Example: A Four-Node Network

Transcription

1 Node Network Flows Flow of material, energy, money, information, etc., from one place to another arc 6 Node Node arc 6 Node Node 5 Node 6 8 Goal: Ship All of the Over the Network to Meet All of the Flows At Nodes Flow can come into a node either by being produced there (supply) or by arriving from other nodes along arcs. Flow can leave a node either by being used there (demand) or by departing to other nodes along arcs. Flow Balance Example: A Four-Node Network Flow Balance Equation at Node At each node, the total flow coming in must equal the total flow going out. Therefore, at each node: x x x x x x supply + total flow arriving from other nodes = demand + total flow departing to other nodes We can write a set of linear equations to describe this. x Define x ij as the flow from node i to node j x x + =

2 Flow Balance Equation at Node Flow Balance Equation at Node Flow Balance Equation at Node x x x x x x x x x x x x x + x = x x + = + x = x + x This Leads to Equations in 5 Unknowns The Associated Matrix Equation: Start With the Augmented Matrix: x + = -x + x + x = - - x + x = x x = - st x - x = x We therefore expect an infinite number of solutions. x Begin row reduction to reduced echelon form: Add Row to Row.

3 This Yields: This Yields: This Yields: nd rd rd all s all s all s Now add Row to Row. Now add Row to Row. Now subtract Row from Row. nd This Yields: all s - This Yields the Reduced Echelon Form: all s - s - From the Reduced Echelon Form, Write Out the Equations for the Basic Variables in Terms of the Free Variables x - x + x = + x - x = x + x = Now subtract Row from Row. The variables x,, and x corresponding to the columns in the matrix are the basic variables. The others, x and x, are the free variables. Now gather the basic variables on the left-hand side.

4 This Yields: Which Is Given in Parametric Form By: What If We Set Both Free Variables to? x = + x - x = - x + x x = - x x x x x = + x - + x - - x = x = x = = x = Basic variables Free variables Free variables This removes their corresponding arcs. No more closed loops remain: yields a spanning tree. Other Spanning Trees are Possible A Third Spanning Tree: A Fourth Spanning Tree: x = x = x = = x = x = = x = x = x = - x = x = x = = x = This results simply from a different ordering of the variables prior to row reduction, yielding different basic and free variables. This tree requires a negative value of flow x, which might not be permitted by other constraints of the problem.

5 Minimum-Cost Solutions Among all the solutions to the flow-balance equations, which should be preferred? A very good way of selecting the optimum solution is by finding the one that has minimum cost. Here, we need additional information regarding the cost of the flow along each arc of the network. Example: Cost Data for Each Arc x $ x $ x $ $6 x $5 These cost data are assumed to be known quantities. Cost of the First Basic Solution $ $ $6 Cost = $6 + $ + $ = $7 Cost of the Second Basic Solution $ $ $ Cost = $ + $ + $ = $6 Less expensive! Could We Have Anticipated This Result? Yes By Looking at the Intermediate Case: Adjust flows to compensate for x = t (+t) $ $ t $ (-t) $6 Incremental cost for Loop -- = t $ + t $ - t $6 = -$t Dial in the free variable t to close Loop -- Net savings! Limits on t When we dial in t, we start with t =, which corresponds to the first basic solution (a spanning tree). As t increases, flow is increasingly diverted from arc to arc, and the total cost of Loop drops. The maximum allowable value of t occurs when the flow on arc is reduced to. We see that this occurs when t =. At this point, we have arrived at another lower cost basic solution (a new spanning tree).

6 This is the Idea Behind the Network Simplex Method User-Defined MATLAB Function Advantages of User-Defined Functions Start with any basic solution (all free variables = ; i.e., a spanning tree). Next, increase just one of the free variables from to see if the total cost of its associated loop drops. If so, keep increasing this free variable until one of the basic variables drops to zero. This is a new lower-cost basic solution. Repeat by dialing in another free variable until a basic solution cannot be improved. This is the minimum cost solution for the entire network. A special type of M-file that runs in its own independent workspace. Receives input data from the calling program through a list of input arguments; processes these data; and then returns the results to the calling program through a list of output arguments. Allow independent testing of subtasks (program modularization). Can be re-used many times in a program. Confine the impact of programming mistakes within the function to only those variables processed within the function. Syntax Dummy vs. Actual Arguments Syntax (continued) function [outarg, outarg,... ] = Dummy arguments (i.e., placeholders) for actual variables returned to the calling program when the function finishes executing. function_name(inarg, inarg,...) Dummy arguments (i.e., placeholders) for actual variables received from the calling program when the function is invoked. We do NOT have to give dummy arguments the same name as the corresponding actual arguments in the main program which calls the user-defined function. Data is passed to and from the calling program via a dummy argument simply by its position in the list of input or output arguments (which matches the position of some actual argument of the calling statement in the main program). function [outarg, outarg,... ] = function_name(inarg, inarg,...) % H comment line % Other comment lines... outarg = outarg = return not required in most functions Executable code wherein each item in the output argument list must appear on the left side of at least one assignment statement.

7 Example: Rectangular to Polar Conversion dummy arguments function [a, b] = rectpolar(c,d) % RECTPOLAR converts rectangular to polar coordinates (angle in degrees) a = sqrt(c.^ + d.^); b = 8/pi * atan(d,c); Calling Statement in the Main Program [r, theta] = rectpolar(x,y); actual arguments MATLAB Protects All Variables in the Calling Program That Are Designated as Input Data to User-Defined Functions Even if a user-defined function modifies its input arguments during execution of the function, it will not affect the original data in the calling program. Built-In Protection Method for Input Data When a user-defined function call occurs, MATLAB first determines if the function modifies any of its input arguments during its execution. If this is so, MATLAB makes a separate copy of the input argument in the function s own workspace. Thus, if the function modifies its input arguments during its execution, these changes are isolated from the calling program. MATLAB Permits the Number of Input and Output Arguments Specified by a Calling Program To Be Determined On the Fly Two special functions are available for embedding within a user-defined function to permit MATLAB to determine the number of input and output arguments specified by the calling program during run-time. nargin and nargout function [outarg, outarg,... ] = function_name(inarg, inarg,...) % H comment line % Other comment lines a = nargin b = nargout... number of actual input arguments used to call function_name number of actual output arguments used to call function_name

8 The Behavior of A User-Defined Function Can Be Adjusted According to nargin and nargout Using nargin and nargout permits the programmer to adjust the behavior of his / her user-defined function according to the number of input and output arguments specified by the calling program. Example: Rectangular to Polar Conversion function [a, b] = rectpolar(c,d) % RECTPOLAR converts rectangular to % polar coordinates (angle in degrees) % If only one input argument is supplied, % assume that d= and proceed normally. if nargin = = d = ; end Additional code a = sqrt(c.^ + d.^); b = 8/pi * atan(d,c); Using nargchk, We Can Check If A Function Is Called With Too Few or Too Many Arguments Function nargchk returns a standard error message if a function is called with too few or too many arguments. Otherwise, an empty string is returned. The syntax is: message = nargchk(min, max, num) where min is the minimum permitted number of arguments; max is the maximum permitted number of arguments; and num is the actual number of arguments. nargchk Is Often Used With error To Terminate Program Execution It is usually desirable to halt program execution and inform the programmer if a function is called with too few or too many arguments. The required syntax is: message = nargchk(min, max, num) error(message); This halts execution if message is not an empty string; returns to the keyboard; and displays message in the Command Window. If message is an empty string, error does nothing and execution continues. Example: Rectangular to Polar Conversion function [a, b] = rectpolar(c,d) % RECTPOLAR converts rectangular to % polar coordinates (angle in degrees) % Check for a legal number of input % arguments. message = nargchk(,, nargin); error(message); New code a = sqrt(c.^ + d.^); b = 8/pi * atan(d,c); warning Is Used to Alert the Programmer Of A Troubling But Non-Fatal Problem It is usually desirable to alert the programmer to a troubling intermediate calculation such as /, but still continue program execution. The required syntax is: warning('message') where message is a character string to be displayed in the Command Window. If the message string is empty, nothing is displayed. In either case, program execution continues.

9 Example: Rectangular to Polar Conversion function [a, b] = rectpolar(c,d) % RECTPOLAR converts rectangular to % polar coordinates (angle in degrees) % Check for (,) input arguments, and % print out a warning message. if c= = & d= = msg = 'x = y = : meaningless angle' warning(msg); end New code a = sqrt(c.^ + d.^); b = 8/pi * atan(d,c);