MODELING INTERACTIVE EFFECTS IN MATHEMATICAL FUNCTIONS FOR BUSINESS SIMULATIONS: A CRITIQUE OF GOOSEN’S INTERPOLATION APPROACH Developments In Business Simulation & Experiential Exercises, Volume 20, 1993 54 MODELING INTERACTIVE EFFECTS IN MATHEMATICAL FUNCTIONS FOR BUSINESS SIMULATIONS: A CRITIQUE OF GOOSEN’S INTERPOLATION APPROACH Steven C. Gold, Ph.D., Rochester Institute of Technology ABSTRACT Goosen (1986) presented an interpolation approach as a simplified and effective way to model complex and non-linear demand and cost functions in computerized business simulations. However the usefulness of the approach becomes limited when one considers the possibility of interactive effects between the dependent and independent variables which are embodied in most business relationships. The important issue of interactive effects are not directly addressed in Goosen’s interpolation method and it is not clear how these types of effects would be systematically included in the general case. The potential usefulness and application of the interpolation method depends on whether or not it could directly and efficiently address the procedures necessary to incorporate interactive effects. PURPOSE A paper by Goosen in the 1 986 issue of the ABSEL Proceedings, titled “An Interpolation Approach To Developing Mathematical Functions For Business Simulations” presents a graphically oriented algorithm for designing functional relationships. Goosen (1986) argues that his interpolation approach is easy to implement and capable of generating results almost identical to more complex curvilinear functions, like those developed by Gold & Pray (1984,1989,1992), Thavikulwat (1989), and Teach (1990) for modeling demand and production relationships. The purpose of this paper is to identify an important issue, namely interactive effects, that need to be more carefully addressed in the “interpolation approach” of Goosen before it can be generally applied to model sophisticated demand, production, and cost relationships in business simulations. INTERACTIVE EFFECTS Sophisticated demand, production, or cost functions must allow for interactive effects to exist between the independent and dependent variables. An interactive effect exists when the relationship between two variables, say Q and X, is influenced by the level of a third variable, say Y. These interactive effects may be linear but, more generally, are non-linear. Interactive effects may be expressed mathematically given a generalized two variable demand function for illustrative purposes: An interactive effect exists between price (P1 and advertising (A) if the partial derivative of Q with respect to P changes with the level of the variable A, that is: and/or the partial derivative of Q with respect to A changes with the level of the variable P: No interactive effect exists if the partial derivatives in both equations 2 and 3 are equal to zero. Interactive effects may also be expressed graphically by noting that the demand function line (or curve) which plots the relationship between price and quantity, assuming advertising expenditures are fixed, shifts to the right as the level of `advertising increases. Similarly, the demand function line or curve which plots the relationship between advertising and quantity, assuming price is fixed, shifts to the left as the price is increased. INTERPOLATION METHOD The five-step interpolation approach outlined by Goosen (1986) does not directly address how interactive effects can be modeled. Step one of the interpolation approach is to sketch on graph paper the desired curvilinear functional relationship between two variables, like price and quantity. (Shifts in the functional relationship are not discussed.) Based on the graph in step one, a set of points on the function are selected (step two), and a schedule is prepared listing the values assigned to the independent variable and the resulting values for the dependent variable (step an interpolation equation is then developed and programmed (steps four and five 4). Goosen’s interpolation equation is: In equation (4), the interpolated value (IV) depends on the plotted values for X (i) and Y(i) which are read from the graph of the function developed in steps 2 and 3 of the interpolation method. The values for Y (i) only depend on the values of X (i) in this methodology. Consequently the interpolated value for the variable V does not directly account for or incorporate the possibility of interactive effects (like those described previously between price and advertising). The modeling of interactive effects would require the graph developed in step 1 of the interpolation method to shift to the right (or left) as the value of an “external” independent variable, like Z (i), were to change. How this type of phenomenon would be integrated into Goosen’s interpolation algorithm is not clear. INTERPOLATION EXAMPLE WITH GOLD-PRAY FUNCTION Goosen provided an example using the interpolation method to emulate the demand function developed by Gold & Pray (1984). The Gold-Pray demand function is a generalized multiplicative market demand model, which incorporates interactive effects between the independent variables. In the example the interpolation method was shown to closely approximate the Gold-Pray function. The interpolation approach was argued to be a simpler and, by implication, a more efficient method for modeling complex functional relationships. The interesting question is how the two methods could be comparable when the interpolation approach does not directly incorporate interactive effects? The answer is that in the example developed by Goosen an interactive effect was added, implicitly, to the interpolation method. The author modified his own interpolation equation as follows: Developments In Business Simulation & Experiential Exercises, Volume 20, 1993 55 In equation 5 the original interpolated value (IV) is multiplied by an interactive effect factor to calculate an “adjusted” interpolated value that now includes an interactive effect. In the example by Goosen the interactive effect factor (IEF) was given as 606.63% and was derived from the Gold-Pray demand function. The modified approach seems simple enough but becomes more complex when carefully evaluated. an interpolation approach would be needed to obtain the interactive effect factor (IEF), i.e. one would have to go through steps 1 to 4 outlined by Goosen just to calculate the IEF before beginning to calculate IV*. Second, their may be more than one interactive effect factor, which is the general case for most business simulations. For example, market demand depends on a set of variables like: substitute price, rival advertising and promotion campaigns, interest rates, sales force, economic growth, inflation rates, and research and development expenditures (to name just a few). All of these factors are highly interrelated and require interactive effects to be modeled in the demand algorithm. Given several interactive effects, the interpolation approach by Goosen would become much more difficult to apply. Third, the interactive effect factor my change with the level of the other independent variables. In the Gold-Pray function the IEF factor for advertising of 606.63% was independent of the level of price. This may not be the desired case in some business simulations. If the advertising elasticity of the firm were presumed to change with the price level, then the IEF factor would change even if advertising expenditures were held constant. This would further complicate the interpolation approach by Goosen. SUMMARY Goosen’s interpolation approach is presented as a simplified approach to modeling complex and non-linear functional forms. However the usefulness of the approach becomes limited when one considers the possibility of interactive effects between the dependent and independent variables in the functional relationships embodied in most business simulations. Interactive effects are not directly addressed in Goosen’s interpolation method and it is not clear how these type of effects would be systematically included in the general case. The potential usefulness and application of the interpolation method depends on whether or not it could directly and efficiently address the procedures necessary to incorporate interactive effects. REFERENCES Gold, S. and Pray, T. (1984), “Modeling Market and Firm Level Demand Functions in Computerized Business Simulations", Simulation & Games: An International Journal of Theory. Design and Research, Vol. 15, No. 3, pp. 346 - 363 Gold, S. and Pray, T. (1989), “The Production Frontier: Modeling Production in Computerized Business Simulations”, Simulation & Games: An International Journal of Theory. Design and Research, Vol. 20, No. 3, pp. 300- 318 Gold, S. and Pray, T. (1 992), Modeling Short-Run Cost and Production Functions in Computerized Business Simulations”, Simulation & Games: An International Journal of Theory, Design and Research, forthcoming Vol. 23, No. 4, pp. 41 7 - 430. Goosen, K., (1986), “An Interpolation Approach to Developing Mathematical Functions for Business Simulations’, Developments in Business Simulations & Experiential Exercises, Vol. 13, pp. 248 -255. Teach, R. D. (1990), ‘Demand Equations which include Product Attributes”, Developments in Business Simulations & Experiential Exercises Vol. 17, pp. 161 - 166. Thavikulwat, P. (1989), “Consumption as the Objective in Computer-Scored Total Enterprise Simulations”, Developments in Business Simulations & Experiential Exercises, Vol. 16, pp. 37-40. 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