O-Matrix is an integrated environment for analysing and visualizing data, and building turnkey scientific and engineering computing solutions. The program includes hundreds of engineering and statistical functions for solving a broad range of technical computing problems. Easy-to-use and flexible plotting commands enable you to rapidly build design prototypes, and implement sophisticated systems.
The foundation of O-Matrix is a high-performance matrix language that is specifically designed for high-performance technical computing. The notation of this language will dramatically reduce your design and implementation efforts, and enable the construction of systems that execute far quicker than other interpreted environments. O-Matrix also provides a compatibility mode that enables you to run MATLABŠ m-files. This enables you to leverage existing m-files, and simplifies the transition to O-Matrix for users experienced with MATLAB.
The O-Matrix environment is interpreted which means your commands are immediately executed as you enter them. Textual output is displayed in the Command window, and plotting commands are displayed in one or more Graphic windows. The environment provides a debugger for debugging, analysing, and profiling complex algorithms.
We will not discuss the specifics of O-Matrix here but instead refer the reader to the O-Matrix website.
lpsolve is callable from O-Matrix via a dynamic linked DLL function. As such, it looks like lpsolve is fully integrated with O-Matrix. Matrices can directly be transferred between O-Matrix and lpsolve in both directions. The complete interface is written in C so it has maximum performance. The whole lpsolve API is implemented with some extra's specific for O-Matrix (especially for matrix support). So you have full control to the complete lpsolve functionality via the omlpsolve O-Matrix driver. If you find that this involves too much work to solve an lp model then you can also work via higher-level script files that can make things a lot easier. See further in this article.
Compile and build omlpsolve: ---------------------------- 1. Under Windows, the Microsoft Visual C/C++ compiler must be installed and the environment variables must be active do that when a command prompt is opened, the cl and nmake commands can be executed. 2. Go to directory lp_solve_5.1\extra\O-Matrix\lpsolve 3. To compile and build omlpsolve, enter the following command: cvc Load the omlpsolve driver in the O-Matrix memory space: ------------------------------------------------------- 1. Under Windows, make sure that the lpsolve51.dll file is somewhere in the path 2. A precompiled library is provided for Windows (omlpsolve.dll). 3. Start O-Matrix 4. Enter the following command in O-Matrix: O> dll <path>\omlpsolve.dll, omlpsolve This can also be added in autoexec.oms to automatically load the omlpsolve driver.
Note that O-Matrix version 5.8 or above is strongly recommended. Lower versions (at least 5.7) should work with this driver, but these versions don't have the ability to print information on the command prompt. For example default while a solve is done, information is printed to the command window. This will only be visible from version 5.8 or above.
O-Matrix is ideally suited to handle linear programming problems. These are problems in which you have a quantity, depending linearly on several variables, that you want to maximize or minimize subject to several constraints that are expressed as linear inequalities in the same variables. If the number of variables and the number of constraints are small, then there are numerous mathematical techniques for solving a linear programming problem. Indeed these techniques are often taught in high school or university level courses in finite mathematics. But sometimes these numbers are high, or even if low, the constants in the linear inequalities or the object expression for the quantity to be optimized may be numerically complicated in which case a software package like O-Matrix is required to effect a solution.
To make this possible, a driver program is needed: omlpsolve (omlpsolve.dll under Windows). This driver must be loaded in O-Matrix and O-Matrix can call the omlpsolve solver.
This driver calls lpsolve via the lpsolve shared library (lpsolve51.dll under Windows). This has the advantage that the omlpsolve driver doesn't have to be recompiled when an update of lpsolve is provided. The shared library must be somewhere in the Windows path.
So note the difference between the O-Matrix lpsolve driver that is called omlpsolve and the lpsolve library that implements the API that is called lpsolve51.
There are also some O-Matrix script files (.oms) as a quick start.
The first thing that must be done, each time O-Matrix is restarted and you want to use lpsolve is load the omlpsolve driver into the O-Matrix workspace. This is done via the dll command. Suppose that omlpsolve.dll is installed in c:\omwin\dll, then the following command must be used to load the driver:
dll c:\omwin\dll\omlpsolve.dll, omlpsolve
That is basically all you need to do. From now on, you can use the library. This until a clear command is given or O-Matrix is restarted. Then this command must be given again to reload the library.
To make things easier, you can edit the file autoexec.oms with your favourite editor (or notepad) in the omwin folder and add above line at the end of this file (before the last end). That will automatically load the lpsolve driver in memory when O-Matrix is started and also when a clear command is executed. So it will appear as if the omlpsolve command is then always available.
To test if everything is installed correctly, enter omlpsolve in the O-Matrix command prompt. If it gives the following, then everything is ok:
omlpsolve O-Matrix Interface version 5.1.0.1 using lpsolve version 5.1.1.3 Usage: [ret1, ret2, ...] = omlpsolve("functionname", arg1, arg2, ...)
Possibly, this is followed with:
No printing capability to command window available. You need to upgrade to version 5.8 for this feature.
Then you are working with an O-Matrix version lower than 5.8. The driver should work, but nothing is printed on the command window when lpsolve has something to report (for example while solving).
However, if you get a message box with the following:
The identifier omlpsolve is not defined.
Then either the dll command that was previous given was unsuccessful (or not given at all) or something was misspelled after the ,
If you get the following:
This application has failed to start because lpsolve51.dll was not found. Re-installing the application may fix this problem.
Then O-Matrix can find the omlpsolve driver program, but the driver program cannot find the lpsolve library that contains the lpsolve implementation. This library is called lpsolve51.dll and should be on your system in a directory that in the PATH environment variable. This path can be shown via the command getenv("PATH")
The lpsolve51.dll files must be in one of these specified directories. It is common to place this in the WINDOWS\system32 folder.
All this is developed and tested with O-Matrix version 5.7. This is the minimum supported release. Older releases are unsupported.
In the following text, O> before the O-Matrix commands is the O-Matrix command line. Only the text after O> must be entered.
To call an lpsolve function, the following syntax must be used:
O> [ret1, ret2, ...] = omlpsolve("functionname", arg1, arg2, ...)
The return values are optional and depend on the function called. functionname must always be enclosed between double quotes to make it alphanumerical and it is case sensitive. The number and type of arguments depend on the function called. Some functions even have a variable number of arguments and a different behaviour occurs depending on the type of the argument. functionname can be (almost) any of the lpsolve API routines (see lp_solve API reference) plus some extra O-Matrix specific functions. Most of the lpsolve API routines use or return an lprec structure. To make things more robust in O-Matrix, this structure is replaced by a handle. This is an incrementing number starting from 0 and the lprec structures are maintained internally by the omlpsolve driver. However you will see not much (if any) difference in the use of it.
Almost all callable functions can be found in the lp_solve API reference. Some are exactly as described in the reference guide, others have a slightly different syntax to make maximum use of the O-Matrix functionality. For example make_lp is used identical as described. But get_variables is slightly different. In the API reference, this function has two arguments. The first the lp handle and the second the resulting variables and this array must already be dimensioned. When lpsolve is used from O-Matrix, nothing must be dimensioned in advance. The omlpsolve driver takes care of dimensioning all return variables and they are always returned as return value of the call to omlpsolve. Never as argument to the routine. This can be a single value as for get_objective (although O-Matrix stores this in a 1x1 matrix) or a matrix or vector as in get_variables. In this case, get_variables returns a 4x1 matrix (vector) with the result of the 4 variables of the lp model.
(Note that you can execute this example by entering command per command as shown below or by just entering example1. This will execute example1.oms.)
O> lp=omlpsolve("make_lp", 0, 4); O> omlpsolve("set_verbose", lp, 3); O> omlpsolve("set_obj_fn", lp, [1, 3, 6.24, 0.1]); O> omlpsolve("add_constraint", lp, [0, 78.26, 0, 2.9], 2, 92.3); O> omlpsolve("add_constraint", lp, [0.24, 0, 11.31, 0], 1, 14.8); O> omlpsolve("add_constraint", lp, [12.68, 0, 0.08, 0.9], 2, 4); O> omlpsolve("set_lowbo", lp, 1, 28.6); O> omlpsolve("set_lowbo", lp, 4, 18); O> omlpsolve("set_upbo", lp, 4, 48.98); O> omlpsolve("set_col_name", lp, 1, "COLONE"); O> omlpsolve("set_col_name", lp, 2, "COLTWO"); O> omlpsolve("set_col_name", lp, 3, "COLTHREE"); O> omlpsolve("set_col_name", lp, 4, "COLFOUR"); O> omlpsolve("set_row_name", lp, 1, "THISROW"); O> omlpsolve("set_row_name", lp, 2, "THATROW"); O> omlpsolve("set_row_name", lp, 3, "LASTROW"); O> omlpsolve("write_lp", lp, "a.lp"); O> omlpsolve("get_mat", lp, 1, 2) 78.2600 O> omlpsolve("solve", lp) 0 O> omlpsolve("get_objective", lp) 31.7828 O> omlpsolve("get_variables", lp) { 28.6 0 0 31.8276 } O> omlpsolve("get_constraints", lp) { 92.3 6.8640 391.293 }
Note that there are some commands that return an answer. To see the answer, the command was not terminated with a semicolon (;). If the semicolon is put at the end of a command, the answer is not shown. However it is also possible to write the answer in a variable. In that case the result is never shown. With or without a terminating semicolon. For example:
O> obj=omlpsolve("get_objective", lp)
Or:
O> obj=omlpsolve("get_objective", lp);
Both will only write the result in variable obj without showing anything on screen. get_variables and get_constraints return a vector with the result. This can also be put in a variable:
O> x=omlpsolve("get_variables", lp); O> b=omlpsolve("get_constraints", lp);
It is always possible to show the contents of a variable by just giving it as command:
O> x { 28.6 0 0 31.8276 }
Don't forget to free the handle and its associated memory when you are done:
O> omlpsolve("delete_lp", lp);
O> omlpsolve("add_constraint", lp, [0.24, 0, 11.31, 0], 1, 14.8);
Most of the time, variables are used to provide the data:
O> omlpsolve("add_constraint", lp, a1, 1, 14.8);
Where a1 is a matrix variable.
Matrices with too few or too much elements gives an 'invalid vector.' error.
Most of the time, omlpsolve needs vectors (rows or columns). In all situations, it doesn't matter if the vectors are row or column vectors. The driver accepts them both. For example:
O> omlpsolve("add_constraint", lp, {0.24, 0, 11.31, 0}, 1, 14.8);
Which is a column vector, but it is also accepted.
An important final note. Several lp_solve API routines accept a vector where the first element (element 0) is not used. Other lp_solve API calls do use the first element. In the O-Matrix interface, there is never an unused element in the matrices. So if the lp_solve API specifies that the first element is not used, then this element is not in the O-Matrix matrix.
Because O-Matrix is all about matrices, all lpsolve API routines that need a column or row number to get/set information for that
column/row are extended in the omlpsolve O-Matrix driver to also work with matrices. For example set_int in the API can
only set the integer status for one column. If the status for several integer variables must be set, then set_int
must be called multiple times. The omlpsolve O-Matrix driver however also allows specifying a vector to set the integer
status of all variables at once. The API call is: return = omlpsolve("set_int", lp_handle, column, must_be_int). The
matrix version of this call is: return = omlpsolve("set_int", lp_handle, [must_be_int]).
The API call to return the integer status of a variable is: return = omlpsolve("is_int", lp_handle, column). The
matrix version of this call is: [is_int] = omlpsolve("is_int", lp_handle)
Also note the get_mat and set_mat routines. In O-Matrix these are extended to return/set the complete constraint matrix.
See following example.
Above example can thus also be done as follows:
(Note that you can execute this example by entering command per command as shown below or by just entering example2.
This will execute example2.oms.)
O> lp=omlpsolve("make_lp", 0, 4); O> omlpsolve("set_verbose", lp, 3); O> omlpsolve("set_obj_fn", lp, [1, 3, 6.24, 0.1]); O> omlpsolve("add_constraint", lp, [0, 78.26, 0, 2.9], 2, 92.3); O> omlpsolve("add_constraint", lp, [0.24, 0, 11.31, 0], 1, 14.8); O> omlpsolve("add_constraint", lp, [12.68, 0, 0.08, 0.9], 2, 4); O> omlpsolve("set_lowbo", lp, [28.6, 0, 0, 18]); O> omlpsolve("set_upbo", lp, [INF, INF, INF, 48.98]); O> omlpsolve("set_col_name", lp, {"COLONE", "COLTWO", "COLTHREE", "COLFOUR"}); O> omlpsolve("set_row_name", lp, {"THISROW", "THATROW", "LASTROW"}); O> omlpsolve("write_lp", lp, "a.lp"); O> omlpsolve("get_mat", lp) { [ 0 , 78.26 , 0 , 2.9 ] [ 0.24 , 0 , 11.31 , 0 ] [ 12.68 , 0 , 0.08 , 0.9 ] } O> omlpsolve("solve", lp) 0 O> omlpsolve("get_objective", lp) 31.7828 O> omlpsolve("get_variables", lp) { 28.6 0 0 31.8276 } O> omlpsolve("get_constraints", lp) { 92.3 6.8640 391.293 }
Note the usage of INF in set_upbo. This stands for "infinity". Meaning an infinite upper bound. It is also possible to use -INF to express minus infinity. This can for example be used to create a free variable.
To show the full power of the matrices, let's now do some matrix calculations to check the solution. It works further on above example:
O> A=omlpsolve("get_mat", lp); O> X=omlpsolve("get_variables", lp); O> B = A * X O> B { 92.3 6.864 391.293 }
So what we have done here is calculate the values of the constraints (RHS) by multiplying the constraint matrix with the solution vector. Now take a look at the values of the constraints that lpsolve has found:
O> omlpsolve("get_constraints", lp) { 92.3 6.864 391.293 }
Exactly the same as the calculated B vector, as expected.
Also the value of the objective can be calculated in a same way:
O> C=omlpsolve("get_obj_fn", lp); O> X=omlpsolve("get_variables", lp); O> obj = C * X O> obj 31.7828
So what we have done here is calculate the value of the objective by multiplying the objective vector with the solution vector. Now take a look at the value of the objective that lpsolve has found:
O> omlpsolve("get_objective", lp) 31.7828
Again exactly the same as the calculated obj value, as expected.
O-Matrix can execute a sequence of statements stored in files. Such files are called oms files because they must have the file type of ".oms" as the last part of their filename (extension).
oms scripts can be compared with batch files or scripts. You can put O-Matrix commands in them and execute them at any time. The oms script is executed like any other command, by entering its name (without the .oms extension).
The omlpsolve O-Matrix distribution contains some example oms scripts to demonstrate this.
You can also edit these files with your favourite text editor (or notepad).
Contains the commands as shown in the first example of this article. To execute and also see which commands are executed in the debug window, use following commands:
O> stop O> trace on example1.oms O> quit O> example1
Note however that execution is much slower when trace is on. It is only used here to see the statements executed.
Contains the commands as shown in the second example of this article. To execute and also see which commands are executed in the debug window, use following commands:
O> stop O> trace on example2.oms O> quit O> example2
Note however that execution is much slower when trace is on. It is only used here to see the statements executed.
Contains the commands of a practical example. See further in this article.
Contains the commands of a practical example. See further in this article.
Contains the commands of a practical example. See further in this article.
Contains the commands of a practical example. See further in this article.
This script uses the API to create a higher-level function called lp_solve. This function accepts as arguments some matrices and options to create and solve an lp model. See the beginning of the file to see its usage:
LP_SOLVE Solves mixed integer linear programming problems. SYNOPSIS: [obj,x,duals] = lp_solve(f,a,b,e,vlb,vub,xint,scalemode,keep) solves the MILP problem max v = f'*x a*x <> b vlb <= x <= vub x(int) are integer ARGUMENTS: The first four arguments are required: f: n vector of coefficients for a linear objective function. a: m by n matrix representing linear constraints. b: m vector of right sides for the inequality constraints. e: m vector that determines the sense of the inequalities: e(i) = -1 ==> Less Than e(i) = 0 ==> Equals e(i) = 1 ==> Greater Than vlb: n vector of lower bounds. If empty or omitted, then the lower bounds are set to zero. vub: n vector of upper bounds. May be omitted or empty. xint: vector of integer variables. May be omitted or empty. scalemode: scale flag. Off when 0 or omitted. keep: Flag for keeping the lp problem after it's been solved. If omitted, the lp will be deleted when solved. OUTPUT: A nonempty output is returned if a solution is found: obj: Optimal value of the objective function. x: Optimal value of the decision variables. duals: solution of the dual problem.
Example of usage. To create and solve following lp-model:
max: -x1 + 2 x2; C1: 2x1 + x2 < 5; -4 x1 + 4 x2 <5; int x2,x1;
The following command can be used:
O> include lp_solve.oms O> [obj, x]=lp_solve([-1, 2], {[2, 1], [-4, 4]}, [5, 5], [-1, -1], [], [], [1, 2]) O> obj 3 O> x { 1 2 }
This script is analog to the lp_solve script and also uses the API to create a higher-level function called lp_maker. This function accepts as arguments some matrices and options to create an lp model. Note that this scripts only creates a model and returns a handle. See the beginning of the file to see its usage:
LP_MAKER Makes mixed integer linear programming problems. SYNOPSIS: lp_handle = lp_maker(f,a,b,e,vlb,vub,xint,scalemode,setminim) make the MILP problem max v = f'*x a*x <> b vlb <= x <= vub x(int) are integer ARGUMENTS: The first four arguments are required: f: n vector of coefficients for a linear objective function. a: m by n matrix representing linear constraints. b: m vector of right sides for the inequality constraints. e: m vector that determines the sense of the inequalities: e(i) < 0 ==> Less Than e(i) = 0 ==> Equals e(i) > 0 ==> Greater Than vlb: n vector of non-negative lower bounds. If empty or omitted, then the lower bounds are set to zero. vub: n vector of upper bounds. May be omitted or empty. xint: vector of integer variables. May be omitted or empty. scalemode: scale flag. Off when 0 or omitted. setminim: Set maximum lp when this flag equals 0 or omitted. OUTPUT: lp_handle is an integer handle to the lp created.
Example of usage. To create following lp-model:
max: -x1 + 2 x2; C1: 2x1 + x2 < 5; -4 x1 + 4 x2 <5; int x2,x1;
The following command can be used:
O> include lp_maker.oms O> lp=lp_maker([-1, 2], {[2, 1], [-4, 4]}, [5, 5], [-1, -1], [], [], [1, 2]) O> lp 0
To solve the model and get the solution:
O> omlpsolve("solve", lp) 0 O> omlpsolve("get_objective", lp) 3 O> omlpsolve("get_variables", lp) { 1 2 }
Don't forget to free the handle and its associated memory when you are done:
O> omlpsolve("delete_lp", lp);
Contains several examples to build and solve lp models. To execute and also see which commands are executed in the debug window, use following commands:
O> stop O> trace on lpdemo.oms O> quit O> lpdemo
Note however that execution is much slower when trace is on. It is only used here to see the statements executed.
Contains several examples to build and solve lp models. Also solves the lp_examples from the lp_solve distribution. To execute and also see which commands are executed in the debug window, use following commands:
O> stop O> trace on ex.oms O> quit O> ex
Note however that execution is much slower when trace is on. It is only used here to see the statements executed.
We shall illustrate the method of linear programming by means of a simple example, giving a combination graphical/numerical solution, and then solve both a slightly as well as a substantially more complicated problem.
Suppose a farmer has 75 acres on which to plant two crops: wheat and barley. To produce these crops, it costs the farmer (for seed, fertilizer, etc.) $120 per acre for the wheat and $210 per acre for the barley. The farmer has $15000 available for expenses. But after the harvest, the farmer must store the crops while awaiting favourable market conditions. The farmer has storage space for 4000 bushels. Each acre yields an average of 110 bushels of wheat or 30 bushels of barley. If the net profit per bushel of wheat (after all expenses have been subtracted) is $1.30 and for barley is $2.00, how should the farmer plant the 75 acres to maximize profit?
We begin by formulating the problem mathematically. First we express the objective, that is the profit, and the constraints algebraically, then we graph them, and lastly we arrive at the solution by graphical inspection and a minor arithmetic calculation.
Let x denote the number of acres allotted to wheat and y the number of acres allotted to barley. Then the expression to be maximized, that is the profit, is clearly
P = (110)(1.30)x + (30)(2.00)y = 143x + 60y.
There are three constraint inequalities, specified by the limits on expenses, storage and acreage. They are respectively:
120x + 210y <= 15000
110x + 30y <= 4000
x + y <= 75
Strictly speaking there are two more constraint inequalities forced by the fact that the farmer cannot plant a negative number of acres, namely:
x >= 0, y >= 0.
Next we graph the regions specified by the constraints. The last two say that we only need to consider the first quadrant in the x-y plane. Here's a graph delineating the triangular region in the first quadrant determined by the first inequality.
O> clear O> X = 0.1:0.05:125; O> Y1 = (15000. - 120*X)/210; O> bar(X, Y1)
Now let's put in the other two constraint inequalities.
O> clear O> X = 0.1:0.05:38; O> mlmode O> Y1 = (15000. - 120*X)/210; O> Y2 = max((4000 - 110.*X)./30, 0); O> Y3 = max(75 - X, 0.); O> Ytop = min(min(Y1, Y2), Y3); O> omatrix O> bar(X, Ytop) O> gtitle("Solution space")
The black area is the solution space that holds valid solutions. This means that any point in this area fulfils the constraints.
Now let's superimpose on top of this picture a contour plot of the objective function P.
O> mlmode meshgrid.m O> [U, V] = meshgrid(0:1:40, 0:1:80); O> Ur = U.row(1) O> Vc = V.col(1) O> Z = 143.*U + 60.*V O> levels = (0:1:11)*1000. O> contour(Z', levels, Ur', Vc'); O> gtitle("Solution space and objective")
The lines give a picture of the objective function. All solutions that intersect with the black area are valid solutions, meaning that this result also fulfils the set constraints. The more the lines go to the right, the higher the objective value is. The optimal solution or best objective is a line that is still in the black area, but with an as large as possible value.
It seems apparent that the maximum value of P will occur on the level curve (that is, level
line) that passes through the vertex of the polygon that lies near (22,53).
It is the intersection of x + y = 75 and 110*x + 30*y = 4000
This is a corner point of the diagram. This is not a coincidence. The simplex algorithm, which is used
by lp_solve, starts from a theorem that the optimal solution is such a corner point.
In fact we can compute the result:
O> x = {[1, 1], [110, 30]} \ {75, 4000} O> print "x =", x x = { 21.875 53.125 }
The acreage that results in the maximum profit is 21.875 for wheat and 53.125 for barley. In that case the profit is:
O> P = [143, 60] * x O> print "Profit, P =", P Profit, P = 6315.63
That is, $6315.63.
Note that these command are in script example3.oms
Now, lp_solve comes into the picture to solve this linear programming problem more generally. After that we will use it to solve two more complicated problems involving more variables and constraints.
For this example, we use the higher-level script lp_maker to build the model and then some lp_solve API calls to retrieve the solution. Here is again the usage of lp_maker:
LP_MAKER Makes mixed integer linear programming problems. SYNOPSIS: lp_handle = lp_maker(f,a,b,e,vlb,vub,xint,scalemode,setminim) make the MILP problem max v = f'*x a*x <> b vlb <= x <= vub x(int) are integer ARGUMENTS: The first four arguments are required: f: n vector of coefficients for a linear objective function. a: m by n matrix representing linear constraints. b: m vector of right sides for the inequality constraints. e: m vector that determines the sense of the inequalities: e(i) < 0 ==> Less Than e(i) = 0 ==> Equals e(i) > 0 ==> Greater Than vlb: n vector of non-negative lower bounds. If empty or omitted, then the lower bounds are set to zero. vub: n vector of upper bounds. May be omitted or empty. xint: vector of integer variables. May be omitted or empty. scalemode: scale flag. Off when 0 or omitted. setminim: Set maximum lp when this flag equals 0 or omitted. OUTPUT: lp_handle is an integer handle to the lp created.
Now let's formulate this model with lp_solve:
O> f = [143, 60]; O> A = {[120, 210], [110, 30], [1, 1]}; O> b = {15000, 4000, 75}; O> lp = lp_maker(f, A, b, [-1, -1, -1], [], [], [], 1, 0); O> solvestat = omlpsolve("solve", lp) O> omlpsolve("get_objective", lp) 6315.63 O> omlpsolve("get_variables", lp) { 21.875 53.125 } O> omlpsolve("delete_lp", lp);
Note that these command are in script example4.oms
With the higher-level script lp_maker, we provide all data to lp_solve. lp_solve returns a handle (lp) to the created model. Then the API call 'solve' is used to calculate the optimal solution of the model. The value of the objective function is retrieved via the API call 'get_objective' and the values of the variables are retrieved via the API call 'get_variables'. At last, the model is removed from memory via a call to 'delete_lp'. Don't forget this to free all memory allocated by lp_solve.
The solution is the same answer we obtained before. Note that the non-negativity constraints are accounted implicitly because variables are by default non-negative in lp_solve.
Well, we could have done this problem by hand (as shown in the introduction) because it is very small and it
can be graphically presented.
Now suppose that the farmer is dealing with a third crop, say corn, and that the corresponding data is:
cost per acre $150.75 yield per acre 125 bushels profit per bushel $1.56
With three variables it is already a lot more difficult to show this model graphically. Adding more variables makes it even impossible because we can't imagine anymore how to represent this. We only have a practical understanding of 3 dimentions, but beyound that it is all very theorethical.
If we denote the number of acres allotted to corn by z, then the objective function becomes:
P = (110)(1.30)x + (30)(2.00)y + (125)(1.56) = 143x + 60y + 195z
And the constraint inequalities are:
120x + 210y + 150.75z <= 15000
110x + 30y + 125z <= 4000
x + y + z <= 75
x >= 0, y >= 0, z >= 0
The problem is solved with lp_solve as follows:
O> f = [143, 60, 195]; O> A = {[120, 210, 150.75], [110, 30, 125], [1, 1, 1]}; O> b = {15000, 4000, 75}; O> lp = lp_maker(f, A, b, [-1, -1, -1], [], [], [], 1, 0); O> solvestat = omlpsolve("solve", lp) O> omlpsolve("get_objective", lp) 6986.84 O> omlpsolve("get_variables", lp) { 0 56.5789 18.4211 } O> omlpsolve("delete_lp", lp);
Note that these command are in script example5.oms
So the farmer should ditch the wheat and plant 56.5789 acres of barley and 18.4211 acres of corn.
There is no practical limit on the number of variables and constraints that O-Matrix can handle. Certainly none that the relatively unsophisticated user will encounter. Indeed, in many true applications of the technique of linear programming, one needs to deal with many variables and constraints. The solution of such a problem by hand is not feasible, and software like O-Matrix is crucial to success. For example, in the farming problem with which we have been working, one could have more crops than two or three. Think agribusiness instead of family farmer. And one could have constraints that arise from other things beside expenses, storage and acreage limitations. For example:
Below is a sequence of commands that solves exactly such a problem. You should be able to recognize the objective expression and the constraints from the data that is entered. But as an aid, you might answer the following questions:
O> f = [110*1.3, 30*2.0, 125*1.56, 75*1.8, 95*.95, 100*2.25, 50*1.35]; O> A = {[120, 210, 150.75, 115, 186, 140, 85], [110, 30, 125, 75, 95, 100, 50], [1, 1, 1, 1, 1, 1, 1], [1, -1, 0, 0, 0, 0, 0], [0, 0, 1, 0, -2, 0, 0], [0, 0, 0, -1, 0, -1, 1]}; O> b = {55000, 40000, 400, 0, 0, 0}; O> lp = lp_maker(f, A, b, [-1, -1, -1, -1, -1, -1], [10, 10, 10, 10, 20, 20, 20], [100, INF, 50, INF, INF, 250, INF], [], 1, 0); O> solvestat = omlpsolve("solve", lp) O> omlpsolve("get_objective", lp) 75398 O> omlpsolve("get_variables", lp) { 10 10 40 45.6522 20 250 20 } O> omlpsolve("delete_lp", lp);
Note that these command are in script example6.oms
Note that we have used in this formulation the vlb and vub arguments of lp_maker. This to set lower and upper bounds on variables. This could have been done via extra constraints, but it is more performant to set bounds on variables. Also note that Inf is used for variables that have no upper limit. This stands for Infinity.
Note that despite the complexity of the problem, lp_solve solves it almost instantaneously. It seems the farmer should bet the farm on crop number 6. We strongly suggest you alter the expense and/or the storage limit in the problem and see what effect that has on the answer.
Suppose we want to solve the following linear program using O-Matrix:
max 4x1 + 2x2 + x3
s. t. 2x1 + x2 <= 1
x1 + 2x3 <= 2
x1 + x2 + x3 = 1
x1 >= 0
x1 <= 1
x2 >= 0
x2 <= 1
x3 >= 0
x3 <= 2
Convert the LP into O-Matrix format we get:
f = [4, 2, 1]
A = {[2, 1, 0], [1, 0, 2], [1, 1, 1]}
b = {1, 2, 1}
Note that constraints on single variables are not put in the constraint matrix. lp_solve can set bounds on individual variables and this is more performant than creating additional constraints. These bounds are:
l = [ 0, 0, 0]
u = [ 1, 1, 2]
Now lets enter this in O-Matrix:
O> f = [4, 2, 1]; O> A = {[2, 1, 0], [1, 0, 2], [1, 1, 1]}; O> b = {1, 2, 1}; O> l = [ 0, 0, 0]; O> u = [ 1, 1, 2];
Now solve the linear program using O-Matrix: Type the commands
O> lp = lp_maker(f, A, b, [-1, -1, -1], l, u, [], 1, 0); O> solvestat = omlpsolve("solve", lp) O> omlpsolve("get_objective", lp) 2.5 O> omlpsolve("get_variables", lp) { 0.5 0 0.5 } O> omlpsolve("delete_lp", lp)
What to do when some of the variables are missing ?
For example, suppose there are no lower bounds on the variables. In this case define l to be the empty set using the O-Matrix command:
O> l = [];
This has the same effect as before, because lp_solve has as default lower bound for variables 0.
But what if you want that variables may also become negative?
Then you can use -INF as lower bounds:
O> l = [-INF, -INF, -INF];
Solve this and you get a different result:
O> lp = lp_maker(f, A, b, [-1, -1, -1], l, u, [], 1, 0); O> solvestat = omlpsolve("solve", lp) O> omlpsolve("get_objective", lp) 2.66667 O> omlpsolve("get_variables", lp) { 0.666667 -0.333333 0.666667 } O> omlpsolve("delete_lp", lp)
These routines are not part of the lpsolve API, but are added for backwards compatibility. Most of them exist in the lpsolve API with another name.
Under Windows, the omlpsolve O-Matrix driver is a dll: omlpsolve.dll
This dll is an interface to the lpsolve51.dll lpsolve dll that contains the implementation of lp_solve.
lpsolve51.dll is distributed with the lp_solve package. The omlpsolve O-Matrix driver dll (omlpsolve.dll) is just
a wrapper between O-Matrix and lp_solve to translate the input/output to/from O-Matrix and the lp_solve library.
The omlpsolve O-Matrix driver is written in C. To compile this code, Microsoft compiler is needed.
Other compilers might also work, but this is untested.
To make the compilation process easier, a batch file can be used: cvc.bat
It may be necessary to edit this file first to change the path where lp_solve and the O-Matrix dll sources are installed.
Change at the beginning lpsolvepath and dllsrcpath. dllsrcpath must point to the folder where dll.h is located.
To make for release, just enter cvc and everything is build.
This compiles three source files: lpsolve.c, omatrix.c and hash.c
Then these are linked with the library lpsolve51.lib to generate the omlpsolve.dll file.
The optional arguments to cvc are used for development. Source files can be provided and then only these are compiled.
For example hash.c should only be compiled once while developing. So specifying
lpsolve.c as first argument will only compile this file and then link everything. This makes the build process a bit faster.
Also the option -DDEMO can be added to add the demo command to test some functionality of lpsolve. This is also only for debugging.
Also the option -DDEBUG can be added. This to print some debug information while executing omlpsolve.
Should only be used for debugging purposes.
Note that the omlpsolve.dll file can be locked by O-Matrix. Then the build process will fail because the dll can not be overwritten. This can be solved by giving the clear command in O-Matrix. This will free the dll.
At this moment, there is no O-Matrix version for this platform.
See also Using lpsolve from MATLAB, Using lpsolve from Scilab, Using lpsolve from Octave