Teach Integer Programming Add-in - Branch and Bound
 The examples for this section are in the Teach IP demo workbook (teachipdem.xls). To solve or change the model you must have the Teach IP add-in loaded. Use the Relink buttons command to establish links to the worksheet buttons.

Problem Definition

Begin the branch and bound exercise by clicking on the Branch/Bound item on the Teach menu. The program presents the problem definition dialog to accept model data.

The program presents a possible name in the Name field such as TeachIP1. There can be multiple IP models in a workbook, and the integer number at the end of the name will advance as IP models are added. The name is used as prefix for a number of named ranges on the worksheet, so once the name is specified it should not be changed. Names must obey the Excel rules for naming ranges. The name should begin with a letter, and include no spaces or punctuation marks (a period is OK). The underline symbol may be used and is often handy for two-part names such as P_1. A name like P1 can't be used because it can be confused with a cell reference.

Fields are available for number of variables, number of constraints and number of integer variables. If the latter is fewer than the number of variables, the variables with the smaller indices are assumed integer. The maximum number of variables is 20 and the maximum number of constraints is 10. The model must be expressed as a maximization. If the Make Random Problem box is checked, the program will provide random coefficients for the objective function and constraints. Random problems have only less than or equal to constraints. All coefficients are integer and positive. A random problem will always have a feasible solution. When the model is displayed the student can change any of the data.

One of four modes of operation is chosen with a button in the Solution Options frame.

• Demonstration Option: In this mode, the program takes the student through the steps of the procedure. Message boxes describe the steps as they occur. The student presses OK buttons to progress.
• Instruction Option: In this mode, the student makes branching decisions and must tell the program when the current incumbent is to be replaced and when an enumeration node has been fathomed. A hint button is available that tells the correct answers. Errors that would cause the algorithm to go astray are not allowed.
• Do it Yourself Option: Here the hint button is disabled and the student must make all the decisions directing the algorithm.
• Run without Stopping Option: The program runs the procedure with no student interaction. The approximate time delay between steps is set in the Delay field. The delay must be an integer number of seconds.

Except when the programming is running under the "without stopping" option, the student is allowed to switch between options.

Example Problem

The add-in builds the model structure shown below. Data has already been entered for the example used for this section (we use the same example as for the cutting plane approach). Yellow ranges are controlled by the program or contain formulas. These areas should not be changed by the student. In contrast to the cutting plane approach, the branch and bound algorithm does not require that the model coefficients be integer.

The format of the display follows that of other mathematical programming add-ins in the collection. The model is linear, and the linear objective coefficients of the variables are placed in the range G9:L9. Simple lower bounds on the variables are placed in the range G10:L10, and simple upper bounds in the range G11:L11. The constraint information starts in row 15. The range G15: L18 holds the linear constraint coefficients. Each constraint has both lower and upper bounds. The lower bounds are in the column range E15:E18, and the upper bounds are in the column range F15:F18. The range D15:D18 holds the constraint values computed for the current solution.

The range in column B, B2:B8 holds information required by the program and should not be changed by the student.

Starting the Procedure

Pressing the Start BB button creates the revised tableau shown below. New rows are provided for the original upper and lower bounds. This is necessary because the program uses the simple lower and upper bound ranges to hold intermediate bounds required by the branch and bound procedure. A new row is also provided for the incumbent solution. As the algorithm finds feasible integer solutions, the best solution is stored in this range. There is also a cell for the incumbent objective value. Initially there is no incumbent and the objective is set to a large negative number. If a known integer solution is initially available, it can be placed in this range.

The solution to the first linear programming relaxation is shown in the figure above. Colors are used to emphasize features of the solution. The light green in the variable value range indicates nonzero variables. Light green in the constraint value range indicates loose constraints, while red shows tight constraints. Not shown in this example is the dark yellow that colors violated constraints.

The program constructs a representation of the enumeration tree. The first vertex is shown in the figure below. Each row describes a vertex of the tree. There are seven columns in the display. The vertex at level 0 represents the problem with no restrictions.

• Level: The number of tree vertices traversed from the top of the tree to the vertex. Level 0 is the top of the tree. This is also the number of simple bounding restrictions placed on the LP.
• Vertex: The index of the tree vertex. Indices are assigned in numerical order as the vertices are created.
• Variable: The branching variable at the vertex.
• Value: The upper or lower bound placed on the variable at this level.
• Up/Down: The direction of branching, -1 for down and +1 for up. Branching up implies that a simple lower bound restriction is placed by this vertex. Branching down adds a simple upper bound restriction.
• Visit: This is the number of times the vertex has been visited. When first encountered the visit number is 1. When encountered through a backtrack step, the number becomes 2.
• Relax: The value of the linear programming relaxation including all the restrictions placed by vertices higher in the tree. If the relaxation has no feasible solution, the word Infeasible is placed in this field.

The Iteration

Begin by pressing the Iterate button on the worksheet. At the top of the tree, there are no restrictions on the variables. In this section, we are referring to the LP solution in the figure above. We describe the example with the dialogs that are used in the Instruction mode. When a decision is required, the dialog is placed at the top of the worksheet in a position that should not block the view of important solution information. It can be moved if necessary. The dialog shows the objective values for the relaxation and incumbent on the left. The student is to enter the index of the branching variable in the Var. field.

For the branch and bound procedure, the branching variable is to be some variable that has a fractional value. Pressing the Hint button on the dialog colors the fractional variables yellow on the display. The index of the variable with the largest fractional value is placed in the Var. field. The fractional value is the difference between the solution value and the nearest integer. For X6, the nearest integer is 2 and the fractional value is 0.4706. Any of the variables with fractional values can be used as the branching variable, and the program will cycle through the options, X2, X5 and X6, as the Hint button is repeatedly clicked. The branching direction is either Up or Down as determined by the button in the Branch frame.

Pressing the OK button makes the selection. The Cancel button dismisses the dialog to return to the worksheet. The Iterate button remains on the worksheet so the student can resume the program. The student can change the solution option by clicking on a different button in the Solution Option frame.

The new LP solution is obtained after the program has branched on variable X6 in the up direction. X6 previously held a number between 1 and 2, but closer to 2., We branch up by placing a lower bound on X6 of 2. Later in the procedure we will branch down on X6 by placing an upper bound on X6 of 1. The red field in the simple lower bound range shows the decision.

A new vertex, row 2 in the figure below, is added to the tree. The visit number is set to 1 because this is the first visit to this vertex. Observe that the added restriction has decreased the value of the optimum LP solution.

Demonstration

 The following links lead through the ten steps of the solution for this example problem. The figures above are repeated in the first step. Iteration 1: First LP Solution. Branch Iteration 2: Branch Iteration 3: Branch Iteration 4: Replace Incumbent and Backtrack Iteration 5: Fathom and Backtrack Iteration 6: Fathom and Backtrack Iteration 7: Branch Iteration 8: Fathom and Backtrack Iteration 9: Fathom and Backtrack Iteration 10: Optimum Integer Solution Tree: Representation of the Branch and Bound Tree

Operations Research Models and Methods
Internet
by Paul A. Jensen