A Survey of Software Packages for Teaching Linear and Integer Programming

Transcription

1 A Survey of Software Packages for Teaching Linear and Integer Programming By Sergio Toledo Spring 2018 In Partial Fulfillment of Math (or Stat) 4395-Senior Project Department of Mathematics and Statistics Faculty Advisor: Dr. Redl: Committee Member: Dr. Cahoy: Committee Member: Dr. Mhoon: Department Chair: Dr. Ryan Pepper

2 Table of Contents Abstract... 3 Introduction... 4 Body... 5 Software Used... 5 TORA... 7 R MATLAB Excel Integer Programming Conclusion Future Work References... 25

3 Abstract The purpose of the project was to survey a different number of software packages that can model and solve linear programming problems and choose the ideal software to teach students with in the Operations Research class being offered at UHD for the fall of 2018 semester. That includes integer and binary integer programming problems as well. The four software packages being surveyed are TORA, R, Excel, and MATLAB. These four were chosen because they have some degree of familiarity for students. TORA is the software that has been used in the past for the class and is also the software used to solve examples in the book used for the course. R is used in many statistics courses, MATLAB in some math, and Excel is something that all students should recognize. TORA is very easy to use, and data manipulation is also simple. Students can solve problems with a number of different algorithms, including the simple method that is a focal point of the class. R offers a wide variety of distinct functions that give varying results and details when solving optimization problems. MATLAB has the simplex function that can be used to solve linear programming problems, but integer and binary problems are a different story. Excel has a solver that allows students to solve all three problem types that is very easy to use and beneficial. In the end, all the software require a bit more work from the student to produce less than what TORA offers. Excel can be beneficial because the students would be able to set up the problems in ways that they can understand better when it comes to placing values. Even though, TORA is the best option, Excel can prove to be extremely beneficial early in the course.

4 Introduction Operations Research is a statistics course that involves making optimal decisions. Many different problem-solving techniques and methods are used to achieve the most efficient results. Linear, Integer, and Mixed Integer Programming problems are used to model and solve problems in operations research which makes them very pivotal topics for the class. Dr. Redl has taught the course in the past and has used the software TORA to solve these types of problems. For the most part, TORA has been useful teaching tool, but there is other software available that could be more beneficial for students learning experience. The best solving tool might not be the best teaching tool for the students and vice versa. The branch and bound method and the simplex method are two algorithms used to solve integer and linear programming problems respectively. It would be ideal to have a software that not just solves the problems but can better help the students understand these algorithms. There are many software packages that could be used to solve linear programming problems that could be evaluated in this project. I felt that I should keep the software that I will use to ones that the students can learn to use without too much effort in time. I also believe that I should assess software that the students have some familiarity with or have at least encountered in the past. Because of this, I decided to focus on Excel, MATLAB, and R in addition to TORA. As a statistics major, I have encountered every one of these and believe that the students who would end up enrolling in the operations research class will at the very least have used most if not all of them. Linear programming uses linear inequalities to model real world problems. The purpose is to find variables that will not only solve the inequalities but find the optimal solution. At times we will be trying to find the maximum values and other times we will search for the minimum. Graphing the inequalities will allow us to form what is called the feasible region. All the values

5 in this region are possible solutions, but only one will be the optimal solution. An important characteristic of the optimum LP solution is that it is always associated with a corner point of the solution space (Taha 18). If possible, it would be best to try find a software that will not only allow us to solve linear programming problems but allow us to also be able to display them graphically. The students can be better aided by the visual representations of these problems in addition to any algebraic methods that Dr. Redl will use. Over the course of this project, I will attempt to model and solve linear programming problems on different software to attempt to find an alternative to the software that is planned on being used in TORA. I will conduct this research knowing that TORA might very well be the best program available but felt that there was a possibility of finding an improvement. The course will be cross listed as both Math 4303 and Stat 4303, which will mean that I will have to take into account that students will not just be from the Statistics department. Ease of use for the students would also need to be a priority. It would not be advisable for Dr. Redl to use a different software that would require significant class time to be able to show the students how to use. If possible, it would be best to find a software that the students would be able to access from home. I will be modeling the acme glass co. problem in each software and discussing my experiences with each one. The acme glass co. problem is a simple two variable linear programming problem. Body Software Used There were four main software that I decided to focus on for the purpose of this project. TORA being the software used by Dr. Redl in his classes was the primary focus. It is a very useful teaching tool that I have used before in the Stat 4303 course. The software is extremely

6 easy to use but very basic with its user interface. The simplicity can be beneficial, but there is also room for improvements. A considerable number of courses in UHD use Microsoft Excel and it has a solver that can be used on linear programming problems. The solver is not installed in Excel by default. It is an add-in that can be installed in a matter of seconds. Taking a look at this solver could very well turn out to be beneficial because Excel is something that mostly every student in a math or statistics related course can be familiar with. It is also available at every computer on campus, which would undoubtedly be a great benefit for students. MATLAB can also be used to solve linear programming problems. There is a function named simplex that will allow students to plug in the coefficients of the problems and other significant values and get a solution with relative ease. Room s735 has the software installed in all its computers. That could be a room that students could go use if they do not have the software themselves since it is a product that would need to be purchased. Because R has been used in a number of different statistics courses that I have taken, I decided that it could be viewed as another option that could be assessed. Students could download the software on their personal computers if needed. Several linear programming problems were given to me by Dr. Redl that I will use to evaluate the different software. TORA and MATLAB files need to be specifically formatted so that the programs can read in the data correctly, but numbers can also be input without needing to reference a specific file. Difficulties inputting data or interpreting the solution outputs will be the key focus for assessing each program. The reason behind this is that I recognize that all of the programs will be capable of solving linear and integer programming problems. Because of this,

7 TORA serves its purpose well but there is chance that one of the other programs can be of more help to the students. TORA Having used TORA in the Decision Math course that Dr. Redl teaches, I became familiar with it and its user interface. I must admit that it does the job that one requires it to do, but it could be improved upon. Problems can either be uploaded via text documents or TORA will prompt the user to input each value manually. TORA requires the text files to be specifically formatted. It would take more effort on behalf of the students to create the text files instead of manual input. The textbook used for the course has a code that will not only allow the students to download TORA but will also include text files that are used in the chapters as examples. For any homework problems, the students would be required to solve, they would be better served manually inputting the different values. Luckily the manual input is easy to use. TORA asks for the number of variables and constraints so that it can generate a table where the user can insert the coefficients for the inequalities or equations. There are two options that the students will have when it comes to solution output. There will be a graphical solution available and an algebraic solution. The algebraic solution will allow the student to solve the problem using a number of different algorithms. The one that I want to focus on is the simplex method. The students can choose to get the final solution the first time of asking or viewing each iteration before receiving the final answer. Choosing to view the iterations, gives the students a tableau with the values that the simplex method chooses with each iteration. The students could solve the problem by hand and compare with the tableau so that they can be sure that they understand how the algorithm helps find the optimal solution. To better understand the user interface that TORA offers, I have placed several screenshots of the process of solving a linear programming problem. The example used is the

8 wyndoor problem; also called the acme glass co. problem. It involves a company that makes doors and windows. The profit is $3 and $5 for door and windows respectively. The values can be any multiple with the same ratio. 300 and 500 for example. We have three constraints in this problem. The constraints represent production plants and available hours to make the doors and windows. Each plant has different production times that represent the coefficients in the inequalities. The plants have certain number of hours available for production weekly. This is represented by the upper bound placed on these inequalities. Plant 1 cannot exceed 4 hours for example. Students can edit any of the values in the boxes colored in the darker blue. Doors and windows can be renamed. The inequalities can be changed to less than or greater than or equal as stated in the light blue column. Nothing in the light blue boxes can be changed other than the one that reads maximize. By clicking it, the problem can be changed in to a minimization problem. The students can then click on the solve menu button to choose their solution method.

9 As stated below, there are many options available to the students to solve the problems. The bounded simplex option is the preferred method for the class. The TORA output produced the tableau with three iterations. Having solved the problem by hand, the values I found for the solution were 0,30, and 36 for iterations 1, 2, and 3 respectively. The solution to the problem was 36 which was also shown by TORA. The solution requires 6 windows and 2 doors to be produced.

10 TORA also features a graphical solution for two-dimensional problems. Although the graphical output does not use the simplex method, having a visual representation of the problem can be of some benefit for the students. The graphical solution will allow the students to click on each constraint function to add it to the graph until the feasible region is clearly visible. They can then solve the problem and there will be an iso-profit line that will be shown moving across the graph until it finds the optimal solution. The figure above is the graphical solution to the wyndoor problem with 2 and 6 being the optimal values for the production of doors and windows. While the graph does not use the simplex method, it can prove to be a valuable tool for students looking to get some type of visual aid. The feasible region is still shown in the graph. The feasible region represents all the points that would be viable solutions to the problem according to the constraints given. Even though every point inside the feasible reason can be used, there is only one optimal solution and it is still shown by the graph.

11 R Solving the same problem with R is not as simple as just filling in some blanks or uploading a file. There are a number of different packages that have functions that can solve linear programming problems. The lpsolve and Rsymphony packages can both be used as well as the boot package. The boot package is the more useful of the three as it has a simplex function, while the others have functions that solve the problems with other means. The book Modeling and Solving Linear Programming with R provided some insight for this project. It was an open access book that anybody can download for free, which was a big bonus. The book is very brief but informative. It was made to be a short introduction into the world or linear programming with R. Unfortunately, the book does not use the boot package to solve the linear programming problems with R. It does give an explanation of the simplex method but does not show any functions used to solve the problems with the simplex method. The boot package seemed like the best option because of the simplex function included. The boot package needs to be installed and the library needs to be added with the following command: Four parameters need to be set for the simplex function. The first is a, a vector with the objective function values. For the wyndoor problem that is 3 and 5. Secondly A1, a matrix with the coefficients of the constraints on the left-hand side of the inequalities. The values need to be inserted from left to right, top to bottom. Third b1, a vector containing the right-hand side of the inequalities being the upper bound for the constraints. The function has a parameter named maxi that needs to be set to true if it is a maximization problem or false if minimizing. Leaving it blank will also default to minimization. It is important that A1 and b1 are specified. Having those

12 two parameters input into the function tells it that the inequalities are less than. If A2 and b2 are used, the function will set the inequalities to greater than. And A3 and b3 set the functions to equalities and not inequalities. Below is the code required to solve the wyndoor problem with the simplex function along with the output. The solution is found but no other information is given. There is no output showing the iterations of the simplex method. Students who have no previous experience with R might need some time to get used to the input method, but it is not a very rigorous process. They should be able to get into the flow of thins with practice. There is another called linprog that uses the simplex method to solve lp problems. The function solvelp provides a few more details in the output, although the input method is slightly different. I was able to use the same variables that I defined with the simplex function and input them in the solvelp function, but in a different order. The order of input is a, b1, and A1. The right-hand side of the inequalities is input before the left-hand side. The left-hand side can still be in matrix format for the function to read it. Another difference was that the signs for the inequalities need to be set inside another parameter named const.dir. This parameter is a vector

13 containing the signs of the inequalities in order of top to bottom. Below is the code that I used along with the output. The output for solvelp certainly gives a few more details than the simplex function. It gives us the solution, the values of the variables of the objective function, the values of the inequalities and how they satisfy the constraints, and the variables including the slack variables. It does not show the actual values obtained with each iteration but is a definite improvement over the

14 simplex function. Again, the input can take some getting used to on the part of the students, but it should not be something that they should struggle with for too long. As far as for the students learning about the simplex method, they can gain some insight with the solvelp function. The output is helpful in showing students how the solution satisfies the constraints. The solution shows how the first factory still has 2 hours free but making any more doors would result in violating some of the other constraints. There is definitely some potential with this specific package. The students could benefit from using a free open source program like R. It would allow them to install the program on any of their personal computers and it is also installed on most computers on campus. One of the drawbacks is the lack of packages that provide graphics for linear programming problems. There is a package called the intpoint package that provides function with a graphical solution, but it uses the interior point method, which is not something that is needed for the class. I have attached an image with code and the graphical output.

15 The graph does plot the linear equations but does not highlight the feasible region enough. It could prove to be useful, but the students could end up somewhat confused by the added points on the inside along with the pattern. The output also shows instead of 36 which could be detrimental to the students learning experience. MATLAB MATLAB is another option that the students have at their disposal on many of the computers on campus. It is a pay to use software if the students would like to have it on their personal computers, but the availability could make that unnecessary. There is a simplex function that Dr. Redl could send the students to use to solve the problems on their own. It solves the problems and produces a tableau somewhat similar to the one provided by TORA. The function is very similar to the solvelp function used in R as far as the numerical input goes. A

16 vector with the coefficients of the objective function is entered first. That is followed by a vector with the right-hand side of the inequalities and a matric with the coefficients of the left-hand side of the inequalities. There is also the parameter inq that is similar to the const.dir variable in the solvelp function. Inq is a vector with the sign of the inequalities or equations. Using the acme problem, the signs for all three constraints is <=. Instead of inputting the sign directly, a -1 is entered to imply less than. The last parameter is set to 0 if the problem needs to be maximized and 1 if it needs to be minimized. Below I have attached screenshots with the code and the tableau that was printed.

17 As shown in the second tableau below, the values are in scientific notation. It is not ideal, but the students should have no problem interpreting the output. The only issue the students might have with the tableau is that it is displayed using scientific notation. The particular file had values of 3000 and 5000 instead of 3 and 5 making the optimal value instead of 36. The same solution for the number of doors and windows to be produced was still achieved. The students should have very little problem with using the simplex function on MATLAB and the tableau could be very useful in Dr. Redl s lectures. The drawback is the lack of graphics packages available for linear programming. The students would have to be fairly competent with MATLAB to even be able to produce a graph with the feasible region highlighted.

18 Excel Excel is the final software I decided to assess in this project. It is something that every student should feel familiar with whether they be statistics or math majors taking the course. Most if not all students should also have access to Excel as it is available on all the computers on campus. There is a solver ad-in that the students can install to help them solve linear programming problems. Students can access this solver by going to their options on the file tab. There is a tab named Add-ins that they click on and that will display a list of the add-ins with the solver being the last one. There is a manage add ins option on the bottom with a button saying go. The students can click on after that, they can click on the check box next to the solver and press ok. After doing all that, the solver add in is added to the data tab in the analyze section. The students will need to set up each problem specifically to use the solver however. They can place the values in any format they like, but when using the solver, they need to make sure that they highlight the correct values. I set up the values for the objective function on top with the inequality constraints under. I also labeled a column as used to show how many hours will be used in each factory when the solver

19 gives us the solutions. Batches produced is set to 0 but will change once the solver finds the solution to the number of doors and windows to produce along with the optimal profit to be made. The sumproduct function will be used to calculate the hours used to make the doors and windows. As shown in the image, D7 will equal to B7*B13+C7*C13. Once the solver finds the number of products to make, the value will be updated to show that it satisfies the constraint given. To use the solver, the student will click on the solver and a panel will open up asking for the objective, decision variables, and the constraints. It will also have options to maximize, minimize, or set a specific value. It also allows the user to specify the solving method, set unconstrained values to nonnegative and even show the results of the iterations. The objective is the profit we want to maximize in cell D13. Changing variable cells will be decision variables that will increase to show the number of doors and windows to make in cells B13 and C13. We can add constraints with the add button and we must simply highlight the

20 region D7-D9 on the left and F7-F9 on the right. We can highlight the region because the inequality constraints al have the same sign. If we had other constraints with different signs, we would have to add them individually cell by cell. Pressing ok takes the user back to the main solver panel, while pressing add will allow the user to keep adding more constraints if needed. After that, we can solve and receive the solution. Showing the iterations will produce the following results:

21 We have 30 as the first possible solution to satisfy the constraints by making no doors and 6 windows. We click continue to check whether there is a better solution and we get a profit of 36 with 2 doors and 6 windows produced. We again click continue to see if there is a better solution and we get the following output: A profit of 36 is the optimal solution and the iterations come to an end. The excel solver is a great tool because of the display of iterations along with the values produced. The constraint

22 drop down list for the signs also allows the user to set integer constraints to turn the problem into an integer programming problem. It seems that there are many benefits to using excel to help students understand the iterative nature of linear programming. The fact that the students must set up the problem properly could be time consuming, but it could also help the students understand what is going on with the problem itself and what the solver is doing to get an optimal solution. Their understanding of the process would improve by setting up each problem. Integer Programming As far as integer programming, much of the process is the same when it comes to using TORA. The student can first click on the integer programming option in the main menu and that will take them to same input screen as the linear programming with the integer box set to yes. Students can also set the upper bound to 1 so that they can make the problem a binary integer programming problem. This models decision making problems like whether or not to make and investment. There are no graphical solutions available in TORA for integer programming. The

23 students can solve the problems using the branch and bound method. This method splits the feasible region into smaller subsets and evaluates them to find the optimal solution. The students have the option to guide the software when solving the problem or to obtain the solution immediately. This is a big help as the students can practice their knowledge of the algorithm when branching the feasible solutions. The lp function in the lpsolve package can be used to solve integer programming problems in R. The function itself is similar to the other functions that solve linear programming problems. The parameters can all be entered the same way. It has a few more parameters that are very important. The int.vec parameter is entered as a vector with the indices of the variables that need to be integer. There is also the all.int parameter that can be set to true when all the variables are required to be integer. The same goes for binary as there are parameters binary.vec and all.bin that function the same way as the two stated above. The optimal value is output. MATLAB didn t offer any ready-made functions for solving integer programming problems. Luckily Excel does have the option. As stated previously, when entering the constraints in the solver, there is a drop-down menu that also has int and bin options. The variables that need to be integer or binary are simply highlighted and Excel can solve it with no problem. Conclusion Each of the four software used can help the students solve the linear programming problems for the operations research class, but not all of them give them the same results. Regarding R, the solvelp function produces the best set of results and details out of all the functions found in the different packages available to R. Most statistics students should be somewhat familiar with R and the interface it provides that they should not struggle too much if

24 they needed to use it. Math majors might find things more difficult at the beginning, but R is very simple to use, and they would not have to deal with a great number of different packages to be overwhelmed. There are limited options to graph the linear programming problems and the ones available have unnecessary information that the students will not need and that could possibly confuse them. MATLAB is a good option because it is used throughout various courses offered at UHD. The tableau can give students insight into the simplex method and the different solutions that are achieved with each iteration. Something that the packages in R do not provide. A small drawback is that students would not be able to download the software onto their own personal computer without having to spend some money. The biggest drawbacks are the lack of graphics packages and binary integer programming solving functions. Excel is an excellent alternative to TORA because of the familiarity that the students would have in using it. I would imagine that at this stage, every undergraduate student has used Excel at one point. Setting up the problems might be more time consuming than the other software, but it would also allow the students to grasp some concepts behind the processes used to solve the linear programming problems. Being able to so freely manipulate the data is also a big plus as is the display of iterations and integer constraint options. In conclusion, any one of these software packages could be a worthy replacement for TORA. But in the end, TORA offers the best ease of use. The students can upload files given to them or simply input the corresponding values into the correct boxes. The simplicity will allow the students to not have to focus on how to use different software and relay all their attention to learning about the different algorithms used to solve the linear programming problems. The TORA output will not only solve the problems but display the information attained with each

25 iteration in a tableau. Students can choose between integer and regular linear programming and they can even produce graphs for two dimensional problems that highlight the feasible region for the problems. The fact that the author of the textbook that will be used in the operations research class is the creator of TORA is a big plus. The download for the software will come with files containing examples used throughout the book. There is some room for improvement for TORA, but it is a more specialized software for linear programming compared to all the other software surveyed making it the best option available at this point. One recommendation is to use Excel when first teaching the students linear programming so that setting up the problem in a way that they understand can help them get a better grasp of the subject. Once the students are comfortable with solving, they could use TORA to input problems and get solutions without using up too much time. Future Work There are two other software packages that would be worth taking a look at in the near future. The first of those is the Gurobi optimizer. Gurobi has a solver for optimization problems and has functionalities with C++, Python, R, Excel, C, MATLAB and Java. The fact that the software can be used with other environments and software could make it an invaluable resource in the future. There is also the Julia language. Julia is a high-level programming language that is increasingly used in the data science community. It is also capable of solving optimization problems and I hope to be able to work with it over the summer of 2018 to get more insight on the language. References Operations Research: An Introduction (8th Edition)-Taha

26 Modeling and Solving Linear Programming with R Sallan Models: Linear Programming Terminology Decision Support Models: Linear Programming I 20week%2011a.pdf