7 Branch and Bound Introduction

1. Formula
   Observation

   Jobs = {$J_1$, $J_2$, $J_3$, $J_4$}

Jobs

The set of all jobs in the job sequencing problem.

Set of four discrete jobs

1. Formula
   Observation

   $P = {10, 5, 8, 3}$

P

The set of profits associated with each job.

Set of four numerical values

1. Formula
   Observation

   $d = {1, 2, 1, 2}$

d

The set of deadlines associated with each job.

Set of four numerical values

1. Formula
   Observation

   $S1 = {J_1, J_4}$

S1

A variable-size solution represented as a subset of jobs.

Subset of Jobs

1. Formula
   Observation

   $S2 = {1, 0, 0, 1}$

S2

A fixed-size solution represented as a binary vector indicating which jobs are done.

Binary vector of length 4

1. Formula
   Observation

   Right side of the whiteboard shows Jobs = {$J_1$, $J_2$, $J_3$, $J_4$}.

Jobs

The set of four jobs considered in the job-sequencing example.

Set of job labels

1. Formula
   Observation

   Right side of the whiteboard shows $P = {10, 5, 8, 3}$.

1. The video does not verbally define what the entries of P represent in this clip.

P

A listed vector associated with the four jobs; the clip does not explain its meaning.

1. Formula
   Observation

   Right side of the whiteboard shows $d = {1, 2, 1, 2}$.

1. The video does not verbally define what the entries of d represent in this clip.

d

A listed vector associated with the four jobs; the clip does not explain its meaning.

1. Formula
   Observation

   Right side of the whiteboard shows $S1 = {J_1, J_4}$.

1. The video does not explain how S1 is obtained or used in this clip.

S1

A subset of jobs written on the board; its role is not explained in this clip.

1. Formula
   Observation

   Right side of the whiteboard shows $S2 = {1, 0, 0, 1}$.

1. The video does not explain the meaning of the $0/1$ pattern in S2 in this clip.

S2

$A 0/1$ vector written on the board; its interpretation is not explained in this clip.

1. Diagram
   Observation

   Tree nodes are numbered 1 through 17 as the instructor expands the state-space tree.

1,2,...,17

Node identifiers used to track the order in which states are expanded in the drawn tree.

Positive integers labeling tree nodes

1. Formula
   Observation

   Jobs = {$J_1$, $J_2$, $J_3$, $J_4$}

Jobs

Set of jobs to be sequenced

Discrete set

1. Audio
   Observation

   The topic is branch and bound. This is one more problem solving strategy, a method by which we can solve a problem.

Branch and Bound is a problem-solving strategy used to solve optimization problems, specifically minimization problems.

1. Applicable to optimization problems

2. Primarily for minimization problems

1. Audio
   Observation

   Suppose I say that I am doing job one and job four, this is my solution. Then the solution can be represented as subset of that jobs.

2. Formula
   Observation

   $S1 = {J_1, J_4}$

A solution is represented as a subset of the available jobs, resulting in a variable size depending on how many jobs are selected.

1. Solution size varies based on the number of selected jobs

1. Jobs

1. Audio
   Observation

   Second method of representing a solution is first job is done, second and third job is not done, fourth job is done. So this is variable size solution, this is fixed size solution.

2. Formula
   Observation

   $S2 = {1, 0, 0, 1}$

A solution is represented as a binary vector where each position corresponds to a job, and the value indicates whether the job is done (1) or not done (0). The size is always equal to the total number of jobs.

1. Solution size is fixed and equals the number of jobs

1. Jobs

1. Audio
   Observation

   In backtracking, we will consider the next level. But here in branch and bound, we will also consider second job. And then third job. And then fourth job.

2. Audio
   Observation

   So it is like breadth first search. We don't go depth wise. We don't follow depth first search. We follow breadth first search for exploring the solutions of a problem.

3. Audio
   Observation

   So the difference between backtracking and branch and bound is backtracking was depth first search and the branch and bound is breadth first search.

The clip contrasts two search strategies on the same job-sequencing state-space tree. Backtracking is described as going down one branch to the next level before considering siblings, whereas branch and bound is described as expanding all alternatives at the current level first. The instructor explicitly summarizes this as backtracking being depth-first search and branch and bound being breadth-first search.

1. Applied to a state-space tree for job sequencing.

2. The comparison is about exploration order, not about bounding rules or cost evaluation.

1. Job-sequencing state-space tree with four jobs

1. Diagram
   Observation

   Title at top center reads Branch and Bound; right side lists Job sequencing with Jobs = {$J_1$, $J_2$, $J_3$, $J_4$}, $P = {10, 5, 8, 3}$, $d = {1, 2, 1, 2}$, $S1 = {J_1, J_4}$, $S2 = {1, 0, 0, 1}$.

2. Audio
   Observation

   Once I take first job, then what are the solution? Job two I can take or job three I can take, then job four I can take.

3. Audio
   Observation

   If I follow one of the route, it says that I am doing first job and fourth job. I am not taking second and third job.

1. The clip does not define the objective function or explain the numerical vectors P and d.

The worked example is a job-sequencing problem with four jobs $J_1$, $J_2$, $J_3$, $J_4$. The instructor draws a rooted tree whose branches correspond to choosing the next job from the remaining jobs. A path from the root records an ordered partial sequence of selected jobs; for example, following the branch labeled $J_1$ and then $J_4$ represents doing job 1 and then job 4 while not yet taking jobs 2 and 3.

1. There are four distinct jobs.

2. Each edge label indicates appending one job to the current partial sequence.

3. The clip only constructs the tree structure and does not evaluate feasibility or optimality.

1. Audio
   Observation

   Now this one level is completed. See, I have considered all the jobs one by one.

2. Audio
   Observation

   Now, once I take first job, then what are the solution? Job two I can take or job three I can take, then job four I can take.

3. Audio
   Observation

   Now which node I should expand? Shall I expand this one or this one? So next is this one.

4. Diagram
   Observation

   After finishing the children of node 1, the instructor expands node 2, then node 3, then node 4, then node 5, and continues across the frontier before moving deeper.

After the root's children are created, the instructor treats that whole level as completed and then expands nodes from left to right across the current frontier. The visible procedure is: finish all immediate choices from the root, then expand node 2, then node 3, then node 4, then node 5, and continue similarly. This demonstrates level-order generation of candidate sequences rather than pursuing one path to full depth first.

1. The tree is explored level by level.

2. At each node, only jobs not already used on the path from the root are appended.

3. The clip shows structural expansion only, not pruning by bounds.

1. Branch and bound uses breadth-first expansion, unlike backtracking
2. Job-sequencing state-space tree with four jobs

1. Audio
   Observation

   So this is a state space tree for the solutions if you are looking like this.

2. Diagram
   Observation

   Whiteboard shows a tree structure with nodes labeled 1 through 14 and edges labeled $J_1$ through $J_4$.

A state space tree represents all possible solutions (permutations of job sequences) as nodes and branches in a tree structure. Each node corresponds to a partial sequence of jobs.

1. Applicable to combinatorial optimization problems like job sequencing.

1. Audio
   Observation

   And if I consider any node, this node says that I am considering second job and fourth job. This node says that I am considering third job and fourth job.

2. Diagram
   Observation

   Instructor points to specific nodes in the tree.

Each node in the state space tree represents a specific partial sequence of jobs. For example, a node might represent the sequence containing the second and fourth jobs.

1. Nodes are generated by adding one job at a time to the sequence.

1. State Space Tree for Job Sequencing

1. Audio
   Observation

   While generating this, I'll put these nodes in the stack... Now which node I should expand next? I should expand fifth node... pop out this one and expand...

2. Diagram
   Observation

   Instructor draws a vertical rectangle representing a stack and writes numbers inside it as nodes are pushed and popped.

The second method uses a stack to manage the exploration of the state space tree. Nodes are pushed onto the stack when generated and popped from the top to be expanded, resulting in a depth-first search order.

1. Requires a Last-In-First-Out (LIFO) data structure.

2. Used to systematically explore the state space tree without recursion.

1. State Space Tree for Job Sequencing

1. Formula
   Observation

   The instructor writes Queue -> FIFO on the whiteboard.

2. Audio
   Observation

   He says that if you are using queue then it is called as FIFO branch and bound.

In this lecture, FIFO branch and bound is defined by using a queue to choose the next node for exploration.

1. Applies to branch-and-bound search when the next node is selected from a queue.

1. Breadth-first expansion rule in this branch-and-bound setting

1. Formula
   Observation

   The instructor writes Stack -> LIFO on the whiteboard.

2. Audio
   Observation

   He says that if you are using stack then it is called as LIFO branch and bound.

In this lecture, LIFO branch and bound is defined by using a stack to choose the next node for exploration.

1. Applies to branch-and-bound search when the next node is selected from a stack.

1. Breadth-first expansion rule in this branch-and-bound setting

1. Formula
   Observation

   The instructor writes Least cost -> LC-BB and later adds Least Cost - BB under the main heading.

2. Audio
   Observation

   He introduces least cost branch and bound and abbreviates it as LC branch and bound.

1. This clip names LC-BB but does not yet give its full selection rule beyond introducing node costs.

The lecture introduces a third branch-and-bound method called Least Cost Branch and Bound, abbreviated LC-BB.

1. Presented as another method within branch and bound.

1. FIFO Branch and Bound
2. LIFO Branch and Bound

1. Audio
   Observation

   But it is useful for solving optimization problem and only minimization problem, not maximization problem. Anyway we can convert a maximization problem into minimization and we can solve it.

Branch and Bound is useful for solving optimization problems, specifically minimization problems. Maximization problems can be converted into minimization problems to be solved using this method.

1. The problem is an optimization problem

1. Audio
   Observation

   So the difference between backtracking and branch and bound is backtracking was depth first search and the branch and bound is breadth first search.

1. This is presented as the instructor's summary for the example, not as a fully general theorem with hypotheses.

In this lecture segment, the instructor states that backtracking explores the solution space by depth-first search, while branch and bound explores it by breadth-first search.

1. The discussion concerns searching a state-space tree for a combinatorial problem such as job sequencing.

For the methods compared in this clip.

1. Audio
   Observation

   So if I follow one of the route, it says that I am doing first job and fourth job. I am not taking second and third job. That is the meaning.

Following a route in the drawn tree corresponds to selecting the jobs along that route in order; for example, the route using $J_1$ then $J_4$ means doing job 1 and job 4 and not taking jobs 2 and 3 along that partial path.

1. The tree is the job-sequencing state-space tree drawn on the board.

For paths in the displayed tree.

1. Audio
   Observation

   The instructor contrasts the previous method with the current one and then names FIFO, LIFO, and least cost methods.

2. Formula
   Observation

   The board lists 1. FIFO-BB, 2. LIFO-BB, 3. LC-BB.

Within this lecture segment, branch and bound is presented in three variants: FIFO-BB, LIFO-BB, and LC-BB.

1. The discussion is about branch-and-bound search methods for the job-sequencing state-space tree.

For the methods introduced in this clip.

1. Audio
   Observation

   He says this we do for all three methods, whether you are using FIFO, LIFO, LC, then only we can solve a minimization problem.

1. The clip does not prove why node costs are necessary; it states this as part of the explanation.

According to the instructor, assigning node costs is required across FIFO, LIFO, and LC methods in order to solve a minimization problem.

1. The problem being solved is a minimization problem.

2. The search is organized as a branch-and-bound state-space tree.

For the three branch-and-bound methods named in this clip.

1. Audio
   Observation

   I will follow this method first, that is subset method, then that is a variable size solution. So for that, I can draw a state space tree and I'll say that I am considering first job.

2. Animation
   Observation

   Drawing a root node and a left branch labeled J1.

1. Expression

   $$$$
   Explanation

   Select the subset method (variable-size solution) to represent the solution.

   Justification

   Stated preference for the subset method.

   Shown in the video
2. Expression

   $$$$
   Explanation

   Draw a root node for the state space tree.

   Justification

   Standard initialization of a state space tree.

   Shown in the video
3. Expression

   $$$$
   Explanation

   Draw a left branch from the root and label it J1.

   Justification

   Represents the decision to consider the first job.

   Shown in the video

The initial steps of constructing a state space tree for the job sequencing problem using the subset method are demonstrated.

1. Audio
   Observation

   But here in branch and bound, we will also consider second job. And then third job. And then fourth job.

2. Diagram
   Observation

   From root node 1, the instructor draws four outgoing edges labeled $J_1$, $J_2$, $J_3$, $J_4$ to nodes 2, 3, 4, 5.

1. Expression

   $$$$
   Explanation

   Start from the root node representing the empty partial sequence.

   Justification

   The tree is being built for job sequencing with four available jobs.

   Shown in the video
2. Expression

   $$$$
   Explanation

   Create one child for each possible first job choice: $J_1$, $J_2$, $J_3$, $J_4$.

   Justification

   The instructor says branch and bound considers the second, third, and fourth job at this stage, and the drawing shows all four first-level branches from the root.

   Shown in the video
3. Expression

   $$$$
   Explanation

   Number the newly created nodes 2, 3, 4, 5 from left to right.

   Justification

   The board visibly labels these nodes after they are drawn.

   Shown in the video

The first level of the state-space tree contains all one-job partial sequences, illustrating breadth-first generation of alternatives.

1. Audio
   Observation

   Now this one level is completed. See, I have considered all the jobs one by one.

2. Audio
   Observation

   Now, once I take first job, then what are the solution? Job two I can take or job three I can take, then job four I can take.

3. Diagram
   Observation

   From node 2, the instructor draws three children labeled $J_2$, $J_3$, $J_4$ and numbers them 6, 7, 8.

1. Expression

   $$$$
   Explanation

   Take the node corresponding to the partial sequence ($J_1$).

   Justification

   This is the leftmost child of the root, labeled by edge $J_1$.

   Shown in the video
2. Expression

   $$$$
   Explanation

   Append each remaining unused job, namely $J_2$, $J_3$, and $J_4$, to form three children.

   Justification

   The instructor explicitly says that after taking the first job, job 2 or job 3 or job 4 can be taken.

   Shown in the video
3. Expression

   $$$$
   Explanation

   Interpret a path such as root → $J_1 \to J_4$ as the partial sequence doing job 1 and then job 4, without jobs 2 and 3 on that path.

   Justification

   The instructor states that following one route means doing first job and fourth job and not taking second and third job.

   Shown in the video

Node 2 expands into all two-job partial sequences beginning with $J_1$.

1. Audio
   Observation

   Now which node I should expand? Shall I expand this one or this one? So next is this one.

2. Audio
   Observation

   I am doing second job, considering second job. But what about the next one? I will be doing third job or fourth job.

3. Audio
   Observation

   See first job already we have discarded. We are not considering it. We are not taking it. Then third job and fourth job. Then here only fourth job.

4. Diagram
   Observation

   The instructor expands node 3 into nodes 9, 10, 11; node 4 into node 12; node 5 into node 13; then proceeds to nodes 14, 15, 16, 17.

1. Some later node numbers are spoken quickly, but the visible numbering on the board supports the overall left-to-right frontier expansion.

1. Expression

   $$$$
   Explanation

   After completing the children of node 2, move to the next unexpanded node on the same frontier, node 3.

   Justification

   The instructor asks which node to expand next and then proceeds to the neighboring node rather than descending further under node 2.

   Shown in the video
2. Expression

   $$$$
   Explanation

   Expand node 3 by appending the jobs not already used on its path, producing children labeled $J_1$, $J_3$, $J_4$.

   Justification

   The audio says that when considering the second job, the next possibilities are third job or fourth job, and the board shows three children under node 3.

   Shown in the video
3. Expression

   $$$$
   Explanation

   Expand node 4 by appending only the remaining unused job $J_4$, because $J_1$ and $J_2$ are already on that path.

   Justification

   The instructor says first job already we have discarded ... then third job and fourth job. Then here only fourth job, matching the single child under node 4.

   Shown in the video
4. Expression

   $$$$
   Explanation

   Continue similarly across the rest of the frontier until all displayed nodes reach depth 4.

   Justification

   The final board shows a complete four-level permutation tree with nodes numbered through 17.

   Shown in the video

The tree is grown by expanding current-level nodes from left to right, confirming the breadth-first procedure described earlier.

1. Audio
   Observation

   Let us look at second method... first node is generated... put these nodes in the stack... pop out this one and expand...

2. Diagram
   Observation

   Instructor sequentially draws nodes 1-9 and updates the stack diagram, showing push and pop operations.

1. Expression

   $$$$
   Explanation

   Generate root node 1 and push its children (nodes 2, 3, 4, 5) onto the stack.

   Justification

   Initial expansion of the root node in the state space tree.

   Shown in the video
2. Expression

   $$$$
   Explanation

   Pop node 5 from the stack. It represents the last job, so it cannot be expanded further.

   Justification

   Depth-first search explores the deepest node first; leaf nodes have no children.

   Shown in the video
3. Expression

   $$$$
   Explanation

   Pop node 4 from the stack and expand it to generate node 6. Push node 6 onto the stack.

   Justification

   Node 4 represents a partial sequence that can be extended by adding the remaining job.

   Shown in the video
4. Expression

   $$$$
   Explanation

   Pop node 6 from the stack. It represents a complete sequence, so it cannot be expanded.

   Justification

   All jobs have been considered in this branch.

   Shown in the video
5. Expression

   $$$$
   Explanation

   Pop node 3 from the stack and expand it to generate nodes 7 and 8. Push both onto the stack.

   Justification

   Node 3 represents a partial sequence that can be extended by adding the remaining jobs.

   Shown in the video
6. Expression

   $$$$
   Explanation

   Pop node 8 from the stack. It represents a complete sequence, so it cannot be expanded.

   Justification

   All jobs have been considered in this branch.

   Shown in the video
7. Expression

   $$$$
   Explanation

   Pop node 7 from the stack and expand it to generate node 9. Push node 9 onto the stack.

   Justification

   Node 7 represents a partial sequence that can be extended by adding the remaining job.

   Shown in the video

The stack-based method successfully explores the state space tree in a depth-first manner, systematically generating and expanding nodes.

1. Audio
   Observation

   He says node 9 cannot be explored further, so take out 2 and explore it.

2. Animation
   Observation

   He draws children from node 2 labeled $J_2$, $J_3$, $J_4$ and numbers them 10, 11, 12.

3. Audio
   Observation

   He says the node numbers are 10, 11, and 12 and in the same order they are pushed into the stack.

1. Expression

   $$$$
   Explanation

   The current live node 9 is terminal for further exploration in the shown tree.

   Justification

   Stated directly by the instructor.

   Shown in the video
2. Expression

   $$$$
   Explanation

   The algorithm therefore removes node 2 from the stack and expands it.

   Justification

   Stated directly by the instructor.

   Shown in the video
3. Expression

   $$$$
   Explanation

   Node 2 generates three children corresponding to choosing the second, third, or fourth job after the first job has already been fixed.

   Justification

   Stated by the instructor while drawing the branches.

   Shown in the video
4. Expression

   $$$$
   Explanation

   These new nodes are numbered 10, 11, and 12 and pushed onto the stack in that same order.

   Justification

   Explicitly stated in the audio and shown on the board.

   Shown in the video

This demonstrates LIFO branch and bound by expanding one selected node fully and pushing all generated children onto a stack.

1. Audio
   Observation

   So let's expand this one. So job two is there. So this is job three and job four. Suppose here the cost is eight and here the cost is seven. Now which one we should explore? This one. But this is the last node second job fourth job. So this is the solution.

2. Diagram
   Observation

   Node $2 (C=12)$ is expanded into Node $6 (J3, C=8)$ and Node $7 (J4, C=7)$. Node 7 is underlined as the solution.

1. Expression

   $$$$
   Explanation

   Identify the node with the minimum cost among the initial children of the root (Node 2 has $C=12$).

   Justification

   Least Cost selection rule

   Shown in the video
2. Expression

   $$$$
   Explanation

   Expand Node 2 to generate its children, Node $6 (J3, C=8)$ and Node $7 (J4, C=7)$.

   Justification

   Branching step in the algorithm

   Shown in the video
3. Expression

   $$$$
   Explanation

   Compare the costs of the newly generated nodes and select the one with the minimum cost (Node 7 has $C=7$).

   Justification

   Least Cost selection rule

   Shown in the video
4. Expression

   $$$$
   Explanation

   Recognize that Node 7 represents a complete sequence (J2, J4) and is therefore the solution.

   Justification

   Termination condition for a leaf node representing a full job sequence

   Shown in the video

The algorithm successfully finds a solution by exploring only the most promising paths based on cost.

1. Audio
   Observation

   See I have taken one simple example here, that is job sequencing with deadlines problem.

2. Formula
   Observation

   Jobs = {$J_1$, $J_2$, $J_3$, $J_4$}, $P = {10, 5, 8, 3}$, $d = {1, 2, 1, 2}$

Represent a solution for a job sequencing problem with 4 jobs.

1. Jobs = {$J_1$, $J_2$, $J_3$, $J_4$}

2. $P = {10, 5, 8, 3}$

3. $d = {1, 2, 1, 2}$

Demonstrate two methods of representing a solution: subset and binary vector.

1. Expression

   $$S1 = \{J_1, J_4\}$$
   Explanation

   Represent the solution as a subset of selected jobs.

   Justification

   First method of representation.

   Shown in the video
2. Expression

   $$S2 = \{1, 0, 0, 1\}$$
   Explanation

   Represent the solution as a binary vector where 1 means the job is done and 0 means it is not.

   Justification

   Second method of representation.

   Shown in the video

$S1 = {J_1, J_4}$ (variable-size), $S2 = {1, 0, 0, 1}$ (fixed-size)

Both representations correctly identify that jobs 1 and 4 are selected, while jobs 2 and 3 are not.

1. Diagram
   Observation

   Whiteboard title: Branch and Bound; right side: Job sequencing with Jobs = {$J_1$, $J_2$, $J_3$, $J_4$}, $P = {10, 5, 8, 3}$, $d = {1, 2, 1, 2}$, $S1 = {J_1, J_4}$, $S2 = {1, 0, 0, 1}$.

2. Audio
   Observation

   Once I take first job, then what are the solution? Job two I can take or job three I can take, then job four I can take.

3. Audio
   Observation

   So if I follow one of the route, it says that I am doing first job and fourth job. I am not taking second and third job.

1. The clip does not compute any objective value, feasibility test, or bound; it only constructs the search tree.

Given four jobs $J_1$, $J_2$, $J_3$, $J_4$, draw the state-space tree used to explore job sequences by branch and bound.

1. Jobs = {$J_1$, $J_2$, $J_3$, $J_4$}.

2. The board also lists $P = {10, 5, 8, 3}$, $d = {1, 2, 1, 2}$, $S1 = {J_1, J_4}$, $S2 = {1, 0, 0, 1}$, but these are not used in the clip.

3. The method being illustrated is breadth-first expansion of the tree.

Show how partial job sequences are generated level by level in the branch-and-bound tree.

1. Expression

   $$$$
   Explanation

   Start at the root and create four children corresponding to choosing each job first.

   Justification

   All one-job partial sequences are placed on the first level.

   Shown in the video
2. Expression

   $$$$
   Explanation

   From the node for ($J_1$), create children for ($J_1$,$J_2$), ($J_1$,$J_3$), and ($J_1$,$J_4$).

   Justification

   After taking job 1, the remaining available jobs are 2, 3, and 4.

   Shown in the video
3. Expression

   $$$$
   Explanation

   Then expand the next frontier nodes in left-to-right order rather than descending immediately under ($J_1$,$J_2$).

   Justification

   The instructor explicitly frames branch and bound as breadth-first search.

   Shown in the video
4. Expression

   $$$$
   Explanation

   Continue until every displayed path reaches length 4, producing the full permutation tree shown on the board.

   Justification

   The final diagram contains nodes numbered up to 17 and all branches end at depth 4.

   Shown in the video

The example yields a four-level state-space tree whose nodes represent partial permutations of the four jobs, expanded breadth-first from the root.

The completed board visibly contains the root, four first-level children, and successive expansions ending with all paths of length 4, matching the spoken description of breadth-first exploration.

1. Diagram
   Observation

   Whiteboard displays a complete state space tree for 4 jobs with nodes numbered 1 to 14.

2. Formula
   Observation

   Jobs = {$J_1$, $J_2$, $J_3$, $J_4$}, $P = {10, 5, 8, 3}$, $d = {1, 2, 1, 2}$

Construct and interpret the state space tree for a job sequencing problem with 4 jobs.

1. Jobs = {$J_1$, $J_2$, $J_3$, $J_4$}

2. Profits $P = {10, 5, 8, 3}$

3. Deadlines $d = {1, 2, 1, 2}$

Understand how nodes represent partial job sequences.

1. Expression

   $$$$
   Explanation

   Draw the root node representing the empty sequence.

   Justification

   Start of the state space tree.

   Shown in the video
2. Expression

   $$$$
   Explanation

   Branch out to create child nodes for each possible first job ($J_1$, $J_2$, $J_3$, $J_4$).

   Justification

   Generating the first level of the tree.

   Shown in the video
3. Expression

   $$$$
   Explanation

   Continue branching to add subsequent jobs, creating nodes for all possible permutations.

   Justification

   Completing the state space tree.

   Shown in the video

The resulting tree has 14 nodes, where each path from the root to a leaf represents a unique permutation of the 4 jobs.

Visual inspection of the drawn tree confirms all permutations are represented.

1. Formula
   Observation

   The right side of the board is headed Job sequencing and lists Jobs, P, d, and Si.

2. Animation
   Observation

   A state-space tree is drawn with root 1 and existing nodes 2 through 9.

3. Audio
   Observation

   The instructor narrates taking out node 2 and exploring it, then pushing nodes 10, 11, 12 into the stack.

1. The clip does not compute an objective value for the job sequence in this interval.

Continue a branch-and-bound search on a job-sequencing state-space tree using a stack.

1. Jobs = {$J_1$, $J_2$, $J_3$, $J_4$}

2. $P = {10, 5, 8, 3}$

3. $d = {1, 2, 1, 2}$

4. Si = {$J_1$, $J_3$}

5. An existing tree already contains nodes 1 through 9.

Expand the next node selected from the stack and update the stack with its children.

1. Expression

   $$$$
   Explanation

   Node 9 is treated as not explorable further.

   Justification

   Stated by the instructor.

   Shown in the video
2. Expression

   $$$$
   Explanation

   Node 2 is taken out of the stack for exploration.

   Justification

   Stated by the instructor.

   Shown in the video
3. Expression

   $$$$
   Explanation

   From node 2, branches are created for the remaining job choices $J_2$, $J_3$, and $J_4$.

   Justification

   Shown by the drawn edges and stated verbally.

   Shown in the video
4. Expression

   $$$$
   Explanation

   The new nodes are labeled 10, 11, and 12 and pushed into the stack in that order.

   Justification

   Explicitly stated and written on the board.

   Shown in the video

The stack receives nodes 10, 11, and 12 after expanding node 2.

The board shows the new children under node 2 and the stack entries 10, 11, 12.

1. Audio
   Observation

   He says now let us look at this method least cost branch and bound method.

2. Animation
   Observation

   He erases the old tree and starts a fresh state-space tree with root node 1.

3. Formula
   Observation

   He writes $C = \infty$ beside the root node.

4. Audio
   Observation

   He says for every node we will calculate the cost and assumes the cost of this one is infinity.

1. The clip ends before the first child cost is computed.

2. The reason for initializing the root cost to infinity is not explained in this excerpt.

Begin a new branch-and-bound search using the Least Cost method on the same job-sequencing instance.

1. Jobs = {$J_1$, $J_2$, $J_3$, $J_4$}

2. $P = {10, 5, 8, 3}$

3. $d = {1, 2, 1, 2}$

4. Si = {$J_1$, $J_3$}

Create a new state-space tree and assign node costs starting from the root.

1. Expression

   $$$$
   Explanation

   The instructor clears the previous tree and announces that he will generate a tree again, specifically a state-space tree.

   Justification

   Stated in the audio and visible on the board.

   Shown in the video
2. Expression

   $$$$
   Explanation

   He draws a new root node labeled 1.

   Justification

   Visible animation on the whiteboard.

   Shown in the video
3. Expression

   $$$$
   Explanation

   He introduces the new idea that every node will have a calculated cost.

   Justification

   Stated directly by the instructor.

   Shown in the video
4. Expression

   $$C = \infty$$
   Explanation

   He assigns the root node cost as infinity.

   Justification

   Written on the board and spoken aloud.

   Shown in the video
5. Expression

   $$$$
   Explanation

   He begins considering the jobs for branching from the root.

   Justification

   Stated in the final seconds of the clip.

   Shown in the video

The LC-BB example is initialized with root node 1 and cost $C = \infty$.

The board shows the new root with $C = \infty$ before the clip ends.

1. Diagram
   Observation

   Whiteboard shows a tree structure with nodes labeled J1, J2, J3, J4 and costs $C=25$, $C=12$, $C=19$, $C=30$, $C=8$, $C=7$.

Find an optimal job sequence using the Least Cost Branch and Bound method given a set of jobs with profits and deadlines.

1. Jobs = {J1, J2, J3, J4}

2. $P = {10, 5, 8, 3}$

3. $d = {1, 2, 1, 2}$

4. Si = {J1, J3}

Determine the sequence of jobs that minimizes the total cost or maximizes profit according to the problem constraints.

1. Expression

   $$$$
   Explanation

   Initialize the root node with an infinite cost ($C = \infty$).

   Justification

   Standard initialization for minimization problems in Branch and Bound

   Shown in the video
2. Expression

   $$$$
   Explanation

   Generate the first level of child nodes representing the inclusion of each job (J1, J2, J3, J4) and assign arbitrary costs ($C=25$, $C=12$, $C=19$, $C=30$).

   Justification

   Branching step to explore different job sequences

   Shown in the video
3. Expression

   $$$$
   Explanation

   Select the node with the minimum cost (Node 2, $C=12$) for expansion.

   Justification

   Least Cost Branch and Bound selection rule

   Shown in the video
4. Expression

   $$$$
   Explanation

   Expand Node 2 to generate its children (Node 6 for J3 with $C=8$, Node 7 for J4 with $C=7$).

   Justification

   Branching step to continue exploring sequences starting with J2

   Shown in the video
5. Expression

   $$$$
   Explanation

   Select the node with the minimum cost among the new children (Node 7, $C=7$).

   Justification

   Least Cost Branch and Bound selection rule

   Shown in the video
6. Expression

   $$$$
   Explanation

   Identify Node 7 as a complete sequence (J2, J4) and declare it the solution.

   Justification

   Termination condition reached

   Shown in the video

The optimal sequence found is J2 followed by J4, with a final cost of 7.

The solution is verified by tracing the path from the root to the selected leaf node, ensuring all branching and selection rules were followed correctly.

1. Diagram
   Observation

   Whiteboard displays 'Branch and Bound' title and 'Job sequencing' problem data.

1. Title 'Branch and Bound'

2. Problem heading 'Job sequencing'

3. Data sets Jobs, P, d

1. Problem data remains visible throughout the clip

Sets the context for the lecture on Branch and Bound applied to job sequencing.

1. Animation
   Observation

   Instructor draws a circle (root node) and a line extending down-left labeled 'J1'.

1. Root node (circle)

2. Left branch labeled 'J1'

1. A new tree structure is drawn on the left side of the board.

1. Existing problem data on the right side of the board remains unchanged.

Visualizes the beginning of the state space tree generation using the subset method.

1. Diagram
   Observation

   At the start, the board already shows the title Branch and Bound, a small initial tree fragment with root node 1 and one edge labeled $J_1$ to node 2, and the right-side job-sequencing data.

1. Title Branch and Bound

2. Root node 1

3. Edge label $J_1$

4. Node 2

5. Right-side list Jobs, P, d, S1, S2

1. The instructor stands beside the board and begins explaining the difference from backtracking.

1. The job data on the right remains visible throughout the clip.

The clip opens with a partially prepared example and then develops the full state-space tree live.

1. Animation
   Observation

   The instructor draws three more edges from node 1 labeled $J_2$, $J_3$, $J_4$ and adds nodes 3, 4, 5.

1. Node 1

2. Edges $J_1$, $J_2$, $J_3$, $J_4$

3. Nodes 2, 3, 4, 5

1. Node 1 gains three additional children.

2. The first level becomes a full set of four one-job choices.

1. The root still represents the empty partial sequence.

2. The right-side job list is unchanged.

This visual step realizes the claim that branch and bound considers all immediate alternatives at the current level.

1. Animation
   Observation

   From node 2, the instructor draws three downward edges labeled $J_2$, $J_3$, $J_4$ and numbers the new nodes 6, 7, 8.

1. Node 2

2. Edges $J_2$, $J_3$, $J_4$

3. Nodes 6, 7, 8

1. Three second-level nodes are added beneath node 2.

1. The path to each new node begins with $J_1$.

2. No pruning or bounding marks are added.

The board now shows all two-job partial sequences that start with job 1.

1. Animation
   Observation

   The instructor next expands node 3 into nodes 9, 10, 11, then node 4 into node 12, then node 5 into node 13, and continues adding deeper nodes until 17.

1. Exact spoken numbering of some late nodes is less clear than the visible board labels.

1. Nodes 3, 4, 5

2. Their children 9-13

3. Later descendants 14-17

1. Expansion proceeds across the current frontier from left to right.

2. The tree deepens level by level until all branches reach length 4.

1. Each edge appends one previously unused job.

2. The overall strategy remains breadth-first rather than depth-first.

The visual process demonstrates the operational meaning of the earlier verbal contrast between backtracking and branch and bound.

1. Diagram
   Observation

   A pre-drawn state space tree is visible on the left side of the whiteboard.

1. State space tree with nodes 1-14

2. Edge labels $J_1-J_4$

1. Instructor points to different nodes to explain their meaning.

1. The structure of the tree remains unchanged during this segment.

The tree visually represents the entire solution space for the job sequencing problem.

1. Diagram
   Observation

   Instructor draws a vertical rectangle (stack) and writes/pops numbers inside it while drawing new tree nodes.

1. Stack diagram

2. Newly drawn tree nodes (1-9)

1. Numbers are written into the stack (push operation).

2. Numbers are erased from the top of the stack (pop operation).

3. New nodes are drawn below existing nodes in the tree.

1. The stack always operates on a Last-In-First-Out basis.

The visual synchronization of drawing tree nodes and updating the stack demonstrates the mechanics of depth-first traversal using an explicit stack.

1. Diagram
   Observation

   A pre-drawn tree titled Branch and Bound occupies the left half of the board, with a Job sequencing data block on the right and a vertical stack below it.

2. Animation
   Observation

   The instructor points to node 9, then moves to node 2 and draws three outgoing branches.

1. Tree nodes 1 through 9

2. Node 2

3. Node 9

4. Vertical stack diagram

5. Job sequencing data block

1. Attention shifts from node 9 to node 2.

2. Three new child nodes are added beneath node 2.

3. The stack is updated with 10, 11, 12.

1. The underlying job-sequencing data on the right remains unchanged.

2. The earlier part of the tree remains visible while the new children are added.

The visual sequence demonstrates how a stack-based branch-and-bound search backtracks from a dead-end node and expands the next available node.

1. Formula
   Observation

   The instructor writes Queue -> FIFO, Stack -> LIFO, and Least cost -> LC-BB.

2. Diagram
   Observation

   The numbered list 1. FIFO-BB, 2. LIFO-BB, 3. LC-BB appears on the board.

1. Queue -> FIFO

2. Stack -> LIFO

3. Least cost -> LC-BB

4. Numbered list of methods

1. New text is added under the tree.

2. The method list is completed with LC-BB.

1. The job-sequencing data block stays on the right.

2. The earlier tree remains visible until it is erased later.

This visual summary classifies the search strategies by the data structure used to select the next node.

1. Animation
   Observation

   The previous tree and stack are erased.

2. Formula
   Observation

   Least Cost - BB is written under the main heading.

3. Animation
   Observation

   A new root node 1 is drawn, followed by $C = \infty$ beside it.

1. Main heading Branch and Bound

2. Subtitle Least Cost - BB

3. New root node 1

4. Cost label $C = \infty$

1. The old state-space tree and stack disappear.

2. A new heading for Least Cost - BB is added.

3. A fresh root node is created and assigned a cost.

1. The Job sequencing data on the right remains visible.

2. The numbered method list remains visible.

The reset signals a shift from explaining FIFO/LIFO node selection to demonstrating cost-based branch and bound from scratch.

1. Animation
   Observation

   The instructor draws a tree structure on the whiteboard, adding nodes and edges sequentially, labeling them with job identifiers and cost values.

1. Root node

2. Child nodes (J1, J2, J3, J4)

3. Grandchild nodes (J3, J4)

4. Edges connecting nodes

5. Cost labels (C=...)

1. Nodes are added to the tree as the algorithm branches.

2. Costs are assigned to each node.

3. The instructor points to specific nodes to explain the selection process.

1. The hierarchical structure of the tree remains consistent.

2. The right side of the board consistently displays the problem parameters (Jobs, P, d, Si) and the list of methods (FIFO, LIFO, LC).

The visual construction of the tree demonstrates how the Branch and Bound algorithm systematically explores the solution space by generating and evaluating partial sequences of jobs.

1. Audio
   Observation

   We don't go depth wise. We don't follow depth first search. We follow breadth first search for exploring the solutions of a problem.

One might assume branch and bound, like backtracking, immediately follows one branch down to the next level.

In this clip, the instructor explicitly says branch and bound does not go depth-wise here; it explores the tree breadth-first by considering all alternatives at the current level before moving deeper.

1. Audio
   Observation

   So if I follow one of the route, it says that I am doing first job and fourth job. I am not taking second and third job. That is the meaning.

A path such as $J_1$ then $J_4$ could be mistaken for implying that all jobs have been handled.

The instructor clarifies that following that route means doing job 1 and job 4 along that partial path, while jobs 2 and 3 are not taken on that route.

1. Audio
   Observation

   After saying stack gives LIFO branch and bound, the instructor immediately adds that you have to follow breadth first search only.

2. Audio
   Observation

   He defines that once a node is picked, all possible nodes from it should be explored before selecting the next node.

1. The clip uses the phrase breadth first search in a branch-and-bound context without contrasting it formally with depth-first search.

One might infer that LIFO branch and bound means going down a single child immediately after another.

In this lecture segment, the instructor explicitly says that even with a stack, you must first generate all possible children of the selected node before choosing the next node.

1. Audio
   Observation

   This is similar to backtracking in the sense that it also uses state space tree for solving the problem.

Branch and Bound is similar to backtracking as both use state space trees, but Branch and Bound is specifically for optimization (minimization) problems.

1. Audio
   Observation

   So the difference between backtracking and branch and bound is backtracking was depth first search and the branch and bound is breadth first search.

The clip uses the same job-sequencing tree to contrast backtracking's depth-first traversal with branch and bound's breadth-first traversal.

1. Audio
   Observation

   Now this one level is completed. See, I have considered all the jobs one by one.

2. Animation
   Observation

   The instructor expands nodes across the frontier from left to right.

The step-by-step construction of the tree is the concrete application of the stated breadth-first expansion rule.

1. Audio
   Observation

   So if I follow one of the route, it says that I am doing first job and fourth job. I am not taking second and third job.

The semantic rule that a path encodes a partial job sequence is part of the definition of the displayed state-space tree.

1. Audio
   Observation

   pop out this one and expand... pop out next node and explore it

2. Diagram
   Observation

   Stack operations dictate the order in which tree nodes are expanded.

The stack-based method is applied to traverse the state space tree, implementing a depth-first search strategy.

1. Formula
   Observation

   The board writes Queue -> FIFO and Stack -> LIFO one below the other.

2. Audio
   Observation

   The instructor asks what is the difference between the previous one and this one and answers by contrasting queue and stack.

FIFO-BB and LIFO-BB are distinguished by whether the next node is taken from a queue or from a stack.

1. Formula
   Observation

   Least cost -> LC-BB is added as a third method after FIFO-BB and LIFO-BB.

2. Audio
   Observation

   The instructor says there is one more method and then explains that every node will have a cost.

1. The clip introduces LC-BB but does not fully develop its selection rule before ending.

LC-BB is presented as an additional branch-and-bound variant that augments the search with node costs, extending the discussion beyond pure FIFO/LIFO node selection.

1. Audio
   Observation

   The instructor introduces cost calculation while discussing least cost branch and bound.

2. Formula
   Observation

   He writes $C = \infty$ at the root of the new LC-BB tree.

The LC-BB demonstration depends on first defining a node cost and a problem-specific cost function.

1. Audio
   Observation

   The instructor says you can use any one and you have to follow breadth first search only.

The stated expansion rule applies to the FIFO method discussed in this clip.

1. Audio
   Observation

   The same breadth-first expansion statement follows directly after the LIFO definition.

The stated expansion rule also applies to the LIFO method discussed in this clip.

1. Audio
   Observation

   Now let us look back again. FIFO method which one we select for exploration this one will expand. LIFO we try to expand this one. Least cost out of this whichever is having least cost we'll try that one.

The Least Cost strategy contrasts with FIFO and LIFO by selecting nodes based on their evaluated cost rather than their generation order.

1. Audio
   Observation

   This is the idea of LC branch and bound and this is much faster than using FIFO and LIFO.

The Least Cost selection rule is applied to achieve the efficiency benefit of avoiding full tree generation and reaching solutions faster.

1. Audio
   Observation

   The topic is branch and bound... useful for solving optimization problem and only minimization problem

1. Branch and Bound Definition

1. Audio
   Observation

   solution can be represented as subset... second method... fixed size solution

2. Formula
   Observation

   $S1 = {J_1, J_4}$, $S2 = {1, 0, 0, 1}$

1. Variable-Size Solution (Subset Method)
2. Fixed-Size Solution (Binary Vector Method)

1. Audio
   Observation

   We don't follow depth first search. We follow breadth first search for exploring the solutions of a problem.

1. Branch and bound uses breadth-first expansion, unlike backtracking
2. Breadth-first expansion rule on the job-sequencing tree

1. Audio
   Observation

   So the difference between backtracking and branch and bound is backtracking was depth first search and the branch and bound is breadth first search.

1. Branch and bound uses breadth-first expansion, unlike backtracking
2. Backtracking is depth-first search and branch and bound is breadth-first search

1. Audio
   Observation

   So if I follow one of the route, it says that I am doing first job and fourth job. I am not taking second and third job.

1. A root-to-node path encodes a partial job sequence
2. Job-sequencing state-space tree with four jobs

1. Audio
   Observation

   Now this one level is completed. See, I have considered all the jobs one by one.

2. Animation
   Observation

   Nodes are expanded left to right across the frontier.

1. Breadth-first expansion rule on the job-sequencing tree
2. Continuing breadth-first expansion across the frontier

1. Diagram
   Observation

   The board lists P, d, S1, and S2, but the spoken explanation in this clip focuses only on tree construction.

1. The clip does not explain these quantities.

1. Job-sequencing state-space tree with four jobs

1. Audio
   Observation

   I'll put these nodes in the stack... pop out this one and expand

1. Stack-Based Depth-First Traversal
2. State Space Tree for Job Sequencing

1. Formula
   Observation

   Queue -> FIFO is written on the board.

2. Audio
   Observation

   The instructor names FIFO branch and bound.

1. FIFO Branch and Bound

1. Formula
   Observation

   Stack -> LIFO is written on the board.

2. Audio
   Observation

   The instructor names LIFO branch and bound.

1. LIFO Branch and Bound
2. FIFO Branch and Bound

1. Audio
   Observation

   The instructor says once you pick up a node you should explore all the possible nodes, then only select the next node.

1. Breadth-first expansion rule in this branch-and-bound setting
2. LIFO Branch and Bound

1. Formula
   Observation

   Least cost -> LC-BB and Least Cost - BB are written on the board.

2. Audio
   Observation

   The instructor introduces least cost branch and bound.

1. Least Cost Branch and Bound (LC-BB)