REDUCING THE COMPLEXITY OF INTERACTIVE VARIABLE MODELING IN BUSINESS SIMULATIONS THROUGH INTERPOLATION Developments In Business Simulation & Experiential Exercises, Volume 20,1993 56 Reducing the Complexity of Interactive Variable Modeling in Business Simulations Through Interpolation Kenneth Goosen, University of Arkansas at Little Rock ABSTRACT This paper, which is prepared in response to a question raised by Steven Gold concerning use of the interpolation approach in modeling, describes how the interpolation approach achieves interaction among variables in a simulation. The theory underlying use of the approach is presented and a five-step algorithm is suggested for implementation. Application of the approach to non- multiplicative models is also addressed. The paper concludes that based on the principle of creating interpolation arrays in terms of percentages, and given a starting base quantity, the interpolation methodology can effectively emulate the interaction of any type of equation that contains multiple interacting variables. INTRODUCTION How a linear interpolation algorithm can be used to achieve results normally associated with complex curvilinear equations was presented by Goosen (1986). In this paper, only a brief mention was made of how the interpolation model could be used to achieve variable interaction. A valid question can be asked concerning whether signification interaction among variables can be achieved by using the interpolation method introduced in Goosen’s paper titled, An Interpolation Approach to Developing Mathematical Functions for Business Simulations.” Because the primary concern in that paper was to introduce the methodology of interpolation, the discussion and illustration of variable interaction was limited to a dependent variable and a change in a single independent variable. The purpose of this paper is to describe how the interpolation methodology can be extended to accomplish interactive effects among any number of variables. Interaction of variables in models using multiplicative and non-multiplicative equations are separately described. TWO CHARACTERISTICS OF MULTIPLICATIVE EQUATIONS A multiplicative model is an equation where each independent variable is treated as a multiplier of the other variables. The equation, V = A x B x C, is an example of the basic form of a multiplicative equation. The inclusion of exponents which also contain the independent variables (e.g., Y = AK1A x Bk2B) does not change the fundamental characteristics of the multiplicative model. A detailed analysis of a simple multiplicative equation, such as Y = A x B, is sufficient to reveal several unique characteristics of multiplicative equations that allow our interpolation methodology to emulate interactive variable behavior. Two basic characteristics of multiplicative equations underlie the theoretical foundation of this approach. The first characteristic concerns the effect of a change in an independent variable on the percentage change of the dependent variable. Successive changes in values for B at a given value of A will result in percentage changes in Y which will be equal to percentage changes in Y for the same changes in B at other assigned values of A. That is, the measured changes in Y resulting from a change in B are a constant percentage for all values of A. Percentage changes in V due to changes in B are basically independent of values for A. An interactive graphical model is presented in Figure 1. Note that for each change in the value of A, a shift in the curve results. Graph A shows values of V for different values of B when A = a1. Graph B shows the shift in the function, A x B, when A = a2. Graph C show an additional shift in the function when A = a3. In Graph A. when A = a1 and the initial quantity is y1 at b1: A change in B to b2 results in a percentage of y2/y1 for the change in V to y2. A change in B to b3 results in a percentage of y3/y1 for the change in V to y3. Note: V2/Y1 ≠ V3/V1 In Graph B, when A = a2 and the initial quantity is y4 at b,: A change in B to b2 results in a percentage of y5/y4 for the change in V to y5. A change in B to b3 results in a percentage of y6/Y4 for the change in V to y6. Note: y5/y4 ≠y6/y4 In Graph C, when A = a3 and the initial quantity is y7 at b1: A change in B to b2 results in a percentage of y8/y7 for the change in Y to y8. A change in B to b3 results in a percentage of y9/y7 for the change in Y to y9. Note: y8/y7 ≠ y9/y7 At each value for A the increases in Y can be summarized as follows: Developments In Business Simulation & Experiential Exercises, Volume 20,1993 57 The significance of these equalities is that a change in B. regardless of the value assigned to A, results in the same percentage change in V. n other words, for each shift in the curve due to a change in A, changes in B will have no effect on the percentage change in Y. The relationship of changes in A relative to B and changes in B relative to A means that interpolation schedules in terms of percentages rather than absolute values can be prepared. The following example shows how a schedule of changes in V values have been converted to a schedule of percentage changes. The quantity schedule is based on the equation Y = A x B The percentages are computed by using the quantities in the bi column as the initial quantities. For example when A = 1 0 the percentage changes in Y resulting from changes in B are 40/40 (1), 60/40 (1.5), 80/40 (2), and 100/40 (2.5). Note that the percentages associated with the B values are the same for each value of A. Consequently, as a practical matter, interpolation in multiplicative equations can be accomplished by using only a single row of percentages. A second unique characteristic concerns multiplicative equations that have maximum or minimum values at certain values for B. The value of B that determines the minimum or maximum is the same regardless of the value for A. Figure 2 shows a graph created by using the Gold/Pray demand model based on the original parameters presented in their 1 984 paper. Note that in Figure 2 whether the price, is $10, $20, or $30, the amount of advertising that maximizes quantity is $200,000. Regardless of the price, the optimal value for advertising is the same. The reason again has to do with percentage relationships. In Figure 2, each change in price produces a constant percentage shift in the demand schedule. A proportional shift in the advertising/quantity schedule occurs for each change in the assigned value for price. The significance of this characteristic is that if the intent in using interpolation is to emulate a multiplicative type model where the model has maximum or minimum values, then care must be taken to see that each sketched array of A values reaches its maximum at the same value of B. ILLUSTRATION OF INTERACTIVE MODELING THROUGH INTERPOLATION From the above graphical and numerical example it is apparent that for multiplicative models the interaction of variables can be expressed in terms of percentages. This fact allows a simple Developments In Business Simulation & Experiential Exercises, Volume 20,1993 58 equation to be developed to achieve the same interactive variable effects inherent in multiplicative models: Specific values for Pa Pb Pc, are determined by interpolation from percentage change arrays. In order to achieve interaction among variables, the interpolation algorithm model we presented in our original paper must be modified to allow the inclusion of percentage arrays. The modified algorithm may be stated as follows: Step 1 - For each independent variable sketch on graph paper the desired function in terms of percentages of change. The sketched curve may be linear or curvilinear. Note: In order to emulate a ` multiplicative function, only one percentage line needs to be drawn; however; the emulation of interactive nonmultiplicative functions through interpolation requires that more than 1 percentage function line be sketched. Step 2 - For each graph, identify points on the function at selected interval increments of the specified independent variable on the X-axis. Determine from the graph the corresponding percentage change. Step 3 - Prepare schedules listing the values assigned to each independent variable and the corresponding percentage values. Step 4 - Develop an interpolation equation that will provide percentage values for all selected values of the independent variables. Step 5 - Compute the value of the dependent variable using the equation V = BQ x Pax Pb x Pc.. Developments In Business Simulation & Experiential Exercises, Volume 20,1993 59 This five-step algorithm is illustrated in Figure 3. The appendix to this paper presents an example of an interpolation computer program makes the required interpolation calculations. The graphs in Figure 4 are prepared from values generated by our interactive interpolation algorithm in Figure 3. INTERACTIVE MODELING OF NON-MULTIPLICATIVE EQUATIONS Regarding all other type of models which here are collectively described as non-multiplicative, the problem of achieving interaction among variables is somewhat more difficult to understand. However, implementation of the interpolation procedure is only slightly more difficult. An example of a non-multiplicative model is the following where A and B are considered to be variables and C, D. and E are constants: Figure 5 shows the behavior of this function for three different values of B. Figure 4 Graphical Illustration of Variable Interaction from Data Created by Interpolation Developments In Business Simulation & Experiential Exercises, Volume 20,1993 60 Note that in Figure 5, the maximum quantity of each curve is at a different value for B. Also, a change in B (e.g., from b1 to b2 and from b1 to b3) at different values of A will not result in proportionate changes in quantity. For each value of B there is a different schedule of percentage changes. To achieve through interpolation the same type of non-multiplicative equation interaction among variables. a schedule of percentage changes such as the following must be prepared, assuming A is the primary variable: In order to determine the effect of changes in B, the interpolation algorithm must identify the value of A first and then interpolate the appropriate array of percentages for the given value of A. The value of A in non-multiplicative equations is important and must be explicitly recognized in the process of interpolation. For example, a value of 1.5 for A requires that an array of percentages at that value be determined by interpolation. The author has developed an effective computer program (see appendix) for this type of interpolation. This program, which is relatively small, allows emulation of non-multiplicative equations to be easily accomplished. Given this computerized interpolation algorithm, the only requirement is that a family of curves be sketched and converted either to percentages or quantity schedules. The 5-step method may be used to create the appropriate change schedules required for interpolation. The complexity issue raised by Goad is greatly diminished once this computerized interpolation algorithm is employed. SUMMARY Based on the principle of creating interpolation arrays in terms of percentages and given a starting base quantity, the interpolation methodology developed can effectively emulate the interaction of any type of equation that contains multiple interacting variables. What is required is the use of the 5-step interpolation procedure to create a percentage change schedule for each variable. The advantage of using interpolation to achieve variable interaction is that the simulation designer can create any type of function that will give the desired results at all levels of activity or decision levels. References Gold, S & Pray T.F., (1984) Simulating Market- and Firm-level Demand Functions in Computerized Business Simulations, Simulations and Games. 1 5, 346-363. Goosen. Kenneth R., “An Interpolation Approach to Developing Mathematical Functions for Business Simulations", Developments in Business Simulation and Experiential Exercises, 13, 248-258 Developments In Business Simulation & Experiential Exercises, Volume 20,1993 61 Table of Contents Volume 20, 1993 Dominant Personality Types and Total Enterprise Simulation Performance Shelf Wars: A Grocery Channel Simulation Shared Cultural Perspectives: An Experiential Exercise Utilizing International Students to Globalize the Classroom An Instrument for Investigating the Effectiveness of Teaching Methods in the Business Policy and Strategy Formulation Course Providing Better Trained Graduates for Accounting Employers The Ambition Gradient Approach to Evaluation of Computer Simulation Game Team Performance Alphatec: A Negotiation Exercise with Logrolling and Bridging Potential Using the Ideafisher Idea Generation System as a Decision Support System in Marketing Strategy Courses A Dynamic Market Share Allocation Model For Computerized Business Simulations Multi-Cultural Adaptability Using Experiential Learning in a Graduate Course Development of Experiential Applications in HRM: Practicing What Preach and Preaching for Practice Linking Students and Business Leaders Through Portfolios Debriefing International Experiential Learning Exercises: Road Signs for Effectiveness Sales Manager: A Simulation Modeling Interactive Effects in Mathematical Functions for Business Simulations: A Critique of Goosen's Interpolation Reducing the Complexity of Interactive Variable Modeling in Business Simulations Through Interpolation Antecedent Biases of Experiential Learners: Trainee Occupation and Subgroup Diversity Pax in Terra Sancta: Simulating the Middle East Peace Negotiations A Multiple Regression Case In Experiential Learning Changes in Ethnocentric/Geocentric Orientation by Business Students after Exposure to a One Summer Course in International Marketing's A Systematic Approach to the Development and Evaluation of Experiential Exercises Entrepreneurs Evaluate Experiential Education A Linear Programming Approach to Open System Total Enterprise Simulations Reflecting Leader Behavior from the Looking Glass, Inc. Simulation Linking Cognitive Styles, Teaching Methods, Educational Objectives and Assessment: A Decision Tree Approach Restructuring Management Education in Post-Communist Countries: How Western Experts Can Help Managerial and Cultural Pre-Conditions for Superior Performance in a Global Setting: An Experimental Study with the Aid of Business Games Multiple Industries in Computerized Business Gaming Simulations Content or Process? - Content and Process! Some Observations and Reflections About Management Education in Central Europe Out-of-Class Experiences to Promote Volunteerism Enacting the Linguistic Consciousness of the Modern Managerial Mind: Post-Modernism and Experiential Learning Intergrating Experiential Exercises into the College Curriculum: The Case of Internationalizing the Business Curriculum Simulation Marketing Oversights Incorporating Advertising Creative Strategy into Computer-Based Business Simulations The Dynamics of a Partnership Between Business and Education Collaborative Education Done Globally Experiential Systems Analysis CADPLAN: A Simulation for Comparative Advertising A Doctoral Symposium: Preparing Students for Conference Behavior Comparing the Simulation with the Case Approach: Again! Total Quality Management: A Model for Continuous Quality Improvement The Quality Audit: An Experiential Exercise for Business Students Extending the Reach of Simulations: DECIDE Heads for the Inner City Lessons Learned from a Customized Management Development Simulation The Foreign Exchange Spot Trading Simulation Using Lotus 1-2-3 to complete a Triple Play in a Simulated Competition International Business Education: Is Enough Being Done? Matching of Student-Teacher Cognitive Style as a Factor in Student Success in an Introduction to Information Systems Course Breathing (More) Life into the Case Approach Lord of the Flies: A Live Case Approach to Leadership Cooperative Case Studies: Experiential Tools for Teaching Business Problem Solving Tools Strategy Simulations in Context: An Evaluation of Key Dimensions The Distribution Channel Game Evaluation of a Simulation Game as an Education Tool for Utility Professionals The Relationship Between Total Enterprise Simulation Performance and Learning Total Quality Management does not Happen by Magic, but it can be Taught Using a Pedagogical Methodology that Utilizes Magic Effectively Preparing Students for Careers in a Global Environment by Integrating Total Quality Management Thoughout the Business Curriculum An Empirical Investigation of Cognitive and Performance Consistency in a Marketing Simulation Game Environment Using MARSGAP with LAPTOP: (A Marketing Simulation Game Analysis Program) with LAPTOP: A Marketing Simulation Adapting TQM Implementation to Organizational Level An MBA Business Simulation: Executive Interaction Experiential Exercises and Pedagogy Track Workshop: Experiencing Cultural Diversity in the Classroom (and the Hotel Meeting Room) Closing the Gap between Corporate and National Culture The Dynamic Manufacturing Company The Use of Experiential Techniques in Corporate Training The State of Simulation Gaming in Easter European Countries- Principally Russia An Experiential Exercise in Cross-Cultural Training Valuing Differences: A Conceptual Framework Demonstration of an Experiential Exercise Effectively Using Experiential Learning to Impart TQM Concepts in a High Technology Environment The Older Worker Questionnaire: An Exercise Concerning Older Worker Stereotypes and Behaviors The Crime Fighting Task Force: An Exercise in Organizational Politics Welcome to the Party! An Expression of Vocational Preference Experiential Exercise for Imparting Cross-Cultural Appreciation Six Swift Simulations on Globalization Overview of BASF Delegate Program