Using Nonlinear Programming and Queuing in Quantitative Decision Making - Essay Example

Using Nonlinear Programming and Queuing in Quantitative Decision Making Abstract This paper discusses the key concepts and issues that are involved in methods used for Quantitative Decision Making. It primarily focuses on the challenges of using the methods discussed in Quantitative Decision Making. The methods discussed are Nonlinear Programming, Decision Analysis, Forecasting, and Queuing. Each subsection begins with a definition or introduction to the corresponding method. The subsection on Nonlinear Programming included a discussion of profit graphs with nonproportional relationships. The subsection on Decision Analysis explained different decision criteria. The subsection on Forecasting has a detailed discussion on forecasting techniques. The subsection on Queuing tackled different queuing systems. The subsections end with the author’s take on how managers can benefit from each model and how the particular methodology addresses actual real-world situations. Keywords: Quantitative Decision Making, Nonlinear Programming, Decision Analysis, Forecasting, and Queuing Introduction Managers have the daunting task of making a multitude of decisions everyday for the respective institutions that they head. Depending on the nature of the variables that a particular situation entails, some decisions are arrived at quite straightforwardly while others need to undergo a series of rigorous processes before they are made. Among these challenging yet indispensable methods are Nonlinear Programming, Decision Analysis, Forecasting, and Queuing. With Hillier and Hillier (2010) as its main reference, the subsections that follow will discuss these methods in detail. Nonlinear Programming versus Linear Programming According to Feiring (1986), Linear Programming is a part of mathematical programming that deals with the competent and effective allocation of limited resources to a number of known activities to obtain the desired goal, which, most commonly concerns maximizing profit or minimizing cost. It is linear in the sense that the criterion (objective function or index) and the constraints (operating rules) of the process can be expressed as linear formulas. When at least one of these formulas is nonlinear in nature, then Nonlinear Programming is used. As a result, while Linear Programming assumes a proportional relationship between activity levels and overall measure of performance, Nonlinear Programming is used to model nonproportional relationships. There are four (4) types of profit graphs associated with nonproportional relationships as listed below. 1. Decreasing Marginal Returns The slope of the graph never increases but sometimes decreases as the level of the activity increases. 2. Piecewise Linear The graph of a piecewise linear function consists of a sequence of connected line segments. Thus, the slope of the profit graph remains the same within each line segment but then decreases at the kink where the next line segment begins. 3. Discontinuities The slope has an almost unpredictable pattern, possible constant then abruptly becomes zero. 4. Increasing Marginal Returns The slope of the graph increases as the level of activity is increased. The application of nonlinear programming to nonproportional relationships makes it more accurate for use by managers since, as Avriel (2003) pointed out, mathematical models involving nonlinear functions usually result from accurate representations of actual real-world situations. However, it is also this drive for accuracy that makes it more difficult to construct nonlinear formulas needed for a nonlinear programming model. Consequently, nonlinear programming models are more difficult, if at all possible, to solve than linear programming models. Decision Analysis Both Linear Programming and Nonlinear Programming, among other mathematical modeling methods, address situations that utilize objective functions that specify the estimated consequences and combination of decision. Of course, actual real-life environments seldom adhere to this degree of predictability. This is when the process of Decision Analysis plays an important role. Decision Analysis involves a methodological approach to rationally addressing management problems that are much more complicated and display higher levels of uncertainty. There are four (4) types of Decision Criteria as identified below together with the steps needed to make a corresponding choice. 1. Decision Making without Probabilities: The Maximax Criterion (Best Situation) The maximax criterion is based upon extreme optimism and focuses only on the best possible outcome. Step 1: Identify the maximum payoff from any state of nature for each decision alternative. Step 2: Find the maximum of these maximum payoffs and choose the corresponding decision alternative. 2. Decision Making without Probabilities: The Maximin Criterion (Worst Situation) The maximin criterion is the criterion for the total pessimist because it puts a focus on the worst that can happen in a given situation. Step 1: Identify the minimum payoff from any state of nature for each decision alternative. Step 2: Find the maximum of these minimum payoffs and choose the corresponding decision alternative. 3. Decision Making with Probabilities: The Maximum Likelihood Criterion (Most Probable Situation) The maximum likelihood criterion says that the focus must be on the most likely state of nature. Step 1: Indentify the state of nature with the largest prior probability. Step 2: Choose the decision alternative that has the largest payoff for this state of nature. 4. Decision Making with Probabilities: Bayes’ Decision Rule (Average Probability) The Bayes’ Decision Rule directly uses the prior probabilities of the possible states of nature and applies the Law of Averages. Step 1: For each decision alterative, calculate the weighted average of its payoffs by multiplying each payoff by the prior probability of the corresponding state of nature and then summing these products. Statistical terminology refers to this weighed average as the expected payoff (EP) for this decision alternative. Step 2: Bayes’ decision rule says to choose the alternative with the largest expected payoff. Decision Trees A Decision Tree is a graphical representation of the decision making process using nodes and branches (Haines, 2009). It provides a quantifiable analysis of the steps involved. The nodes are the forks or junction points on a decision tree. There are two kinds of nodes, Decision Nodes and Event Nodes or Chance Nodes. Decision nodes are represented by squares, which indicate that a decision needs to be made at that point in the process. Event Nodes or Chance Nodes are represented by circles, which indicate that there is a random event at that point. Branches are simply lines that connect one node to another, indicating that it is possible to go from one decision or event to the next. Firms generally rely on managers to make the right decisions. Thus, managers have to possess knowledge in going about making these crucial decisions. Managers have to be very analytical and must be able to correctly identify the decision criteria that a particular situation calls for. Forecasting As rational beings who are at ease with the rules of logic, human beings are generally uncomfortable with uncertainty. Managers are most familiar with this predicament. Aside from the periodic task of determining the optimal use and allocation of their respective firms’ resources, managers likewise have to plan out strategies that will benefit their organizations in the long run. This is where forecasting, or the process of analyzing available current and historical data to determine future trends (Armstrong, 2001), comes in handy. There are five types of Forecasting Techniques that aid in estimating many future aspects of a business or other operation. Also, for the sake of succeeding discussion, let us use the following values showing the demand for items sold by Company X over a period of five months: (Month 1: 30, Month 2: 45, Month 3: 57, Month 4: 43, Month 5: 35). 1. Last-Value Forecasting Method (Naïve Method) The Last Value Forecasting Method simply uses last period’s actual data value to forecast data for the next period. Table 1. Last-Value Forecasting Method Month Actual Demand Forecast 1 30 2 45 30 3 57 45 4 43 57 5 35 43 6 35 Table 1 represents an example of the Last-Value Forecasting Method. 2. Average Forecasting Method The average forecasting method uses the average of all monthly activities to date as the forecast for the next month. Using our sample data, we have the following computation: Forecast for Month 6 = (30 + 45 + 57 + 43 + 45) / 5 months = 42. 3. Moving-Average Forecasting Method The Moving-Average Forecasting Method provides a middle ground between the last-value and averaging method by using the average of the monthly activities for only the most recent months as the forecast for next month. To get a three-month moving average forecast for the sixth month using our given data, we take the average of the values from Months 3, 4 and 5: Forecast for Month 6 = (57 + 43 + 45) / 3 months = 45. 4. Exponential Smoothing Forecasting Method The Exponential Smoothing Forecasting Method is so named because the weight falls off exponentially as the data becomes older. Exponential Smoothing gives the greatest weight to the last month and then progressively smaller weights to the older months. The formula used for Exponential Smoothing is: New forecast = previous forecast – alpha * (actual demand – previous forecast). Using the sample data, an initial forecast of 35, and an alpha of .7, the forecast for Month 2 will be computed as follows: New forecast for Month 2 = 35 + .7(30 – 35) = 31.5. The forecast for Month 3 will be: New forecast for Month 3 = 31.5 + .7(45 – 31.5) = 40.95. The process repeats until one reaches the desired period. Table 2 shows the forecasted values up to Month 6. Table 2. Exponential Smoothing Forecasting Method Month Actual Demand Forecast 1 30 2 45 31.5 3 57 40.95 4 43 52.18 5 35 45.76 6 38.23 5. Exponential Smoothing with Trend Exponential Smoothing may further be extended when data exhibits a linearly upward or downward trend. To input the effect of trending, use a two-dimensional graph with activities measured along the vertical axis and time measured along the horizontal axis. After plotting the activities data month to month, this method finds a line passing through the midst of the data as closely as possible. Extending the line into future months provides the forecast of activities for future months. It should be noted that it is not enough to simply forecast data. One must also determine the accuracy of the data by finding the difference between the actual or real and the predicted or forecast value, or the forecast error. Using the data from the exponential data table, we find the forecast errors for each month as shown in Table 3. Table 3. Forecast Errors. Month Actual Demand Forecast Forecast Error 1 30     2 45 31.5 13.5 3 57 40.95 16.05 4 43 52.18 - 9.18 5 35 45.76 - 10.76 6   38.23   Once forecast errors are computed, forecast accuracy may be determined by using the Mean Absolute Deviation (MAD) and the mean square error (MSE). The MAD may be computed by taking the average of the absolute values of the forecast errors. Using the data from Table 3, MAD can be computed as such: MAD = ( |45 – 31.5| + |57 – 40.95| + |43 – 52.18| + |35 – 45.76| ) / 4 = 12.37. This indicates that the forecaster is off by an average of 12.37 units per forecast. In a similar manner, MSE also measures forecast accuracy and is computed by dividing the squares of the forecast errors by N – 1 (where N is the number of periods used in the computation). Thus, the MSE for the same set of values as the one recently mentioned is: MSE = ( |45 – 31.5|2 + |57 – 40.95|2 + |43 – 52.18|2 + |35 – 45.76|2 ) / (4 – 1) = 213.3. One has to take note that there are cases when it is more important to make a forecasting model that has the ability to change quickly to respond to changes in data patterns than an accurate forecasting model. Thus, a forecaster’s worth will largely be dependent on making sure that the forecasting method chosen reflects the relative balance between responsiveness and accuracy. Queuing Models Any firm or organization is familiar with queues or waiting lines. As Sharma (2006) pointed out, waiting line problems arise because of either of two reasons. First, there is too much demand on the facilities so that there is an excess of waiting time or inadequate number of service facilities. Second, there is too less demand, in which case there is too much idle facility time or too many facilities. In either of these cases, it is the manager’s task to come up with a queuing model that will either schedule arrivals or provide facilities or both so as to obtain an optimum balance between costs associated with waiting time and idle time. Queuing theory deals with models that describe the transformations of random flows of customers by servers during the process of servicing (Kalashnikov, 1994). Generally, the dynamics of queuing models is governed by sequences of random variables which make up the components of a queuing model. A queuing model starts with the input data which are sequences and variables that define the operation of components. In an analogous manner, the output data for queuing models are often random sequences and stochastic processes. Examples are the sequence of interdeparture times or the output flow, the sequence of waiting times or customer sojourn times, and many others. Other than the mentioned sequences, structural information is needed to define the queuing model: arrival discipline, queue discipline, service discipline, and so forth. This information determines the type of mapping of input data into output data. If this mapping, together with the actual composition of input data and output data, is given, then the queuing model or queuing system is defined. There are different kinds of queuing systems used in actual real-world applications. Among these are Commercial Service Systems, Internal Service Systems, and Transportation Service Stations. A commercial service system is a queuing system where a commercial organization provides a service to customers from outside the organization. A barber shop is an example of a commercial service system where the customers are those people who want to have their hair cut by barbers who act as servers in this queuing system. As its name suggests, an internal service system is a queuing system where the customers receiving service are internal to the organization providing the service. An example is a company’s maintenance system. The repair crew act as servers that cater to the customers which, in this case, are the machines that need to be repaired. Other queuing systems fall under the category of Transportation Service System. This system involves transportation so that either the customer or the server(s) are vehicles. One example of such a system is manifested in a highway tollbooth. The cars and other automobiles act as customers serviced by the cashiers. For as long as clients need to be serviced, managers will always be haunted by queuing problems. As such, queuing theory will always be a manager’s aid in perfectly matching customers and service facilities while balancing the cost of offering the service and the cost incurred due to the delay in offering the service. Summary As technological advancements leap by bounds, firms and organizations face more and more pressure each day to keep their head above water. They will be more keen on hiring managers that have a clear system in mind when tackling their departments’ challenges. Quantitative analysis will play a much larger role in the manager’s decision making process. Thus, managers who overcome the challenges posed by the methods of Nonlinear Programming, Decision Analysis, Forecasting, and Queuing will truly be invaluable assets to the companies they belong to. References: Armstrong, J. S. (2001). Principles of forecasting: A handbook for researchers and practitioners. New York, NY: Springer Science + Business Media, Inc. Avriel, M. (1976). Nonlinear programming analysis and methods. New Jersey, NJ: Prentice-Hall. Feiring, B. R. (1986). Linear programming: An introduction. Sage university paper series on quantitative applications on the social sciences, Series No. 07-60. Beverley Hills, CA. Haines, S. (2009). The product manager’s desk reference. USA: McGraw-Hill Professional. Hillier and Hillier. (2010). Introduction to management science:A modeling and case studies approach with spreadsheets (4th ed.). NY: McGraw-Hill/Irvin Publishing Company. Kalashnikov, V.V. (1994). Mathematical methods in queuing theory. The Netherlands: Kluwer Academic Publishers. Sharma, S. C. (2006). Operations research: Inventory control and queuing theory. New Delhi: Discovery Publishing House. Read More