancing its utility for the prediction of minimum and maximum bounds of the PDI value.

Later herein, in the section entitled "Description of the Preferred Embodiment" the PDI model is applied to a hypothetical military setting. However, it can be used in any application for the measurement of discrete spatial population density for density points of 100 or less. Also, this PDI model need not be limited to people. It can be applied to any representable two dimensional configuration of points as diverse as stars, microscopically observable data such as living micro-organisms, automobile traffic, aerial photographic data and the like. By rational extension, the model can be applied to three dimensional configurations such as multi-floored structures.

Accordingly, the principle objects of the present invention are:

(1) To provide a process which aids in the laying out of the locations of personnel and equipments in a functional organization in a quadrilateral area, and which enables the persons tasked to perform the layouts to predict minimum and maximum bounds of a population density index value.

(2) To provide a process which aids in the laying out of the locations of personnel and equipments in a functional organization in a quadrilateral area, and which provides a high level of precision in the discrimination of adverse crowding effects as between compared layout solutions.

(3) To provide a process which aids in the laying out of locations of personnel and equipments in a functional organization in a quadrelateral area, and which is applicable for a range of personnel/equipment data items between 2 and 100.

(4) To provide a process which aids in the laying out of locations of personnel and equipments in a functional organization in a quadrilateral area, and which is universal in its applicability to subject matters that can be modeled as discrete spatial population density points.

BRIEF DESCRIPTION OF THE DRAWINGS

A more complete understanding of the invention and many of the attendant advantages thereto will be readily appreciated as the same becomes better understood by reference to the following detailed description when considered in conjunction with the accompanying drawings wherein:

FIGS. 1A, 1B, and 1C are different examples of population density configurations depicting different positioning, respectively, of the same number of density points in a uniform lattice;

FIG. 2 depicts the population demographer's model of population density for each of FIGS. 1A, 1B, and 1C;

FIG. 3 is a depiction of four density points plotted on an X, Y Cartesian coordinate system with a fixed origin 0;

FIGS. 4A and 4B depict three density points, with identical average Euclidean distances between them but positioned, respectively, on different sized areas;

FIG. 5 is an example of a 2.times.2 "unit" lattice containing four connected density points;

FIG. 6 depicts a 4.times.3 "unit" lattice containing twelve density points;

FIGS. 7A, 7B, 7C, and 7D each represent a 5 ft.times.5 ft area containing twelve points; FIG. 7A depicting a minimum density situation, FIG. 7B depicting a maximum density situation, FIG. 7C depicting an actual observed density situation, and FIG. 7D depicting a hypothetical lattice translation of FIG. 7C;

FIG. 8 is a bar graph presenting values of PDI for four major phases of a hypothetical submarine scenario and comparing PDI values for one phase in submarines of two different sizes; and

FIG. 9 is a flow chart which presents the process of spatially locating equipments in a submarine attack center.

DESCRIPTION OF THE PREFERRED EMBODIMENT

For the remainder of this disclosure, reference will be made particularly to FIGS. 5-8.

The process which aids in laying out of personnel and/or equipment locations within a functional organization in accordance with the present invention includes provision of a novel modeling technique, which is performed in four steps. First, given the sample size of density points and geometric area of interest, the points are plotted in a lattice or uniform, that is, two dimensional grid, distribution, that is, in a checkerboard arrangement with every consecutive horizontal and vertical point being equidistant. Secondly, using this plotted distribution, two theoretical indices are calculated--a lower bound density index and an upper bound density index. Thirdly, the data regarding the personnel and/or equipments which are the subjects of the layout are collected, and the population density index values are calculated. The inventor has proven mathematically that the actual PDI (PDI.sub.act) is bounded by the minimum and maximum PDIs. Fourthly, the "effective interperson distance" index is calculated based on the "effective interpoint distance" index (or where personnel crowding is the principle consideration the "effective interperson distance" index ), and the research findings are compared to the model indices.

The model enables an evaluation of the different layout solutions explored as part of a given layout task, and enables comparisons with solutions provided in connection with other layout tasks. The lattice or uniform distribution is an effective visual aid for demonstrating how population density changes with dynamic human or equipment positioning being recorded at will.

The lower bound of the population density index can be calculated for any uniform arrangement of points so long as the number of points does not exceed 100. The lower bound is based on a lattice of the integers called a "unit lattice". In general, a unit lattice means a uniform distribution of n points in area A such that n=A. This implies that the interpoint distance of consecutive horizontal and vertical points is always equal to 1. A nonunit lattice will mean that n and A are not equal. In the special cases of n=2 and n=3 objects, "unit lattice" means either a unit line segment (n=2) or, for n=3, a Euclidean equilateral triangle (perimeter=3 units), each constructed in the interior of A. A linear dimension is herein designated in feet.

FIG. 5 is an example of a 2.times.2 unit lattice. In FIG. 5, note that the area is 4 ft.sup.2 and that the number of points is 4, or n=A. The horizontal and vertical distance between each of the consecutive points is equal to 1. This is derived from a simple relation that provides interpoint distances of lattices. Namely, if .delta. denotes the interpoint distance, then ##EQU6## where .delta. is in feet.

The above example illustrates the approach that is used for approximating interpoint distances for any lattice of n points uniformly distributed in area A. The general formula to do this is given by: ##EQU7##

The next step, calculating the average Euclidean distance of all possible pairs of points in FIG. 5, is given by: ##EQU8##

The average Euclidean distance for a unit lattice is called .DELTA. to distinguish it from the general Euclidean distance given by d in the general population density index formula: ##EQU9##

The population density index value for the unit lattice of FIG. 5 can now be calculated. To do this, first substitute the numerical value of .DELTA. (i.e., 1.14) from the above calculation, into the general population density index formula: ##EQU10## to wit: ##EQU11##

Table 1 shows computed values of .DELTA. for representative unit lattice distributions for example sizes not exceeding 100 density points.

                  TABLE 1
    ______________________________________
    EUCLIDEAN DISTANCE VALUES FOR SELECTED
    UNIT LATTICES (IN FT)
    Lattice                Lattice
    (n = Area) .sup.-- .DELTA.
                           (n = Area)
                                     .sup.-- .DELTA.
    ______________________________________
    2 .times. 1
               1.00        7 .times. 4
                                     2.97
    2 .times. 2
               1.14        7 .times. 5
                                     3.19
    3 .times. 1
               1.00        7 .times. 6
                                     3.43
    3 .times. 2
               1.42        7 .times. 7
                                     3.68
    3 .times. 3
               1.63        8 .times. 2
                                     2.97
    4 .times. 2
               1.71        8 .times. 3
                                     3.09
    4 .times. 3
               1.90        8 .times. 8
                                     4.20
    4 .times. 4
               2.14        9 .times. 2
                                     3.29
    5 .times. 2
               2.01        9 .times. 3
                                     3.41
    5 .times. 3
               2.19        9 .times. 9
                                     4.72
    5 .times. 4
               2.41        10 .times. 2
                                     3.62
    5 .times. 5
               2.65        10 .times. 3
                                     3.72
    6 .times. 2
               2.32        10 .times. 4
                                     3.88
    6 .times. 3
               2.48        10 .times. 5
                                     4.07
    6 .times. 4
               2.69        10 .times. 6
                                     4.27
    6 .times. 5
               2.92        10 .times. 7
                                     4.50
    6 .times. 6
               3.18        10 .times. 8
                                     4.74
    7 .times. 2
               2.65        10 .times. 9
                                     4.98
    7 .times. 3
               2.78        10 .times. 10
                                     5.24
    ______________________________________

                  TABLE 2
    ______________________________________
    Application of Model To Four Stages Of Hypothetical
    Antisubmarine Warfare Scenario* (Area = 436 ft.sup.2)
    Stage n     .sup.-- .DELTA.
                       .delta.
                           PDI.sub.min
                                 PDI .sub.max
                                        PDI.sub.act
                                              .delta..sub.eff
                                                   .delta./.delta..sub.eff
    ______________________________________
    Transit
          11    2.32   6.0 119   715    140   5.1  1.18
    Con-  13    2.65   5.6 121   676    160   4.2  1.33
    tact
    Pros.
    Battle
          20    2.78   4.6 173   789    250   3.2  1.44
    Stats.
    Firing-
          20    2.78   4.6 173   789    350   2.3  2.10
    point
    Proce-
    dures
    ______________________________________
     *PDI values scaled by 10.sup.4.
      Assumes 1ft uniform interpoint distance.

• Inventors
  O'Brien, Jr., Francis J.;
• Assignee
  The United States of America as represented by the Secretary of the Navy (Washington, DC)
• Published
  Aug-10-1993
• Current US Classes:
  703/2
  705/8
• Application #
  754789
• International Classes
  G06F 015/20
• Field of Search
  364/400 364/401 364/402
• Examiner
  Envall, Jr.; Roy N.
• Agent
  McGowan; Michael J., Lall; Prithvi C., Oglo; Michael F.