Branch
and Bound

Begin the branch and bound exercise by clicking on the Branch/Bound item on the OR/MS 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 twopart names such as P_1. A names 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.
Except when the programming is running under the "without stopping" option, the student is allowed to switch between options. The addin 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 addins 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. Pressing on 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 node is shown in the figure below. Each row describes a node of the tree. There are seven columns in the display. The node at level 0 represents the problem with no restrictions.
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 refering 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 with 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 node, 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 node. Observe that the added restriction has decreased the value of the optimum LP solution.

Operations
Research Models and Methods
by Paul A. Jensen and Jon Bard, University of Texas, Copyright
by the Authors