I'm not sure how "good" I am at it - but I do teach quant methods now and have done so on and off for the past 20 years. Regards - Andy
I have been looking at this particular problem for a couple of hours and have some formulations. However, I am a little confused on how to formulate a particular portion of the equasion. I have $1K to spend on newspaper and radio ads. I have to spend a minimum of $250 on each media. I need to spend twice the amount on newspaper than on radio. An index has been set to measure audience exposure on a scale of 0-100. Radio is 80, and newspaper is 50. If I use X1 = Radio X2 = Newspaper How do I quantify the indexes into this equation? Would it read 80X1 + 50X2 = $1000 With the constraints being: X1 >/= 250 (2) X2 >/= 250 Any suggestions? I find math classes very hard to take on line. I asked the professor but received a less than understandable response.
ok here's a hint. There can't be a solution to the problem that you've formulated. The equality can't hold if the the constraints involving X1 and X2 also hold.
To express the problem mathematically, decision variables are required to represent each of the two types of ads. Assigning X1 to radio ads and X2 to newspaper ads, the objective function can be expressed as MAX: 80X1 + 50X2, as the objective is to maximize audience exposure. The constraints on the results in terms of the decision variables must be identified. A minimum of $250 must be spent on each media; these constraints can be expressed as X1 > 250 and X2 > 250. At least (I am assuming that it is at least and not exactly) twice the amount spent on radio must be spent on newspaper. This can be expressed as X2 > 2X1. The total budget for ad expenditures is $1000.00. This constraint is expressed as X1 + X2 < 1000 (I am assuming that you don’t have to spend exactly $1000.00). A nonnegativity condition specifies that the decision variables cannot take any negative values, as it is not possible to spend a negative amount of money. This is expressed as X1, X2 > 0. The complete linear programming model, therefore, would be as follows: MAX: 80X1 + 50X2 Subject to: X1 > 250 X2 > 250 X2 > 2X1 X1 + X2 < 1000 X1, X2 > 0 Microsoft Excel and its built-in tool, Solver, can be used to solve the optimization problem.
A couple of questions to spark your thinkiing: 1. What does X1 and X2 stand for? Is it minutes of advertising or dollars of advertising? The answer to this question should help you in solving the problem 2. What is your objective here? Are you minimizing cost or maximizing exposure? 3. When I see 80X1 + 50X2 = $1000 I think you've confused the objective function with a constraint. 4. How about " I need to spend twice the amount on newspaper than on radio." It would seem you need a constraint to enforce this. Regards - Andy
Wow - thanks for all of the great responses. X1 = dollars spent on radio ads X2 = dollars spent on newspaper ads I used the figure 80X1 + 50X2 = $1,000 based on the hypothesis that the 80 and 50 would come into play at this point in the equasion because the 80 and 50 correspond to the audience exposure ratings. Should I put these audience exposure ratings as a constraint, or part of the actual solving equasion? I am using Management Scientist as the tool on these programs. I would rather use Excel (as I understand this program better), but I have to format my answer in MS per the professor's instructions. I can usually finish an assignment in one sitting. I have been working on the five questions for this week for no less than 16 hours this weekend. I think this class is going to be the kicker - hoepfully I can keep my head above water for the 8 weeks of this class and the 12 weeks of the actual QMS class next quarter. After that, it is just writing (marketing classes are easy!) W.
Here is some feedback... Ok - so X1 and X2 have to be <= 1000. Are you sure that want to work directly with dollars of radio and TV? Or do you want to work with minutes of radio and # of ads? I ask this because I'm not sure of the 80 and 50 figure - do they apply to dollars spent on advertising? Or to minues/# of ads? Also, think about the idea of twice the newspaper spending compared to radio. How would you express in algebra? How can you turn this into a constraint? I think that you have the objective function (80 x1 and 50 x2) confused with a constraint - namely that X1 and X2 have to be = 1000. I suspect that the 80 and 50 are part of the objective function. But a question - does the 80 and 50 apply to dollars spent on radio and newspaper? Or is it # of minutes of radio or # of ads? ok - this is his preference - although one can use Solver to solve this problem.
A Mathematical Programming Language If anyone wants to pursue LP applications as well as Integer Programming, you may want to down load AMPL off the internet site. It is a lot more versatile than Excel Solver and you can also download the CPLEX 8.0 optimization package to go with it. Best part is that it is all free of charge. I've been using the WinAmp interface for the last year and it isn't bad for being free. Dick
the 80 and 50 are not dollars -- but indexes of media effectiveness (based on a scale of 0-100, where 100 means the media is totally effective). It does not appear to have anything to do with money or minutes. The problem is based on dollars, not minutes. Thanks! W.
Gus has it right (if X2 > 2X1 holds). You can also download from the internet the package Lindo, which I think it makes it extremely easy. If you give me an email address I can send you a student version. Regards PD Does anyone know if this LP is used in the real world? And if so, could you provide examples of companies that use it?
Simplex Algorithm Hand cranking the Simplex algorithm takes time, but Lindo will make this problem easy! Lindo just applies the Simplex algorithm at lightening speed. I used Lindo and Lingo quite extensively in my graduate program. http://planetmath.org/encyclopedia/SimplexAlgorithm.html http://www.lindo.com/cgi/frameset.cgi?leftdwnld.html;downloadf.html
