	TABLE 1

	Notation	Represents

	Decision Variable: Table Games (DV_TG1)	Blackjack
	Decision Variable: Table Games (DV_TG2)	Craps
	Decision Variable: Table Games (DV_TG3)	Roulette
	Decision Variable: EGDs (DV_EGD1)	Nickel EGD
	Decision Variable: EGDs (DV_EGD2)	Quarter EGD
	Decision Variable: EGDs (DV_EGD3)	Dollar EGD

Based on the availability of data, the analysis period in this example, as selected at88, is 36 months. Then, time series revenue data including total revenue generated and total units on the floor per period for 36 months is retrieved, at90, from the FRS4 ofFIG. 1 for DV_TG1(i.e., all blackjack tables). Next, at92, this data is analyzed to determine, using regression analysis techniques for a linear equation, the corresponding mathematical expression that represents the data most accurately. That expression, in this case, is determined to be Equation 1:

y=$12,180*DV_TG1 (Eq. 1)

wherein:
y is monthly revenue generated from blackjack tables on the casino floor (DV_TG1).

Next, at94, as was discussed in greater detail, above, in connection withFIGS. 5A-5B, by reviewing statistical properties (e.g., r²statistic) associated with this example linear equation and any other equation types (not shown), it is determined that Equation 1 represents a “best fit” equation and, thus, represents the proper demand function for DV_TG1.

Then, at96, it is determined if all demand functions have been determined. Since there are six decision variables, in this example, the process is repeated, at90,92 and94, until the demand function for each of the six decision variables is determined. In this example, the six demand functions, as determined, are shown in Table 2:

TABLE 2

Equation	Decision Variable	Demand Function

1	DV_TG1	y = $12,180 *DV_TG1
2	DV_TG2	y = $19,980 * DV_TG2
3	DV_TG3	y = $14,340 * DV_TG3
4	DV_EGD1	y = $2,460 * DV_EGD1
5	DV_EGD2	y = $2,490 *DV_EGD2
6	DV_EGD3	y = $930 * DV_EGD3

Although six linear equations are shown in Table 2, it will be appreciated that a wide range of counts and of different types of equations may be employed, such as, without limitation, polynomial, logarithmic, exponential, or power. Although monthly revenue is disclosed, smaller or larger time periods may be employed.

Next, at98, it is determined whether the objective function is to optimize profit. For this example, the answer is yes. Hence, at100, information is generated on the cost functions that exist within the casino operation that are impacted by the decision variables. Otherwise, if the answer were no, then, the procedure would resume at116 ofFIGS. 5A-5B. This path is discussed below in connection with Example 8.

The relevant gaming departments to be analyzed are determined at100. For this example, those include thetable games department18 and theEGD department16 ofFIG. 1, although casino management could also choose to incorporate the cost functions associated with other departments (not shown) that are related to other gaming units, such as14, and other gaming activities (e.g., casino cage; soft count). Then, at102, time series data including total expenses incurred by thetable games department18 and theEGD department16 along with corresponding number of total gaming units on thecasino floor6, respectively, for each of the example 36 months is retrieved from the FRS4. At104, a distinction is made between fixed and variable costs for thetable games department18 and theEGD department16, and like data is aggregated, at106, in order to produce four separate time series that represent fixed table games costs, variable table game costs, fixed EGD costs and variable EGD costs. As fixed costs are, by definition, not a function of total units on the casino floor, the fixed cost functions for the EGD and

table games departments

16,18 consist simply of constants, at108. These constants, for this example, are determined as shown in Equations 7 and 8:

FC_TG=$16,280 (Eq. 7)

wherein:
FC_TGrepresents the monthly fixed costs in the table games department.

FC_EGD=$21,150 (Eq. 8)

wherein:
FC_EGDrepresents the monthly fixed costs in the EGD department.
The remaining time series data represents variable costs for thetable games department10 and theEGD department12. These time series are analyzed, at110, to determine the mathematical expression that represents the data most accurately. The expression for variable cost table games data, in this example, is determined to be Equation 9:

VC_TG=$9,459*ΣDV_TG. (Eq. 9)

wherein:
VC_TGrepresents the monthly variable costs generated from the table games department as a function of the total number of table game units on the casino floor (ΣDV_TG).

As was discussed in greater detail, above, in connection withFIGS. 5A-5B, by reviewing statistical properties (e.g., r²statistic) associated with the example linear Equation 9, it is determined, at112, that Equation 9 represents a “best fit” equation and thus represents the proper variable cost function for the table games department (VC_TG).

Next, at114, it is determined if all variable cost functions have been determined and the process is repeated, at110 and112, in order that the variable cost function for the EGD department is determined. In this example, the fixed cost constants and variable cost functions are summarized by Table 3:

TABLE 3

Equation	Cost	Function

7	FC_TG	FC_TG= $16,280
8	FC_EGD	FC_EGD= $21,150
9	VC_TG	VC_TG= $9,459 *Σ DV_TG
10	VC_EGD	VC_EGD= $420 * Σ DV_EGD

Although four linear equations are shown in Table 3, it will be appreciated that a wide range of counts and of different types of equations may be employed, such as, without limitation, polynomial, logarithmic, exponential, or power. Although monthly revenue is disclosed, smaller or larger time periods may be employed.

Next, at116, constraints are determined that realistically represent the state of the casino operation. In this example, the constraints are determined and translated into mathematical expressions, at118. For example, the casino floor may be limited to 24,000 square feet. Subsequently, all decision variables are assigned a square footage amount (i.e., the area that they require to be operational). For example, these assignments may be: DV_TG1: 40 sq. ft.; DV_TG2: 100 sq. ft.; DV_TG3: 80 sq. ft.; DV_EGD1: 20 sq. ft.; DV_EGD2: 20 sq. ft.; and DV_EGD3: 20 sq. ft.

In this example, the decision variable values are represented by integers (e.g., there cannot be half of a table game; there cannot be a third of an EGD) and are non-negative.

Finally, given casino management's understanding of their target gaming patrons, minimums and maximums may be imposed on the selection problem in order to assure that the solution is reasonable and, thus, serves the needs of the casino patrons. These minimum and maximum counts are defined for this example as shown in Table 4:

TABLE 4

Minimum	Decision Variable	Maximum

10	DV_TG1	20
1	DV_TG2	10
1	DV_TG3	5
25	DV_EGD1	100
25	DV_EGD2	1000
25	DV_EGD3	500

All demand, cost and constraint expressions for this selection problem are reviewed, at120, to determine if all expressions are linear. This is to determine which optimization process is employed—linear programming or non-linear programming. In this example, all expressions are linear, at122, and, thus, the procedure resumes at124. At124, suitable linear programming techniques are applied to solve for the values of the decision variables that will maximize the objective function and optimize profitability given the set constraints. This provides the solution shown in Table 5:

	TABLE 5

	DecisionVariable	Value

DV

_TG1	20
	DV_TG2	10
	DV_TG3	5
	DV_EGD1	100
	DV_EGD2	500
	DV_EGD3	490

Then, at126, this solution is reviewed for reasonableness. Since this is a workable mix of blackjack, craps, and roulette tables, and nickel, quarter and dollar slot machines that will fit within the 24,000 square foot casino, it is accepted and the example is done at140.

Example 7

This example shows an unreasonable solution and the constraints being adjusted or added at128 ofFIG. 5B. This example is similar to Example 6 except that Table 6 replaces Table 4 and Table 7 replaces Table 5.

As shown in Table 6, given the initial constraints provided in Table 7, the solution at124 ofFIG. 5B suggests that the casino should employ594 blackjack tables and only one of each of the values of the other decision variables (e.g., craps, roulette, Nickel EGD, Quarter EGD, Dollar EGD). This solution is considered unreasonable at126, and the constraints are adjusted at128 to be those shown in Table 4 of Example 6, which would produce a reasonable solution at124 and126.

TABLE 6

Minimum	Decision Variable	Maximum

1	DV_TG1	2000
1	DV_TG2	2000
1	DV_TG3	2000
1	DV_EGD1	2000
1	DV_EGD2	2000
1	DV_EGD3	2000

	TABLE 7

	Decision Variable	Value

	DV_TG1	594
	DV_TG2	1
	DV_TG3	1
	DV_EGD1	1
	DV_EGD2	1
	DV_EGD3	1

Example 8

This example is similar to Example 6, except that revenue is optimized rather than profit, Table 2 is replaced by Table 8, Table 3 is not employed, Table 4 is replaced by Table 9, and Table 5 is replaced by Table 10:

TABLE 8

Equation	Decision Variable	Demand Function

1′	DV_TG1	y = $24,000 *DV_TG1
2′	DV_TG2	y = $19,980 * DV_TG2
3′	DV_TG3	y = $14,340 * DV_TG3
4′	DV_EGD1	y = $2,460 * DV_EGD1
5′	DV_EGD2	y = $2,490 *DV_EGD2
6′	DV_EGD3	y = $930 * DV_EGD3

TABLE 9

Minimum	Decision Variable	Maximum

10	DV_TG1	100
2	DV_TG2	100
2	DV_TG3	100
25	DV_EGD1	100
25	DV_EGD2	500
25	DV_EGD3	500

	TABLE 10

	DecisionVariable	Value

DV

_TG1	100
	DV_TG2	100
	DV_TG3	100
	DV_EGD1	25
	DV_EGD2	50
	DV_EGD3	25

Example 9

This example is similar to Example 6, except that revenue is optimized rather than profit, Table 2 is replaced by Table 11, Table 3 is not employed, Table 4 is replaced by Table 12, and Table 5 is replaced by Table 13:

TABLE 11

Equation	Decision Variable	Demand Function

1″	DV_TG1	y = $24,000 *DV_TG1
2″	DV_TG2	y = $19,980 * DV_TG2
3″	DV_TG3	y = $14,340 * DV_TG3
4″	DV_EGD1	y = $2,460 * DV_EGD1
5″	DV_EGD2	y = $DV_EGD2²+ $400 * DV_EGD2+ $50
6″	DV_EGD3	y = $930 * DV_EGD3

TABLE 12

Minimum	Decision Variable	Maximum

	TABLE 13

	DecisionVariable	Value

DV

_TG1	100
	DV_TG2	88
	DV_TG3	2
	DV_EGD1	27
	DV_EGD2	500
	DV_EGD3	25

In this example, the demand function (Equation 5″) for EGD₂is clearly non-linear (i.e., contains a term with a power greater than 1). Therefore, suitable non-linear programming techniques are employed, at134, to solve for the values of the decision variables that will maximize the objective function (optimize revenue) given the set constraints. In this example, the values are accepted and the example is done at142.

Example 10

This example shows detailed single decision variable variations in historical and current demand. Demand functions are generated from data readily available from theCMS2 and FRS4 ofFIG. 1. Generally, such data is available on a daily basis and can often be obtained on an hourly or by-shift basis. Table 14 shows a typical partial (for economy of disclosure, only some of the data at some of the dates are shown) data series used to determine the demand function for Blackjack tables:

TABLE 14

Date	No. Units (Var. X)	Blackjack Revenue (Var. Y)

Jan. 5, 2004	20	$9,320
Jan. 4, 2004	20	$9,010
Jan. 3, 2004	20	$8,885
Jan. 2, 2004	20	$8,600
Jan. 1, 2004	20	$9,576
Dec. 31, 2003	20	$8,045
Dec. 30, 2003	22	$11,000
. . .	. . .	. . .
Mar. 16, 2000	28	$12,010
Mar. 15, 2000	28	$11,980

Running a regression analysis on the data from Table 14 will reveal the demand function. As this data series includes both generally current revenue data (e.g., yesterday; one week ago; last month) as well as historical data (e.g., last year; two years ago), the resultant curve will show the effects of changes in revenue (variable Y) on changes in number of units (variable X) over both the short and long term. In this way, the demand function incorporates both short-term demand information as well as historical demand information.

Examples 11-14 consider other alternative example constraints that can be imposed in the disclosed method, at116.

Example 11

If the decision variables are units defined by their manufacturer (e.g., Bally Gaming System EGDs; International Gaming Technology EGDs), then constraints may be developed to, for example, force a solution requiring at least a certain minimum number of units for one or more manufacturers in order to take advantage of quantity purchasing discounts. Hence, there may be a plurality of different electronic gaming devices, with one of the decision functions being a minimum count of the different electronic gaming devices from a particular manufacturer. The constraint, thus, associates a discount from the particular manufacturer with the minimum count, and ensures that the discount is received by providing the minimum count with the optimal values, as determined.

Example 12

If the decision variables are units defined by their personality or “theme” (e.g., Double Diamond; Wheel of Fortune), then constraints may be developed to, for example, force a solution requiring a certain maximum number of units for each theme to ensure that there are not too many of one type and thus provide a variety of choices available to the gaming patron. Hence, for example, by employing as one of the decision functions a maximum count of the different EGDs of a particular one of the personalities or themes (e.g., Double Diamond), then the constraint is employed to require that maximum count when the optimal values are determined.

Example 13

Constraints may be developed to force the solution set to adhere to a certain market positioning (e.g., relatively more table games and relatively less EGDs; relatively more EGDs and relatively less table games). Hence, the constraint may require, for example, a greater count of the EGDs with respect to the table games, when the optimal values are determined.

Example 14

Constraints may be developed to consider the physical space available for unit placement. Here, for example, the casino floor may have a physical area, and there may be a physical area associated with each of the different gaming units. The constraint may require that the total combined physical area of the different EGDs and the total combined physical area of the different table games is less than or equal to the total physical area of the casino floor when the optimal values are determined.

Example 15

Although Equations 7-10 of Table 3 show total fixed costs and total variable costs for each of the EGD and table games departments, it is possible to determine an historical fixed cost function and an historical variable cost function for each of the different classes of gaming units. For example, the FRS4 ofFIG. 1 may provide, directly or indirectly, fixed and variable cost functions for each one of thegaming units8, or for each of the classes of the gaming units.

Example 16

Although theCMS2 and FRS4 ofFIG. 1 are shown, data collection with respect to relevant data associated with the gaming units or unit characteristics or classifications may originate from a wide range of sources (e.g., manufacturers; suppliers; taxing authorities; regulatory authorities; industry organizations).

Example 17

Step

90 ofFIG. 5A may obtain the current and historical times series data for the decision variables from a user interface of a financial reporting system, such as, for example, a graphical user interface, a text user interface, a DOS user interface, or any other electronic or automated user interface (e.g., of theCMS2 or FMS4 ofFIG. 1), wherein a user retrieves revenue (“win”) and total unit count time series data by classes (e.g., decision variables).

Example 18

Example 19

As an alternative to or in addition to even steps90-96 ofFIG. 5A, if participation gaming units are/were on the casino floor at any time during the selected analysis period ofstep88, then a separate time series is retrieved that shows total revenue from participation games, along with the number of games on the casino floor for the same time period.

Example 20

As an alternative to or in addition to even steps100-114 ofFIG. 5A, if participation gaming units are/were on the casino floor at any time during the selected analysis period ofstep88, then a separate time series is retrieved that shows total participation royalties (i.e., costs to the casino) and other costs, along with the number of games on the casino floor for the same points in time. These royalties and other costs are segregated between fixed and variable costs at104. A fixed cost function for participation games is determined at108. A variable cost function for participation games is defined using the regression analysis techniques atstep110. Depending on the nature of the variable cost data series, the resultant best-fit equation (e.g., least squares) fromstep112 may be, for example, linear, polynomial, logarithmic, exponential or power.

The disclosed linear and non-linear programming techniques for a gambling enterprise as applied to particular gaming unit types and classes is extremely advantageous to gambling operators. These techniques, more specifically, permit the gambling industry to predict and optimize revenue or profit for a particular gaming unit type or class and to determine exactly what combination of gaming unit types or classes should be deployed on the casino floor for maximum revenue or profit.

While for clarity of disclosure reference has been made herein to the exemplaryuser interface display158 for displaying optimal counts, it will be appreciated that such counts may be stored, printed on hard copy, be computer modified, or be combined with other data. All such processing shall be deemed to fall within the terms “display” or “displaying” as employed herein.

While specific embodiments of the invention have been described in detail, it will be appreciated by those skilled in the art that various modifications and alternatives to those details could be developed in light of the overall teachings of the disclosure. Accordingly, the particular arrangements disclosed are meant to be illustrative only and not limiting as to the scope of the invention which is to be given the full breadth of the claims appended and any and all equivalents thereof.