VISUAL MODELING OF BUSINESS SIMULATIONS Developments in Business Simulation & Experiential Learning, Volume 27, 2000 VISUAL MODELING OF BUSINESS SIMULATIONS Victor Perotti, RIT Thomas Pray, RIT ABSTRACT This paper presents a visual modeling technique which will aid designers of business simulations. Three demand examples are presented using the visualization software, Mathematica. INTRODUCTION Over the past twenty years ABSEL model builders have worked diligently to improve the algorithms that drive business simulations. ABSEL researchers such as Gold, Pray Goosen, Teach, Decker, LaBarre, Thavikulawat, and Carvallo have published numerous papers on different aspects of improving the realism and the reliability of business games. Still, many designers, even after developing equations, struggle with: (i) how to select starting values, (ii) how to gauge sensitivity of the parameters used in the model, and (iii) how to ensure the system is robust. This paper is intended to illustrate a modeling and visualization tool known as Mathematica that can be used by designers of business simulations to gain a better understanding of their models. The paper begins with a brief summary of the ABSEL and Simulation and Gaming literature on algorithm development and then gives a general description of Mathematica describing, what it is and how it can be used. Finally, it offers a “visual modeling technique.” Three commonly used demand models are analyzed. Visual exploratory analysis is performed on (i) the Cobb Douglas power function, (ii) the Gold and Pray Demand System (1984), and (iii) the product attribute model of Gold and Pray (1997). Mathematica is utilized to identify both stability and lack of stability of a system of equations. The paper concludes with some suggestions on the use of the software package and caveats associated with the methodological approach suggested by the authors. BUSINESS SIMULATION ALGORITHMS A review of the modeling literature illustrates that there has been considerable work in algorithm enhancement of business games. For example, in the operations arena, Thavikulaw (1993) proposed a linear set of equations to model production processes. Gold and Pray (1989), Gold (1990), and Gold (1992) have developed models for cost and production functions embodied in business games. Pray and Methe (1991) put forth a model for new product development with generalized demand and production functions. Quality modeling became popular in the 1990s with the work of Thavikulwat (1992), Mergen and Pray (1992) and Teach (1992). All of these authors demonstrated methods and algorithms for modeling quality that could be added to existing or to new simulations. In the area of marketing, many articles have been written about how to model demand and other non-price determinants of demand. Pray and Gold (1982) with their classic article “Inside the Black Box” investigated the demand robustness of a number of commonly used business games. Articles soon followed by Teach (1984), Gold and Pray (1984), Goosen (1986) and Decker, LabBarre and Adler (1987). These authors’ work on algorithms moved the modeling of demand to a higher level. Further extensions by Golden (1987), Lambert and Lambert (1988) and Thavikulwat (1988,89) tested the reliability of various models and raised new issues about how demand should be modeled. Market segmentation was addressed 34 Developments in Business Simulation & Experiential Learning, Volume 27, 2000 formally by Teach (1990), Carvalho (1991, 95) and Gold and Pray (1997,98). As can be seen by this brief review, the leading business games designers have shared their design contributions with the field. But all of these algorithms described in the literature are just mathematical models and thus have certain limitations and shortcomings. Some algorithms are highly sensitive to the starting parameters selected. Others require the decision variables to be constrained in a narrow range for the simulation to behave in a manner that is consistent with theory. Some models have discontinuities, which can also yield unreasonable results. To identify the shortcomings in a model, one alternative is to look at lots of numerical results while manipulating a few variables. A better alternative, in many cases, would be to look at a visualization of the model and to thus explore its behavior across a variety of situations. VISUALIZATION Visualization is a process in which images are created to gain new insight into abstract data or complex functions. Much of the visualization work to date has been for scientific applications like weather forecasting, fluid dynamics, or Magnetic Resonance Imaging (MRI). However, recent applications like data mining, process streamlining and network analysis have driven new demand for visualization in the Business environment. For simulation designers, visualization offers a range of tools and techniques that will allow them to explore their complex, many- dimensional models. The methodology to be presented here offers a relatively easy “visual method” of testing and verifying the overall effectiveness of the algorithm, and provides insights into where difficulties may arise with actual usage. The drudgery of hours of mathematical sensitivity analysis can be avoided with the visual approach promoted by Mathematica. MATHEMATICA Mathematica, by Wolfram Research, is a software tool that allows the creation, solution, visualization and distribution of complex mathematical models. Almost every conceivable mathematical operation, analysis or function can easily be conducted with this software. In general, Mathematica finds an analytical solution for a huge variety of equations. When this is not possible, several different approximation techniques are available to provide numerical solutions for a user- specified error tolerance. The built-in mathematical functions often enable solutions for problems that would be very difficult or impossible for most users without this tool. WORKING WITH MATHEMATICA The Mathematica interface is an electronic notebook, where one can include ideas, partial results, and graphics. Users develop mathematical models by evaluating individual lines of Mathematica code, thus creating partial results that can be combined over and over again to develop more complex models. Visualization functions are available at every step of the development process to help a model-builder verify the behavior of the model. Because of the variety of tools and functions available, a developer will frequently discover unanticipated behaviors that can enhance his understanding or help him avoid future problems with a model. Once most of the development has been completed, the notebook interface can be used to explore the model both numerically and visually. This exploratory mode of interaction with Mathematica can take the form of a set of “what-if” scenarios that allow the user to explore the full complexity of the work. Charts, graphics and animations can be created automatically to contribute to the user’s understanding. Sharing such a model online is simple since the freely available MathReader 35 Developments in Business Simulation & Experiential Learning, Volume 27, 2000 software allows anyone to explore and interact with a notebook. Figure 1 show portions of Mathematica in action. A Cobb Douglas demand function with the elasticities and scale parameters are set so that the demand would be 6000 units at the starting values. Varying only the price then generates a two-dimensional demand plot. FIGURE 1 WORKING WITH MATHEMATICA LEARNING MATHEMATICA Like any new piece of software, a new user to Mathematica will have to spend some time learning both the notebook interface and the language used to write equations and functions. While the notebook interface is quite easy, the language of available commands is vast and can be intimidating. Fortunately, one can accomplish most analyses by exploring just the small subset of the language that is applicable to a specific problem. The documentation for Mathematica is very well written and illustrated, and additional reference guides make developing significant models possible even for beginners. VISUAL MODELING TECHNIQUE What follows is a demonstration of the visual modeling and exploratory techniques using three different demand models. Cobb Douglas Market Demand Function - A Stable Function with Constant Elasticity The first model selected is the Cobb Douglas function. This function was first deployed as a way to describe production functions in microeconomics, but can be easily modified to fit the demand side. To demonstrate the visual modeling aspects, we will simulate a simple demand function where price (P) and marketing 36 Developments in Business Simulation & Experiential Learning, Volume 27, 2000 15 20 25 30 35 40 45 Price 2500 5000 7500 10000 12500 15000 17500 Demand (M) are the independent variables, and demand (Q) is the response or dependent variable. The functional form is as follows: Q=aP-epMem (1) DIMINIS M 500 100 6000 7000 8000 9000 10000 Demand where: the elasticities for price and marketing are ep and em respectively. “a” is scaling coefficient. To verify the coding of the model we set the price elasticity (ep) at –1.2, the marketing elasticity (em) at .3, and the scaling coefficient “a” at 44257. As in Figure 1, using these values, the demand will start about 6000 units for a price of $25 and marketing expenditure at $500. We then varied price from $3 to $45 and marketing from $200 to $3000. The results are demonstrated below using Mathematica. FIGURE 2 THE COBB DOUGLAS DEMAND CURVE Figure 2 shows a two dimensional plot - a classic Marshallian demand curve. In this example price was varied from $3 to $45 while marketing was held constant at $500. The plot shows that demand is maximized at 17,500 and that the quantity demanded appears to be asymptotic to the x-axis. In the next illustration, Figure 3, the diminishing returns to marketing are clearly seen as we vary marketing from $200 to $3000 while holding price constant at $20. It is interesting to note that demand reaches zero and that for this model it is possible to gener small levels of m THE D 10 20 Pri 0 10000 20000 30000 Demand 10 20 Pri To look at syste Mathematica allo the Figure 4, we Figure 4 illustr demand function behavior over th note is that at hi dollar expenditur demand very mu shortcoming of that high-priced successful with th 37 FIGURE 3 HING RETURNS TO ARKETING 0 1500 2000 2500 3000Marketing ate “negative demand “ for very arketing. FIGURE 4 EMAND SURFACE 30 40 ce 1000 2000 3000 Marketing 30 40 ce ms of more than two variables, ws three-dimensional plots. In vary both price and marketing. ates the non-linearity of the and the relative stability of e range. What is interesting to gh prices, over $40, even large es in marketing will not increase ch. This may be construed as a the model, possibly indicating niche strategies may not be is demand model. Developments in Business Simulation & Experiential Learning, Volume 27, 2000 A contour map, Figure 5, provides another way to look at the effects of two variables. In this graphic, the lighter areas represent higher levels of demand. Again, notice the lack of sensitivity when price is high. Although difficult to see when looking at tables of numbers, this shortcoming of the demand function becomes apparent with the visual approach. In fact, the diminished impact of marketing expend discovered by the a plots. Gold and Pray - Demand Function Gold and Pray elasticity demand shortcomings of us Their system inv described in detai Pray [1984]. The follows: Qt where: Qt = mar average price at ti expenditure at tim parameters k where To solve for the parameters of the market demand equation the administrator must specify the desired exogenous elasticities of each independent demand variable at two different levels (i.e. P,M). The elasticity formulas are as follows: Ept = g2+g3Pt (1 + ln Pt) (3) t) (4) THE 10 15 20 500 1000 1500 2000 2500 3000 Marketing asticity at time t, Emt = g4+g5Mt(1 + ln MFIGURE 5 CONTOUR MAP where: Ept = price el iture at high prices was not uthors until they viewed these A Variable Elasticity Market (1984) developed a variable model to overcome the ing the Cobb Douglas system. olves 10 equations and is l with examples in Gold and algorithm to be simulated is as = g1Pt -(g2+g3Pt)Mt +(g4-g5Mt) (2) ket demand at time t, Pjt = me t, Mjt = average marketing e t, and gk = market demand k = 1 through 5. 25 30 35 40 45 Price Emt = marketing expenditure elasticity at time t. Selecting two levels for each elasticity (Ep, and Em) and the corresponding demand variable (P and M) over a reasonable range gives two equations with two unknowns and allows simultaneous solution of the system parameters gk (for k=2,5). The selection of g1 determines the initial market size. To demonstrate, the following values were set: FIGURE 6 THE PARAMETERS Starting value Final value Price $ 25 $ 35 Ep .95 3.00 Marketing $500 $1500 Em .4 .15 With these parameters, the price elasticity of demand at the market level will increase from an inelastic .95 to a highly elastic 3.00 when the price increases from $25 to $35. Likewise, the marketing elasticity declines as more money is allocated to marketing. With these values and a scaling coefficient, market demand is about 6000 units when price is $25 and marketing $500. Using Mathematica and varying price over the relevant range from $25 to $35 and marketing from $500 to $1500 yields some very interesting results. 38 Developments in Business Simulation & Experiential Learning, Volume 27, 2000 FIGURE 8 PRICE ELASTICITIES 28 30 32 34 Price -2.5 -2.25 -1.75 -1.5 -1.25 Elasticity The 3-D plot of Figure 7 demonstrates that the gross behavior of the Gold and Pray model is consistent with the theory reflected in the Cobb Douglas function. It is interesting to note that this variable elasticity behaves similarly to that in the Cobb Douglas model in that at the higher prices, demand is not very responsive to increases in marketing. FIGURE 7 GOLD AND PRAY DEMAND SURFACE 26 28 30 32 34 Price 600 800 1000 1200 Marketing 4000 5000 6000 7000 Demand 26 28 30 32 34 Price The Gold and Pray model can be further verified by using Mathematica to calculate the arc elasticities for price and marketing. Indeed, this line graph shown in Figure 8 shows the price elasticities are consistent with expectations. However, the model and theory are in agreement only if the price and marketing values are constrained to be within the relevant ranges of Figure 6. 39 Developments in Business Simulation & Experiential Learning, Volume 27, 2000 Figure 9 shows that the marketing elasticity is consistent with theory over the range $500 to $1500. However, above $1500 negative returns occur to marketing. Instability of Function - Price Issues Mathematica makes it readily apparent that the demand system is robust over the constrained range of decision inputs. Increasing the price and marketing out of the range reveals some fascinating behavior about the model. Notice in Figure 10 that at prices from $4 to $20 the function behaves in a manner that is inconsistent with demand theory. The model behaves consistently with theory for all prices above $20. What is interesting however, is that at lower prices say from $5 to $18 dollars, price increases cause demand to increase! We don’t think the designers expected the model to behave as an economic G INSTABILI MAR 500 10 6500 6750 7000 7250 7500 7750 Demand FIGURE 9 MARKETING ELASTICITY 500 1000 1500 Marketing −0.1 0 0.1 0.2 0.3 0.4 Elasticity Figure 11 dep advertising vary price fixed at appears to be co the relevant ra marketing occu $1500. The neg some simulation realism of negat The Gold and P The final system examples. To d level demand fo used for each between $25 and expenditures, b elasticities of th ,based on the sca from 0 to 5. INSTABILITY IN THE FUNCTION FIGURE 10 10 20 30 40 Price 1000 2000 3000 4000 5000 6000 Demand Qjt = g1Pjt - where: Q segment j at time of all products average marketin segment j at tim 40 FIGURE 11 TY OF THE FUNCTION- KETING ISSUES iffen good. 00 1500 2000 2500 3000Marketing icts quantity demanded with ing from $200 to $3000 and $25. The marketing response nsistent with expectations over nge. But negative returns to r outside the upper limit of ative returns may be helpful in situations. But the theoretical ive returns can be challenged. ray Attribute Model is described via a set of simple etermine the market- and firm- r each segment, Equation 5 is segment. Price is set to vary $35 and advertising/marketing etween $500 and $1200. The e gravity flow distances (dijt) les, are controlled over a range (g2+g3Pjt)Mjt +(g4-g5Mjt)Djt -(g6+g7Djt) (5) jt = market demand for the t, Pjt = harmonic average price in segment j at time t, Mjt = g expenditure for all products in e t, Djt = average distance of all Developments in Business Simulation & Experiential Learning, Volume 27, 2000 products for the segment j at time t, gk = market demand parameters k. FIGURE 13 THREE-DIMENSIONAL VIEW 2 4 6 8 1000 1200 1400 1600 Demand Marketing Period Gold and Pray (1999) demonstrated how distance could be employed in the model to simulate attributes desired by the customer. In their example they illustrated that Firm 3 would gain significant market share by introducing a new product based on attributes desired by the customer. In Figures 12 and 13 we replicated Gold and Pray’s example of the new product introduction but added a new element. We assumed the firm would increase price from $25, to say $28, because of the added features. Mathematica was then utilized to visually present firm’s demand with ceterus paribus. Notice the rapid decline in demand at period 4, but the demand quickly responds, demonstrating the model allows for firms to increase their price substantially with the new features. Figure 13 shows the result in a three-dimensional perspective. Summary and Conclusions The purpose of this paper was to present a visual modeling technique that we recommend to designers of business simulations. Utilizing visual representation will aid designers in creating their models by showing the parameter configurations where the model might behave unstably. Furthermore, the interactive nature of the Mathematica tool allows designers to try many different configurations quickly to help identify those that will be used in the final algorithm. FIGURE 12 FIRM 3’S DEMAND PRICE INCREASED IN PERIOD 4 2 3 4 5 6 7 8 Period1000 1200 1400 1600 Demand In the first of three examples presented, the visual modeling technique helped identify the shortcomings of the Cobb Douglas as a model for demand modeling. Specifically, it was noted that the impact of marketing expenditure is greatly diminished at high prices. In our second example, the Gold and Pray demand system was shown to resolve some of the shortcomings (by allowing the elasticities to vary), but was highly unstable outside the preset parameters. The final illustration took the Gold and Pray (1999) attribute model and illustrated what happened to firm-level demand when the firm introduced a new product and simultaneously increased the product price. It suggests that the third model may alleviate some of the shortcomings of the previous two models. References available upon 41 Table of Contents Volume 27, 2000 Internet International: A Simulation Exercise for Understanding Technological Innovation and customer Service In a Rapidly Growing Internet Server Company Simulations and Learning: Dialog and Directions Endnote Activity: A Tool for Integration of Course Content and Communication Skill Practice Incorporating Video as a Teaching Strategy in Interpersonal Communication Vision Quest: An Alternative Approach to Industry Analysis for MBA Courses in Strategic Strategic Management: An Evaluation of the Use of Three Learning Methods Trainer, Mentor, Educator: What Role for the College Business Instructor in the Next Century? Using the Internet and Shareware to Facilitate Computer Simulation in Distance Learning Classes Visual Modeling of Business Simulations Teaching about Information with Management Games A Self-Evaluation Based on the Discussion and Decision in Experts' Business Gaming The Restaurant Game Using Journals to Enhance Computer Simulation Based Learning Exercises to Facilitate Better Student Writing in the Undergraduate Strategy Class Identifying, Resolving, and Managing Common Ethical Dilemmas in the Workplace: An Experiential Approach Integrating the Digital Revolution into the Classroom The Wheel of Learning: An Integrative Business Curriculum Experiment The Changing Nature of Simulation Research: A Brief ABSEL History Perspectives on Simulation & Gaming's Review Process Experiential Learning Across Disciplines: Mixing International Business and Accounting Simulating Governmental Effects on Economic Development Internationalizing the Introduction to Business Course Using an International Text and Domestic Simulation with a Twist Using Stock Value as the Performance Measure in a Business Simulation Game Introducing Cross-Elasticities in Demand Algorithms Validating a Model of Currency Valuation An Exercise for Exploring the Relationship between Jungian Psychological Types and Organizational Politics Exercise: Preparing Financial Reports Using the Group Categorizing Technique Effect of Trust and Cultural Beliefs on Negotiation Processes: Data from an Experiential Role Play Experiential Learning Gets Stamp of Approval From the Boyer Commission Talent Search 2000 - An Experiential Activity to Help Strengthen Skills in Employee Recruitment and Selection Clemson University's Collaborative Learning Environment Initial Data on a Test Bank Assessing Total Enterprise Simulation Learning Changing the Assessment Paradigm: Using Student Portfolios To Assess Learning from Simulations How We Learn and Why We Don't: The Cognitive Profile Model: A Workshop in Teaching to Reach Your Students Knowing Thyself: A Portfolio Approach to Student Self-Assessment Collaborative Learning and Web-Based Instruction in a Cognitive Apprenticeship Model Teamwork Attributes in a Classroom Simulation Virtual Teams: Meeting the Next Challenge for Experiential Education New Age Learning: Nuance or Nonsense Developing Charisma: An Experiential Exercise in Leadership Problems and Solutions in Going Web-based with an Agribusiness Simulation Creating a Comprehensive Web-Enhanced Classroom Your Class is in Session, Now What? The Challenges of Teaching On-line An Application of Process Control Charts for Attributes as a Form of Classroom Assessment for Experiential Learning Work Goals and Life Aspirations: Do You Have What it Takes to be an Entrepreneur An Exercise to Develop Initiative: Possible Dream? The Ball Point Pen Assembly Company Management Game Review System Development Total Enterprise Simulations and Optimizing the Decision Set: Assessing Student Learning Across Decision Periods Facilitating Learning in the New Millennium with the Complete Online Decision Entry, System (CODES) The Marketing Management Experience The Right Venue for Your Simulation One More Time: Overall Dominance in Total Enterprise Simulation Performance A Profile of ABSEL Conference Attendees Learning Readiness: An Underappreciated Yet Vital Dimension in Experiential Learning Active Learning in a Professional Undergraduate Curriculum The Problem Is - They Think Differently! Cultures Integration in Mergers and Acquisitions: Putting Managers Together in a Business Simulation The Global Business Game: A Strategic Management and International Business Simulation