part © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in Business Analytics: Data Analysis and Chapter Decision Making 14 Optimization Models Introduction  A wide variety of problems can be formulated as linear programming models, but there are some that cannot  Some models require integer variables, or they are nonlinear in the decision variables  Once the models are formulated, Solver can be used to solve them  However, integer and nonlinear models are inherently more difficult to solve  Solver must use more complex algorithms and is not always guaranteed to find the optimal solution © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Worker Scheduling Models  Many organizations use worker scheduling models to schedule employees to provide adequate service  LP can be used to schedule employees on a daily basis, as shown in the next example © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Example 14.1: Worker Scheduling.xlsx (slide of 2)  Objective: To develop an LP model that relates five-day shift schedules to daily numbers of employees available, and to use Solver on this model to find a schedule that uses the fewest number of employees and meets all daily workforce requirements  Solution: The number of full-time employees that a post office needs each day is given in the table on the bottom left  Union rules state that each full-time employee must work five consecutive days and then receive two days off  The post office wants to meet its daily requirements using only full-time employees  The variables and constraints for this problem appear in the table on the bottom right © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Example 14.1: Worker Scheduling.xlsx (slide of 2)  Add an integer constraint in Solver to ensure that the number of employees starting work on some days is not a fraction  The spreadsheet model with this integer constraint is shown below  When you solve this problem, you might get a different schedule that is still optimal  Such multiple optimal solutions are not at all uncommon and are good news for a manager, who can then choose among the optimal solutions © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Modeling Issues  The postal employee scheduling example is called a static scheduling model because we assume that the post office faces the same situation each week  In reality, demands change over time A dynamic scheduling model is necessary for such problems  A scheduling model for a more complex organization has a larger number of decision variables, and optimization software such as Solver will have difficulty finding a solution  Heuristic methods have been used to find solutions to these problems  The scheduling model can be expanded to handle part-time employees, the use of overtime, and alternative objectives such as maximizing the number of weekend days off © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Blending Models  In many situations, various inputs must be blended to produce desired outputs  In many of these situations, blending models can be used to find the optimal combination of outputs as well as the mix of inputs used to produce desired outputs  Examples of blending problems:  A company using a blending model would run the model periodically (each day, for example) and set production on the basis of current inventory of inputs and the current forecasts of demands and prices  Then the model would be run again to determine the next day’s production © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Example 14.2: Blending Oil.xlsx (slide of 2)  Objective: To develop an LP model for finding the revenue-maximizing plan that meets quality constraints and stays within the limits on crude oil availabilities  Solution: Chandler Oil has 5000 barrels of crude oil and 10,000 barrels of crude oil available  Chandler sells gasoline and heating oil, which are produced by blending the two crude oils together  Each barrel of crude oil has a quality level of 10, and each barrel of crude oil has a quality level of  Gasoline must have an average quality level of at least 8, whereas heating oil must have an average quality level of at least  Gasoline sells for $75 per barrel, and heating oil sells for $60 per barrel  The variables and constraints required for this model are listed below © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Example 14.2: Blending Oil.xlsx (slide of 2)  The spreadsheet model for this problem is shown below © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Logistics Models  In many situations a company produces products at locations called origins and ships these products to customer locations called destinations  Each origin has a limited capacity that it can ship, and each destination must receive a required quantity of the product  Logistic models can be used to determine the minimum-cost shipping method for satisfying customer demands © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Set-Covering Models  In a set-covering model, each member of a given set (set 1) must be “covered” by an acceptable number of another set (set 2)  The objective in a set-covering problem is to minimize the number of members in set necessary to cover all the members in set  Set-covering models have been applied to areas as diverse as airline crew scheduling, truck dispatching, political redistricting, and capital investment © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Example 14.10: Locating Hubs1.xlsx (slide of 3)  Objective: To develop a binary model to find the minimum number of hub locations that can cover all cities  Solution: Western Airlines wants to design a hub system that connects flights to and from cities within 1000 miles of each hub  The table below lists the cities that are within 1000 miles of other cities © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Example 14.10: Locating Hubs1.xlsx (slide of 3)  The variables and constraints for this set-covering model are listed in the table below   There is a binary variable for each city to indicate whether a hub is located there Then the number of hubs that cover each city is constrained to be at least one © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Example 14.10: Locating Hubs1.xlsx (slide of 3)  The spreadsheet model for Western and a graphical representation of the optimal solution are shown below © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Nonlinear Optimization Models  In many optimization models, the objective and/or the constraints are nonlinear functions of the decision variables  Such an optimization model is called a nonlinear programming (NLP) model  When you solve an NLP model, it is very possible that Solver will obtain a suboptimal solution  This is because a nonlinear function can have local optimal solutions that are not the global optimal solution  A local optimal solution is one that is better than all nearby points  The global optimum is the one that beats all points in the entire feasible region  There are mathematical conditions that guarantee that the Solver solution is the global optimum, but these are difficult to understand  A simpler way is to run Solver several times, each time with different starting values in the changing cells © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Example 14.11: Peak-Load Pricing.xlsx (slide of 2)  Objective: To use a nonlinear model to determine prices and capacity when there are two different daily usage patterns, peak-load and off-peak  Solution: Florida Power and Light must determine the price per kilowatt hour (kwh) to charge during both peak-load and off-peak periods  The daily demand for power during each period (in kwh) is related to price as follows:  Here, Dp and Pp are demand and price during peak-load times, and D0 and P0 are demand and price during off-peak times  Assume that it costs FPL $10 per day to maintain one kwh of capacity  The variables and constraints for this model are shown below © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Example 14.11: Peak-Load Pricing.xlsx (slide of 2)  The spreadsheet model is shown below © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Portfolio Optimization Models  Portfolio optimization models are used to determine the percentage of assets to invest in stocks, gold, and Treasury bills  To any work with investments, the following formulas must be understood:  All investors want to choose portfolios with high return (measured by the expected value in the first equation), but they also want portfolios with low risk (usually measured by the variance in the second equation)  The second equation can be rewritten slightly by using covariances instead of correlations, as shown below  This makes calculating the portfolio variance very easy with Excel’s ® matrix function © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Matrix Functions in Excel  Two built-in Excel matrix functions, MMULT and TRANSPOSE, simplify the calculation for the variance of portfolio return  The matrix is a rectangular array of numbers  A matrix is an i x j matrix if it consists of i rows and j columns  An example of a x matrix is:  If matrix A has the same number of columns as matrix B has rows, it is possible to calculate the matrix product of A and B, denoted AB  The Excel MMULT function performs matrix multiplications in a single step © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part The Portfolio Selection Model  Most investors have two objectives in forming portfolios: to obtain a large expected return and to obtain a small variance (to minimize risk)  The problem is inherently nonlinear because the portfolio variance is nonlinear in the investment amounts  The most common way of handling this two-objective problem is to specify a minimal required expected return and then minimize the variance subject to the constraint on the expected return  Financial analysts typically estimate the required means, standard deviations, and correlations for stock returns from historical data  However, there is no guarantee that these estimates are relevant for future returns © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Example 14.12: Portfolio Selection.xlsx (slide of 3)  Objective: To use NLP to find the portfolio that minimizes the risk, measured by portfolio variance, subject to achieving an expected return of at least 12%  Solution: Perlman & Brothers intends to invest a given amount of money in three stocks  The means and standard deviations of annual returns have been estimated as shown in the first table below  The correlations between the annual returns on the stocks are listed in the second table © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Example 14.12: Portfolio Selection.xlsx (slide of 3)  The company wants to find a minimum-variance portfolio that yields an expected annual return of at least 0.12 (that is 12%)  The variables and constraints for this model are presented below  One interesting aspect of this model is that it is not necessary to specify the amount of money invested © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Example 14.12: Portfolio Selection.xlsx (slide of 3)  The portfolio optimization model is shown below © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Modeling Issues  Typical real-world portfolio selection problems involve a large number of potential investments However, the basic model does not change at all  If a company is allowed to short a stock, the fraction invested in that stock is allowed to be negative  To implement this, eliminate the nonnegativity constraints on the changing cells  An alternative objective might be to minimize the probability that the portfolio loses money © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Keys to Solving Most Optimization Problems  Determine the changing cells  Set up the model so that you can easily calculate what you wish to maximize or minimize  Set up the model so that the relationships between the cells in the spreadsheet and the constraints of the problem are readily apparent  Optimization models not always fall into ready-made categories  A model might involve a combination of the ideas discussed in the production scheduling, blending, and aggregate planning examples © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part ... oil and 10,000 barrels of crude oil available  Chandler sells gasoline and heating oil, which are produced by blending the two crude oils together  Each barrel of crude oil has a quality level... level of 10, and each barrel of crude oil has a quality level of  Gasoline must have an average quality level of at least 8, whereas heating oil must have an average quality level of at least... least  Gasoline sells for $75 per barrel, and heating oil sells for $60 per barrel  The variables and constraints required for this model are listed below © 2015 Cengage Learning All Rights Reserved