AN INTERPOLATION APPROACH TO DEVELOPING MATHEMATICAL FUNCTIONS FOR BUSINESS SIMULATIONS Developments in Business Simulation & Experiential Exercises, Volume 13, 1986 AN INTERPOLATION APPROACH TO DEVELOPING MATHEMATICAL FUNCTIONS FOR BUSINESS SIMULATIONS Kenneth R. Goosen, University of Arkansas at Little Rock ABSTRACT Simulation designers are often required to define the precise nature of functional relationships with only vague guidelines from theory or practice. This paper presents a methodology for the easy implementation of self-designed functional relationships. The approach presented does not require any special parameters or exponents; yet it is capable of generating results almost identical to sophisticated curvilinear functions. The method described in this paper relies on interpolation as the means of obtaining dependent variable values from a graphical representation of a functional relationship. INTRODUCTION The design of general management business and functional area simulations involves writing scenarios, selecting decisions, formatting financial statements, creating fictional historical data, and developing mathematical functions. The results of all these activities except the mathematical equations are visible through description in the student simulation manual. The scenario with its descriptive details gives the simulation a static appearance of reality; however; it is the set of mathematical equations embedded in a computer program that provides the dynamics of simulated realism. Game designers for the most part have relied upon mathematical equations to create quantitative relationships among business simulation variables. A major problem with mathematically defined functional relationship equations is that these equations tend to control the magnitude and breadth of simulation parameters. In some cases the equations have given undesirable results at certain decision points and the game designer has been forced to create artificial limits to restrict certain decisions. Analysis by Gold and Pray of currently used games showed that game designers have experienced considerable difficulty in developing equations that give the desired results over a wide range of decision values. /1/ A game normally contains a number of different functional relationships. For many relationships the exact functional behavior is not prescribed by theory or indicated by empirical data. For relationships involving credit, wage rates, commission rates, salaries, etc. equations that give the results desired by the game designer or required by theory may not exist or be difficult to develop. Also, in instances where theoretical functional relationships have been developed, the mathematical complexity of the equations may be beyond the expertise of the game designer to implement. Because game designers are often required to define the precise nature of functional relationships with only vague guidelines from theory, an easy method of implementing self-designed functional relationships is desirable. This paper presents such a methodology. The approach presented does not require any special parameters or exponents; yet it is capable of generating results almost identical to sophisticated curvilinear functions. The method presented in this paper requires that the game designer first graphically define the desired functional relationship. The graphical representation of the functional relationship may have multiple inflection points and multiple minimum and maximum points, if this is desired by the game designer. The method described in this paper relies on interpolation as the means for obtaining dependent variable values for every possible independent variable value. The technique does not in any way prescribe the nature of the functional relationship. However, the methodology presented can duplicate and improve the results of mathematical functional relationships that tend to deteriorate at certain levels of activity. The advantage of the interpolation approach presented is that it gives the game designer complete control output over the entire prescribed range of sales and production activity. BASIC THEORY OF DEVELOPING FUNCTIONAL RELATIONSHIPS Various types of equations may be found within the simulation computer program. They vary from the simple to the complex. While some are linear in nature, others may be curvilinear. Some are fairly complex equations which serve to determine units manufactured and units sold. Others simply serve as counters to determine balances for financial statements. Of the various types, the functional, equations are those most critical to simulating reality of economic forces. Functional equations are equations in which the independent variable (usually a decision value) is an input into a function which determines the value of the dependent variable. The resulting dependent variable value is either an output for financial statements or an input value for other functional equations. The existence of functional equations creates opportunities for students to make optimal decisions. Without these equations, the results of game play would have no relationship to reality. Increasing values for price or advertising, e.g., would have no adverse consequences. In an earlier paper, Goosen /2/ established the importance of functional relationships. “If a business simulation is to be realistic, the same general relationships found in the real world must be found within the simulation. In order for a simulation to have the needed air of realism and dynamism, mathematical functions must be created to simulate these economic forces.” Examples of decisions that require functional relationships with points of inflection or minimum and maximum values are price, advertising, research and development, wage rate, credit, salesmen commissions. 248 Developments in Business Simulation & Experiential Exercises, Volume 13, 1986 A major problem in the development of functional relationships is the creation of relationships that (1) are supported by theory, and (2) have minimum and maximum values or inflection points. Very little research has been published concerning the development of functional equations for business games. A notable exception is the work done by Gold and Pray /3/. Gold and Pray have done an outstanding job in several papers presented at ABSEL in setting forth procedures for the development of market demand equations. Their approach represents a major advancement in business simulation modeling. Their work should be read and understood by serious game designers. However, their model is an application to a specific area, market demand. The interpolation approach in this paper is general and can be applied to any decision area of a simulation. An ideal functional relationship is one in which for a range of desired decision values (the independent variable) a maximum or minimum value exists or contains points of inflection. In some functions, inflection points are necessary in order to simulate real world behavior. Mathematically, an inflection point is the point in a line where there is a change in the curvature from concave to convex or conversely. For total cost curves this means that the related marginal cost and average cost curves have minimum values. A graphical example of a total cost curve and the associated marginal cost and average cost curves is shown in Figure 1. Points of inflection are desirable in business simulation functional relationships because they create increasing and decreasing returns. For example, this means that successive equal increases in advertising would have differing effects. At some point the additional cost of advertising would exceed the additional revenue. AN INTERPOLATION TECHNIQUE FOR DEVELOPING MATHEMATICAL FUNCTIONS Satisfactory mathematical equations that have inflection points or maximum and minimum values at the desired points over a desired range of values are difficult to develop. In many cases equations that appear suitable only give desired results over a limited range of values. Furthermore, in order to control the output values within the desired limits of economic activity, the creation of parameters (constants) is often necessary. A satisfactory functional relationship is one that (1) provides a realistic response to a decision value, and (2) produces realistic values over the entire range of sates and production activity. The interpolation method covered in this paper gives the simulation designer control over desired output values over a specified range of activity. Although some of the elegance and symmetry of a true curvilinear functional equation containing inflection points is sacrificed, the results of the interpolation approach are not significantly different. Therefore, the simplicity of design and the ability to control desired results justifies the loss of mathematical elegance for mathematical manipulation. The interpolation method to be described is a technique that attempts to approximate functional equations that have minimum or maximum values or that have points of inflections. Interpolation is an operation for determining a number or value not listed in a table or array of numerical values by deriving the missing value on the basis of proportions computed from the numbers that are listed in the table. The interpolation model for computing functional relationship values is described in Appendix A. A computer program for applying this model is presented in Appendix B. The interpolation method for developing mathematical functions for business simulations involves the following steps: 1. Prepare a graph having the desired calibrations and draw or sketch the desired curvilinear functional relationship. 2. Select points at regular intervals on the drawn functional relationship in order to prepare an interpolation schedule. 3. Prepare a schedule of interpolation values by measuring the coordinate values of the points where each linear line connects to another linear line. These ordered paired values may be called range values. 4. For any independent variable value not listed in the table of ordered pairs of values, compute the missing dependent value by using the interpolation equation presented in Appendix A. Illustration Assume that a functional relationship between the wage rate and production rate per hour is desired. Furthermore, assume that up to a given wage rate the returns are increasing followed by decreasing returns with increases in wages. To accomplish step 1, create a graph and draw the desired functional relationship as shown in Figure 2A. Then as illustrated in Figure 2B, determine points on the drawn functional relationship to serve as the basis for an interpolation schedule. The effect of this process is to create a linear approximation of the functional relationship as illustrated in Figure 2C. For step 3, identify the corresponding coordinate values of each point (see Figure 2B) and then prepare a table showing arrays for the X (wage rate) and Y (PRPH) values, as illustrated in Figure 2D. Values not explicitly shown in the table of X and Y values may be computed by interpolation, as shown in Figure 2E. The interpolation values were computed based on the interpolation equation presented and explained in Appendix A. The interpolation value, IV, for PRPH at a wage rate of $1.8 was computed as follows: 249 Developments in Business Simulation & Experiential Exercises, Volume 13, 1986 As long as decision for the wage rate remains in the $1 to $7 range, the PRPH for any value of the wage rate can be determined. The interpolation method, therefore, produces a set of dependent and independent variable values that can serve as substitutes for values generated from a true curvilinear functional, equation. The interpolation schedule in Figure 2D does, in fact, contain an inflection point. Average cost per unit is minimized at a wage rate of $4 and $5. Graphically, the interpolated average cost per unit schedule may be presented as follows: AN INTERPOLATION MODEL FOR MARKET DEMAND The interpolation technique just illustrated can be used to develop a satisfactory market demand model involving price and advertising as variables. A good market demand model involving a multiplicative type mathematical equation has been presented by Gold and Pray./3/ An interpolation- based market demand model which will give results similar to the Gold and Pray model will now be described. In the next section the results from the interpolated model will be compared to results computed by the Gold and Pray model using the same decision values. The purpose of this comparison is to demonstrate that the interpolating approach can achieve almost identical results without the need for parameters or exponents. The Gold and Pray model is summarized in Figure 3. This model is described by Gold and Pray as a generalized multiplicative model; that is, a change in a multiplier in the equation does not change other values in the equation which serve as multiplicands. The equation for market demand which includes price, marketing (advertising), and research and development as independent variables is based on an exponential function in which the independent variables also appear as variables in the exponent. In the Gold and Pray equation, the exponents are linear equations containing price, advertising, and research and development as variables. Gold and Pray defined P, R, and M as exponentially smoothed values; however, this assumption is not essential to the analysis that follows: Following the procedure used by Gold and Pray in their paper, the equation for R & D will be dropped for illustrative purposes and the analysis will concentrate on price and advertising. The Gold and Pray demand model contains 5 parameters for price and marketing: g1, g2, g3, g4, g5. The values for these parameters are determined by the designer’s specification concerning the elasticities associated with the demand. In their paper, Gold and Pray assumed the following elasticities for two levels of prices and marketing. 250 Developments in Business Simulation & Experiential Exercises, Volume 13, 1986 Based on these assumptions, the following values for g1, g2,…..g5 were computed by Gold and Pray: Given these parameters, Gold and Pray then computed the total market demand at different prices and different levels of advertising. The results of their calculations are presented in Figure 4. These values generated by the Gold and Pray demand model, have been presented graphically in Figures 5 and 6. At a price of $20, the elasticity of demand is 1. At this price sales is maximized at $12,429.845. At an advertising level of $190,000 market demand is maximized at 12,483,417 units. The Gold and Pray price/quantity demand curve is curvilinear and therefore not of the linear type most frequently illustrated in principles of economics textbooks. Implicit in the Gold and Pray model is a functional relationship between advertising and market demand. This relationship is illustrated in Figure 6. Notice that Figure 6 shows the relationship between advertising and quantity to have a maximum value and that a point of inflection exists indicating increasing and decreasing returns. An unique characteristic of the Gold and Pray Demand model is that when advertising is incremented, the percentage increase in market demand is constant at all levels of price. To demonstrate that the Gold and Pray model involves constant percentage in total marker demand changes at different levels of price, the Gold and Pray model was used to generate the data appearing in Figure 7. In Figure 7, total market demand was computed at advertising levels of $50,000, $90,000, $130,000 and $170,000. All percentage changes are computed with quantity at $50,000 of advertising as the base amount. Notice that an increase in advertising from $50,000 to $130,000 had an 823% increase in total market demand at all levels of prices. This means that a change in advertising simply shifts the demand curve to the right with the slope of the line decreasing with each incremental increase in advertising. For each price, the percentage increase in quantity is the same. 251 Developments in Business Simulation & Experiential Exercises, Volume 13, 1986 The effect of advertising on total demand at advertising levels of $50,000 $130,000 and $170,000 is graphically illustrated in Figure 8. Notice that a change in advertising shifts the demand curve to the right such that a family of curves results. The fact that the Gold and Pray model implicitly multiplies demand determined by different prices by a constant percentage for a given change in advertising means that the effect of marketing expenditures can be stated simply as percentage change rather than in terms of units. This fact is quire significant because the interpolation approach can be used to duplicate the results of the Gold and Pray model in a very simple and straight forward manner without the need for an exponential, model. Therefore, from an interpolation viewpoint, the equation for total market demand can be mathematically stated as: The observation that a change in advertising affects quantity at all prices by an equal percentage applies to research and development since the function for research and development is identical to advertising. Therefore, if R & D is set to zero, then the equation becomes: PQ and PC are values that can be computed by interpolation from the table of values for price and advertising. COMPARISON OF THE COLD/PRAY AND INTERPOLATION MODELS To demonstrate that the Pray/Gold model and the interpolation model give comparable results, the above interpolation demand model will be used to compute total market demand at different prices. Advertising initially will be assumed to be $50,000. Since the interpolation technique requires arrays of paired values, the range points to be used in the interpolation model are shown in Figure 9. The range points are derived from data generated by the Gold and Pray model. Since the range points are points on the demand line created by the Gold and Pray demand model, the interpolation model will produce identical market demand values at these points. However, the question to be answered is: For values not explicitly given in the interpolation table of values, how closely will the interpolated values correspond to values generated by the Gold and Pray demand model? Based on the Gold and Pray model and the illustrative data used by them, the following tables of paired values can be listed as the range values for the following interpolation schedules. The above interpolation schedules may be presented graphically as illustrated in Figure 10. Suppose that price in a game competition ranges from $20 to $40 and that advertising is $110,000. What values for quantity would result through interpolation? Fig. 11 shows these amounts. Corresponding amounts are shown computed from Gold and Pray model. The interpolation approach closely approximates the results of the multiplicative demand model of Gold and Pray. In fact, when the difference in demand is computed as percentage of the quantity derived from the Gold and Pray model, the highest percentage variation is less than 12%. A much higher degree of accuracy can 252 Developments in Business Simulation & Experiential Exercises, Volume 13, 1986 be achieved by decreasing the distance between the range values. If the interpolation had been between prices of $40 and $30 rather than $40 and $20 the maximum percentage variation would have been less than 2% at all levels of price. The interpolation method described in this paper is not intended to be a method that produces results identical to other mathematical models. The fact that the interpolation method give results which nearly duplicate values from more sophisticated mathematical methods gives the interpolation method considerable merit and appeal. The usefulness of the interpolation method in the final analysis does not depend on whether it can duplicate the results of other methods. Its validity and usefulness has to be judged in terms of whether it gives the simulation designer the desired results without undue complexity. The author has found the interpolation method described in this paper extremely valuable in the design of mathematical functions for his own simulations. SUMMARY An approach for approximating functional equations having minimum and maximum values and points of inflection has been presented. Using data from the multiplicative model presented by Gold and Pray, a demand model based on the interpolation process was used. The results were compared to data generated by the Gold and Pray model using identical price and advertising amounts. The differences in results did not vary acre than 12%. The interpolation approach does not involve complex mathematics but gives results comparable to methods that do. The major advantage of the interpolation method is that it gives the simulation designer control over desired results without the need for difficult-to--compute parameters required in models such as the Gold and Pray Demand model. Using the interpolation method presented in this paper, points of inflection along with minimum and maximum values are easy to create. REFERENCES /1/ Pray, Thomas F. and Steven Gold, “Inside the Black Box: An Analysis of Underlying Demand Functions in Contemporary Business Simulations”, Developments in Business Simulations and Experiential Exercises, Vol. 9, 1982 /2/ Goosen, Kenneth R., “A Generalized Algorithm for Designing and Developing Business Simulations,” Developments in Business Simulations and Experiential Exercises, Vol. 8, 1981. /3/ Gold, Steven and Pray Thomas F., “Simulating Market And Firm Level Demand - A Robust Demand System”, Developments in Business Simulations and Experiential Exercises, Vol. 10, 1983. 253 Developments in Business Simulation & Experiential Exercises, Volume 13, 1986 APPENDIX B BASIC COMPUTER PROGRAM FOR COMPUTING INTERPOLATED VALUES The programming of the interpolation approach in this paper is fairly simple. Following is an example of the programming required to implement the interpolation method in an actual business simulation The data in this particular example is obtained from the illustration in the paper comparing the Gold/Pray demand model to the interpolation method. This particular program may be used to implement any graphical representation of a functional relationship. The only significant changes would be in statements 100, 110, 130, and 150. The interpolation method is generic, and consequently, only minor changes in the program below are required to make it applicable to other types of functional relationships. 254 Developments in Business Simulation & Experiential Exercises, Volume 13, 1986 255 Table of Contents Volume 13, 1986 How to Use Microcomputers in Human Resource Management Courses What's My Line? An Exercise in Job Analysis, Description and Classification Relationships Between Team Cohesion Dimensions and Business Game Performance Four Factors Affecting Group Performance in Business Policy Simulations Enhancing Strategic Management Goal-Setting Skills in the Business Policy Course Using a Business Simulation in the Principles of Management Course - Learning Outcomes and Perceptions Personal Computers: Drexel's Experience Using Psychological Type to Enhance Negotiation Skills The Effects of Conflict Handling Styles on Negotiation Behavior: An Analysis of an Experiential Exercise An Experiential Test of Bargaining Decisions and Representation Self-Grading of Examination by Students, A Viable Alternative to Traditional Procedures Perceived Instructor Enthusiasm and Student Achievement An Experiential Approach to Teaching Subordinate-Oriented Communication Assessing the Effects of a Computerized Study Guide in Macroeconomic Principles: A Statistical Analysis A Sales Management Simulation for the PC an Integrative Tool for the Sales Management Course Teaching Re-evaluation of Salespersons Through the Use of a Simulation Game A Test of Student Performance and Attitudes and Performance Under Varying Game Conditions Developing and Testing Airways: A Marketing Simulation Experiential Learning Revisited: Some Thoughts on Designing More Adaptive Management Education Programs Developing the Competencies of Creativity and Accurate Self-Assessment Self-View and Norms: Some Longitudinal Laboratory Results Strategic Planning with an Experiential Case The ABSEL Research Consortium: Preliminary Steps and Position Papers An Analysis of ABSEL Conference papers (1974-1985) The Creation and Operation of a Data-base System for ABSEL Research Funding for a Research Consortium on Simulation and Experiential Learning Simulation - Indoctrination or Learning The Dilemma In Evaluating Classroom Innovations Personality Variables on Group Cohesion, Team Participation and Total Learning Justifications for and Problems in Developing and Using Computerized Experiential Activities Management Competency Models and the Life-Long Learning Project: What Role for ABSEL External Validity of Business Games Restructuring the University - An Experiential Exercise The Surrogate Class Exercise Images of Effectiveness: A Classroom Exercise The Algebran Industry: A Tutorial Framework Another Chance: A Multi-Leveled, Macro Business Game for Micro-Computers A Water Quality Management Simulation Game Acquisitions? Divestitures? Progress Report on a Conglomerate Game Enterprise: A Multi-Purpose Management Development Simulation Marketer: A Microcomputer Simulation in a High Tech Industry A Pedagogical Analysis of Alternative Tax Elections for Exploration Costs in the Mining Industry Using of Multiple Microcomputers Application Programs to Teach Fundamental Business Concepts and Practices Demonstrating the Role of Simulation in Strategic Operations Planning: A Case Study in Bank Check Processing Location Analysis The Subjective Side of the Decision Support System: A Pitfall for the Panacea A Group-Based Procedure for Revealing Judgmental Heuristics and Biases An Experiential Exercise in Bayesian Decision-Making Values for Selected Parameters in Physical Distribution Simulation and Games Computerized Business Management Simulations for Tyros The Management Decision Laboratory at New York University An Example: The Use of Management Games on Microcomputers by Computer Novices Enhancing Mainframe Simulation Via Microcomputers: Designing Decision Support Systems Integrating Personal Computers Into a Course as a Decision Support Tool A Decision Support System for Capital Funds Forecasting Performance and Attitudinal Affects of a Decision Support Package in a Business Game Using the Real World: Interviews with Woman Mangers The Cooperatively-Authored Student Handbook International Buying: An Experiential Exercise The Research/Teaching Interface: Turning a Pretest into an Experiential Exercise Building Microcomputer Business Simulations Simulation with Discrete and Continuous Mathematical Modeling An Interpolation Approach to Developing mathematical Functions for Business Simulations The Design of Simulation Models Using Thought Organizers The Assessment Center as a Teaching/Learning Device An Approach to Meeting the AACSB Guidelines for Introducing Skills and Personal Characteristics (SACs) Using an Interclass Simulating and Role-Playing Pedagogy Experiential Learning about the World of Work: A Program for Primary and Secondary School Educators Assessment of Relevant Personal Characteristics of Potential Entrepreneurs