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Linear Programming Problem (LPP)

Graphical Method to Solve LPP

Problem Statement:

Maximize Z = 3x1 + 4x2
Subject to the constraints:
x1 + 2x2 ≤ 4
3x1 + 2x2 ≤ 6
x1, x2 ≥ 0 (Non-negativity constraint)

  • Step 1: Find the coordinates
    We start by finding the points where the constraints intersect the axes, as these points will help us plot the lines representing the constraints.
    For the constraint x1 + 2x2 ≤ 4:
    To find the points where the line intersects the x1 and x2 axes:
    • If x2 = 0, then x1 = 4, giving the point (4, 0).
    • If x1 = 0, then x2 = 2, giving the point (0, 2).
    For the constraint 3x1 + 2x2 ≤ 6:
    • If x2 = 0, then x1 = 2, giving the point (2, 0).
    • If x1 = 0, then x2 = 3, giving the point (0, 3).
  • Step 2: Draw the lines and identify the feasible region
    We now plot the lines x1 + 2x2 = 4 and 3x1 + 2x2 = 6 on a graph using the points found earlier.
    The feasible region is the area where both constraints overlap, and it represents the possible values of x1 and x2 that satisfy all constraints.
    Since both constraints have ≤, the feasible region will be the area below both lines, towards the origin (0, 0).
    Graph of Feasible Region
  • Step 3: Identify the corner points of the feasible region
    The optimal solution of an LPP in the graphical method will always lie at one of the corner points (vertices) of the feasible region.
    The corner points in this example are:
    • O: (0, 0) – the origin
    • A: (2, 0) – where 3x1 + 2x2 = 6 intersects the x1-axis
    • C: (0, 2) – where x1 + 2x2 = 4 intersects the x2-axis
    • B: The intersection of the two constraints x1 + 2x2 = 4 and 3x1 + 2x2 = 6. To find B, we solve both equations simultaneously:
      Solve the system:
      x1 + 2x2 = 4
      3x1 + 2x2 = 6
      Subtract the first equation from the second:
      (3x1 + 2x2) - (x1 + 2x2) = 6 - 4
      This simplifies to:
      2x1 = 2, so x1 = 1.
      Substitute x1 = 1 into x1 + 2x2 = 4:
      1 + 2x2 = 4, so x2 = 3/2.
      The intersection point B is (1, 3/2).
  • Step 4: Evaluate the objective function at each corner point
    Now, we substitute the coordinates of each corner point into the objective function Z = 3x1 + 4x2 to find the value of Z at each point:
    • At O: Z = 3(0) + 4(0) = 0
    • At A: Z = 3(2) + 4(0) = 6
    • At B: Z = 3(1) + 4(3/2) = 3 + 6 = 9
    • At C: Z = 3(0) + 4(2) = 8
  • Step 5: Determine the optimal solution
    Since the problem is a maximization problem, the optimal solution will be the point that gives the maximum value of Z.
    From the calculations, we see that the maximum value of Z is 9, which occurs at point B (1, 3/2).
    Thus, the optimal solution is:
    • x1 = 1
    • x2 = 3/2
  • Step 6: Minimization (if required)
    If we were tasked with minimizing Z instead of maximizing it, the minimum value would occur at the origin O (0, 0), where Z = 0.

Graphical Method to Solve LPP (Minimization Problem)

Problem Statement:

Minimize Z = 3x1 + 4x2
Subject to the constraints:
x1 + 2x2 ≥ 4
3x1 + 2x2 ≥ 6
x1, x2 ≥ 0 (Non-negativity constraint)

  • Step 1: Find the coordinates
    Since the constraints have “≥” signs, the feasible region will be away from the origin.
    To plot the constraints, we first find the points where the lines intersect the axes.
    For the constraint x1 + 2x2 ≥ 4:
    • If x2 = 0, then x1 = 4, giving the point (4, 0).
    • If x1 = 0, then x2 = 2, giving the point (0, 2).
    For the constraint 3x1 + 2x2 ≥ 6:
    • If x2 = 0, then x1 = 2, giving the point (2, 0).
    • If x1 = 0, then x2 = 3, giving the point (0, 3).
  • Step 2: Draw the lines and identify the feasible region
    We now plot the lines x1 + 2x2 = 4 and 3x1 + 2x2 = 6 using the points found earlier.
    The feasible region in this case will be above both lines (away from the origin), as the constraints are in the form of “≥”.
    The common area between the two constraints will give us the feasible region where both constraints are satisfied.
    Graph of Feasible Region
  • Step 3: Identify the corner points of the feasible region
    Just like in the maximization case, the optimal solution for a minimization problem also lies at one of the corner points (vertices) of the feasible region.
    The corner points of the feasible region are:
    • P: (4, 0) – where x1 + 2x2 = 4 intersects the x1-axis
    • Q: (0, 3) – where 3x1 + 2x2 = 6 intersects the x2-axis
    • R: The intersection of the two constraints. To find this point, we solve both equations simultaneously:
      Solve the system:
      x1 + 2x2 = 4
      3x1 + 2x2 = 6
      Subtract the first equation from the second:
      (3x1 + 2x2) - (x1 + 2x2) = 6 - 4
      This simplifies to:
      2x1 = 2, so x1 = 1.
      Substitute x1 = 1 into x1 + 2x2 = 4:
      1 + 2x2 = 4, so x2 = 3/2.
      The intersection point R is (1, 3/2).
  • Step 4: Evaluate the objective function at each corner point
    Now we substitute the coordinates of each corner point into the objective function Z = 3x1 + 4x2 to find the value of Z at each point:
    • At P: Z = 3(4) + 4(0) = 12
    • At Q: Z = 3(0) + 4(3) = 12
    • At R: Z = 3(1) + 4(3/2) = 3 + 6 = 9
  • Step 5: Determine the optimal solution
    Since the problem is a minimization problem, the optimal solution will be the point that gives the minimum value of Z.
    From the calculations, we see that the minimum value of Z is 9, which occurs at point R (1, 3/2).
    Thus, the optimal solution is:
    • x1 = 1
    • x2 = 3/2
  • Note: Unbounded Solution (if the question was for maximization)
    If this problem were a maximization problem, the solution would be unbounded.
    This is because, as we move away from the origin in the feasible region, the value of Z increases indefinitely.
    Thus, there is no maximum value for Z, and we say the solution is unbounded.

Alternative solutions

Problem Statement:

Maximize Z = 3x1 + 2x2
Subject to the constraints:
x1 + 2x2 ≤ 4
3x1 + 2x2 ≤ 6
x1, x2 ≥ 0 (Non-negativity constraint)

  • Step 1: Find the coordinates
    We start by finding the points where the constraints intersect the axes, which will help us plot the lines representing the constraints. For the constraint x1 + 2x2 ≤ 4:
    To find the points where the line intersects the x1 and x2 axes:
    • If x2 = 0, then x1 = 4, giving the point (4, 0).
    • If x1 = 0, then x2 = 2, giving the point (0, 2).
    For the constraint 3x1 + 2x2 ≤ 6:
    • If x2 = 0, then x1 = 2, giving the point (2, 0).
    • If x1 = 0, then x2 = 3, giving the point (0, 3).
  • Step 2: Draw the lines and identify the feasible region
    We now plot the lines x1 + 2x2 = 4 and 3x1 + 2x2 = 6 on a graph using the points found earlier.
    The feasible region is the area where both constraints overlap, representing the possible values of x1 and x2 that satisfy all constraints.
    Since both constraints have ≤, the feasible region will be the area below both lines, towards the origin (0, 0).
    Graph of Feasible Region
  • Step 3: Identify the corner points of the feasible region
    The optimal solution of an LPP in the graphical method will always lie at one of the corner points (vertices) of the feasible region.
    The corner points in this example are:
    • O: (0, 0) – the origin
    • A: (2, 0) – where 3x1 + 2x2 = 6 intersects the x1-axis
    • C: (0, 2) – where x1 + 2x2 = 4 intersects the x2-axis
    • B: The intersection of the two constraints x1 + 2x2 = 4 and 3x1 + 2x2 = 6. To find B, we solve both equations simultaneously:
      Solve the system:
      x1 + 2x2 = 4
      3x1 + 2x2 = 6
      Subtract the first equation from the second:
      (3x1 + 2x2) - (x1 + 2x2) = 6 - 4
      This simplifies to:
      2x1 = 2, so x1 = 1.
      Substitute x1 = 1 into x1 + 2x2 = 4:
      1 + 2x2 = 4, so x2 = 3/2.
      The intersection point B is (1, 3/2).
  • Step 4: Evaluate the objective function at each corner point
    Now, we substitute the coordinates of each corner point into the objective function Z = 3x1 + 2x2 to find the value of Z at each point:
    • At O: Z = 3(0) + 2(0) = 0
    • At A: Z = 3(2) + 2(0) = 6
    • At B: Z = 3(1) + 2(3/2) = 3 + 3 = 6
    • At C: Z = 3(0) + 2(2) = 4
  • Step 5: Determine the optimal solution
    In this case, we have two points (A and B) that give the same maximum value of Z:
    • A (2, 0): Z = 6
    • B (1, 3/2): Z = 6
    This indicates that there are multiple optimal solutions. Therefore, the solution is classified as an alternative optimal solution since the maximum Z can be achieved at both points.
  • Step 6: Discuss the type of solution
    Since both points A and B yield the same maximum value for Z (which is 6), we can conclude that:
    • This problem has alternative solutions, as there are multiple combinations of x1 and x2 that maximize the objective function.
    • Moreover, if we extend the feasible region further, we may find that there are infinite solutions along the line segment connecting points A and B. Hence, the solution can also be described as having infinite optimal solutions.

Graphical Method to Solve LPP (No Solution Case)

Problem Statement:

Maximize Z = 6x1 + 4x2
Subject to the constraints:
2x1 + 4x2 ≤ 4
4x1 + 8x2 ≥ 16
x1, x2 ≥ 0 (Non-negativity constraint)

  • Step 1: Find the coordinates
    We start by finding the points where the constraints intersect the axes.
    For the constraint 2x1 + 4x2 ≤ 4:
    To find the points where the line intersects the x1 and x2 axes:
    • If x2 = 0, then 2x1 = 4, giving the point (2, 0).
    • If x1 = 0, then 4x2 = 4, giving the point (0, 1).
    For the constraint 4x1 + 8x2 ≥ 16:
    To find the points where the line intersects the axes:
    • If x2 = 0, then 4x1 = 16, giving the point (4, 0).
    • If x1 = 0, then 8x2 = 16, giving the point (0, 2).
  • Step 2: Draw the lines and identify the feasible region
    We now plot the lines 2x1 + 4x2 = 4 and 4x1 + 8x2 = 16 on a graph using the points found earlier.
    The feasible region for the first constraint is the area below the line, while for the second constraint, it is the area above the line.
    Since one constraint is ≤ and the other is ≥, we are looking for an intersection of these two regions.
    Graph of No Solution
  • Step 3: Analyze the feasible region
    Upon examining the plotted constraints, we see that there is no overlap between the feasible regions defined by the two constraints.
    Thus, there is no common area where both conditions are satisfied simultaneously. This indicates that:
    • The problem has no feasible solution.
    • The constraints are contradictory, as one requires values below a certain level while the other requires values above a certain level.
  • Step 4: Conclusion
    In this case, since there is no feasible region where all constraints are satisfied, the linear programming problem does not have a solution.
    It illustrates the importance of properly formulating constraints to ensure they can coexist without conflict in optimization problems.

Simplex Method to Solve LPP

Maximize Z = 5x1 + 3x2

Subject to the constraints:

3x1 + 5x2 ≤ 15
5x1 + 2x2 ≤ 10
x1, x2 ≥ 0

  • To solve an LPP using the Simplex Method, we must ensure that the objective function is in maximization form. If it is a minimization problem, we first convert it to a maximization problem.
  • If the given objective function is in the form of minimization (Min Z), we can convert it into a maximization problem by using the following relationship:
    • Max Z' = -Min Z
    • For example, if the objective is to minimize Z = 5x1 + 3x2, we convert it to:
      • Max Z' = - (5x1 + 3x2)
      • This simplifies to: Max Z' = -5x1 - 3x2
    • Now, we can proceed to solve the maximization problem using the Simplex Method.
  • Next, we need to balance the constraints. For constraints with ≤ signs, we add slack variables to convert inequalities into equalities. For example, if we have a ≤ constraint, we add a slack variable. Similarly, if we have a ≥ constraint, we subtract a surplus variable to achieve equality.
  • Now, converting the given problem:
    Maximize Z = 5x1 + 3x2 + 0x3 + 0x4 (introducing slack variables x3 and x4)
    Subject to:
    3x1 + 5x2 + x3 = 15
    5x1 + 2x2 + x4 = 10
    All variables x1, x2, x3, x4 ≥ 0
  • The variables x3 and x4 are known as the basic variables, as they form the identity matrix in the system of equations. The basic variables usually include the slack variables we introduced. The objective now is to make the decision variables (x1, x2) the basic variables through iterations.
  • We now create the initial Simplex tableau as follows:
                
               | Cj  |   5  |   3  |   0  |   0  |
      CB | Xb  |  B  |  x1  |  x2  |  x3  |  x4  |
    ----------------------------------------------
      0  | x3  |  15 |  3   |  5   |  1   |  0   |
      0  | x4  |  10 |  5   |  2   |  0   |  1   |
                
            
    Here:
    • CB represents the cost of the basic variables in the objective function (initially zero for slack variables).
    • Xb represents the basic variables.
    • Cj represents the cost coefficients of the decision variables x1, x2, and slack variables x3, x4 in the objective function.
  • Next, we perform the Simplex iteration. We want to bring x1 and x2 into the basis:
            
              | Cj  |   5  |   3  |   0  |   0  |
     CB | Xb  |  B  |  x1  |  x2  |  x3  |  x4  |
    ----------------------------------------------
     0  | x3  |  15 |  3   |  5   |  1   |  0   | (Row 1)
     0  | x4  |  10 |  5   |  2   |  0   |  1   | (Row 2)
    ----------------------------------------------
          Zj  |     |  0   |  0   |  0   |  0   |
          Cj  |     |  5   |  3   |  0   |  0   |
     Zj - Cj  |     | -5   | -3   |  0   |  0   |
            
        

    To calculate Zj, we use the formula Zj = CB × B, where:

    • CB is the cost coefficient of the basic variables in the basis (here, x3 and x4).
    • B is the column for the variable we are calculating Zj for.
    For example, for the x1 column:
    Zj(x1) = (CB(row1) × B(row1,x1)) + (CB(row2) × B(row2,x1))
    Substituting the values:
    Zj(x1) = (0 × 3) + (0 × 5) = 0
    Similarly, Zj for x2, x3, and x4 will be calculated. In this iteration, since all the current basic variables (x3 and x4) have cost coefficients of 0, all Zj values will be 0.

    After calculating Zj, we compute Zj - Cj, which shows the difference between the calculated and actual costs. The most negative value, -5 (in the x1 column), indicates that x1 should enter the basis.

    To decide which variable to remove from the basis (x3 or x4), we compute the minimum ratio of b values and the coefficients in the x1 column:

            
              | Cj  |   5  |   3  |   0  |   0  |
     CB | Xb  |  b  |  x1  |  x2  |  x3  |  x4  | min ratio
    --------------------------------------------------------
     0  | x3  |  15 |  3   |  5   |  1   |  0   | 15/3 = 5
     0  | x4  |  10 |  5   |  2   |  0   |  1   | 10/5 = 2 → smallest ratio
            
        

    As 2 is the smallest ratio, x4 is the outgoing variable, and x1 will replace x4 in the basis.

  • Now, we pivot on the key element (5 in the x1 column, row 2). After performing row operations, the updated tableau looks like this:
                                    
               | Cj  |   5  |   3  |   0  |   0  |
      CB | Xb  |  b  |  x1  |  x2  |  x3  |  x4  |
    ----------------------------------------------
      0  | x3  |   9 |  0   | 19/5 |  1   | -3/5 | (Row 1)
      5  | x1  |   2 |  1   |  2/5 |  0   |  1/5 | (Row 2)
    ----------------------------------------------
                                    
                                
    Row 2 was divided by 5, and row 1 was updated using row 1 - 3 * (new row 2).
  • Next, we compute Zj - Cj again:
                
               | Cj  |   5  |   3  |   0  |   0  |
      CB | Xb  |  b  |  x1  |  x2  |  x3  |  x4  | min ratio
    --------------------------------------------------------
      0  | x3  |   9 |  0   | 19/5 |  1   | -3/5 | 45/19 → smallest ratio
      5  | x1  |   2 |  1   |  2/5 |  0   |  1/5 | 10/2
    --------------------------------------------------------
           Zj  |     |  5   |  2   |  0   |  1   |
           Cj  |     |  5   |  3   |  0   |  0   |
      Zj - Cj  |     |  0   | -1   |  0   |  1   |
                
            
    The next entering variable is x2 (as -1 is the most negative value in Zj - Cj), and the outgoing variable is x3 (smallest ratio). We perform another pivot operation.
  • After performing the row operations, the final tableau is:
                                    
               |   Cj  |   5  |   3  |   0   |   0  |
      CB | Xb  |    b  |  x1  |  x2  |  x3   |  x4  |
    --------------------------------------------------------
      3  | x2  | 45/19 |  0   |   1  |  5/19 |-3/19 | (Row 1) new
      5  | x1  | 20/19 |  1   |   0  | -2/19 |  5/19 | (Row 2)
    --------------------------------------------------------
           Zj  |       |  5   |  3   |  5/19 | 16/19
           Cj  |       |  5   |  3   |   0   |   0
      Zj - Cj  |       |  0   |  0   |  5/19 | 16/19
                                    
                                
    Since there are no more negative values in the Zj - Cj row, the optimal solution has been reached.
  • Finally, the optimal solution is:
    x1 = 20/19, x2 = 45/19
    Optimal value of Z = 5(20/19) + 3(45/19) = 225/19 ≈ 11.84

Maximize Z = 3x1 + 2x2 + 5x3

Subject to the constraints:

x1 + x2 + x3 ≤ 9
2x1 + 3x2 + 5x3 ≤ 30
2x1 - x2 - x3 ≤ 8
xi ≥ 0

  • Convert the constraints by adding slack variables:
    Maximize Z = 3x1 + 2x2 + 5x3 + 0x4 + 0x5 + 0x6
    Subject to:
    x1 + x2 + x3 + x4 = 9
    2x1 + 3x2 + 5x3 + x5 = 30
    2x1 - x2 - x3 + x6 = 8
    xi ≥ 0
  • Create the initial Simplex tableau:
                
               | Cj  |   3  |   2  |   5  |   0  |   0  |   0  |
      CB | Xb  |  b  |  x1  |  x2  |  x3  |  x4  |  x5  |  x6  |
    ----------------------------------------------------------------------
      0  | x4  |   9 |   1  |   1  |   1  |   1  |   0  |   0  | (Row 1)
      0  | x5  |  30 |   2  |   3  |   5  |   0  |   1  |   0  | (Row 2)
      0  | x6  |   8 |   2  |  -1  |  -1  |   0  |   0  |   1  | (Row 3)
                
            
  • Perform the Simplex iteration by identifying the entering and exiting variables:
                
               | Cj  |   3  |   2  |   5  |   0  |   0  |   0  |
      CB | Xb  |  b  |  x1  |  x2  |  x3  |  x4  |  x5  |  x6  |
    ------------------------------------------------------------------------
      0  | x4  |   9 |   1  |   1  |   1  |   1  |   0  |   0  | (Row 1)
      0  | x5  |  30 |   2  |   3  |   5  |   0  |   1  |   0  | (Row 2)
      0  | x6  |   8 |   2  |  -1  |  -1  |   0  |   0  |   1  | (Row 3)
    ------------------------------------------------------------------------
           Zj  |     |   0  |   0  |   0  |   0  |   0  |   0  |
           Cj  |     |   3  |   2  |   5  |   0  |   0  |   0  |
      Zj - Cj  |     |  -3  |  -2  |  -5  |   0  |   0  |   0  |
                
            
    The most negative value in Zj - Cj is -5, so x3 enters the basis. We calculate the minimum ratio to determine the exiting variable:
    Row 1: 9/1 = 9
    Row 2: 30/5 = 6 (smallest ratio)
    Row 3: 8/-1 (ignored because it’s negative)
    Thus, x5 exits, and x3 replaces x5 in the basis.
  • Perform the pivot on the key element (5 in Row 2, x3 column), and update the tableau:
    for row 2 we are dividing it by 5
    for row 1 we are doing row 1 - row 2 (new)
    for row 3 we are doing row 3 + row 2 (new)
                
               | Cj  |   3  |   2  |   5  |   0  |   0  |   0  |
      CB | Xb  |  b  |  x1  |  x2  |  x3  |  x4  |  x5  |  x6  |
    -------------------------------------------------------------------
      0  | x4  |   3 |   3/5|   2/5|   0  |   1  |  -1/5|   0  | (Row 1)
      5  | x3  |   6 |   2/5|   3/5|   1  |   0  |   1/5|   0  | (Row 2)
      0  | x6  |  14 |  12/5|  -2/5|   0  |   0  |   1/5|   1  | (Row 3)
    --------------------------------------------------------------------
                
            
  • Compute Zj - Cj again:
                
               | Cj  |   3  |   2  |   5  |   0  |   0  |   0  |
      CB | Xb  |  b  |  x1  |  x2  |  x3  |  x4  |  x5  |  x6  |
    ---------------------------------------------------------------------
      0  | x4  |   3 |   3/5|   2/5|   0  |   1  |  -1/5|   0  | (Row 1)
      5  | x3  |   6 |   2/5|   3/5|   1  |   0  |   1/5|   0  | (Row 2)
      0  | x6  |  14 |  12/5|  -2/5|   0  |   0  |   1/5|   1  | (Row 3)
    ---------------------------------------------------------------------
           Zj  |     |   2  |   3  |   5  |   0  |   1  |   0  |
           Cj  |     |   3  |   2  |   5  |   0  |   0  |   0  |
      Zj - Cj  |     |  -1  |   1  |   0  |   0  |   1  |   0  |
                
            
    The most negative value in Zj - Cj is -1, so x1 enters the basis. We calculate the minimum ratio to determine the exiting variable:
    Row 1: 3/(3/5) = 5 (smallest ratio)
    Row 2: 30/2 = 15
    Row 3: 14/(12/5) = 5.833
    Thus, x4 exits, and x1 replaces x4 in the basis.
  • now we wil put 1 at key element place which is 3/5 and at row 2 and row 3 we will put 0 below the key element.
    for row 1 we will do row 1 * 5/3
    for row 2 we will do row 2 - 2/5 * row 1(new)
    for row 3 we will do row 3 - 12/5 * row 1 (new)
                                    
               | Cj  |   3  |   2  |   5  |   0  |   0  |   0  |
      CB | Xb  |  b  |  x1  |  x2  |  x3  |  x4  |  x5  |  x6  |
    -------------------------------------------------------------------
      3  | x1  |   5 |   1  |   2/3|   0  |  5/3 |  -1/3|   0  | (Row 1) new
      5  | x3  |   4 |   0  |   1/3|   1  | -2/3 |   1/3|   0  | (Row 2)
      0  | x6  |   2 |   0  |   -2 |   0  |  -4  |   1  |   1  | (Row 3)
    --------------------------------------------------------------------
      Zj - Cj  |     |   0  |   5/3|   0  |   5/3|   2/3|   0  |
                                    
                                
    As now all Zj - Cj values are postive, hence we have our optimal solution
    so, x1 = 5, x2 = 0 and x3 = 4
    We put this in Z cost function we get max Z = 35

Big M method

The Big M Method is an approach used to solve linear programming problems (LPP) involving artificial variables when the problem includes greater-than-or-equal-to (≥) or equality (=) constraints. It allows converting such constraints into a standard form suitable for solving using the Simplex method.

Min Z = x1 + x2
Constraints:
2x1 + x2 ≥ 4
x1 + 7x2 ≥ 7
x1 ≥ 0

  • First, we have to convert it into a maximization function.
    For that, we will multiply it by -1.
    Max z' = -x1 - x2
    Since we have ≥, we need to subtract a surplus variable.
    2x1 + x2 - x3 + 0x4 = 4
    x1 + 7x2 + 0x3 - x4 = 7
    Now, the problem is that we don't have a unit matrix, so we add 2 more variables called artificial variables.
    Max z' = -x1 - x2 + 0x3 + 0x4 - Mx5 - Mx6
    2x1 + x2 - x3 + 0x4 + x5 + 0x6 = 4
    x1 + 7x2 + 0x3 - x4 + 0x5 + x6 = 7
    This M is a very large value, which is why we call it the Big M method.
  •                                 
             |  Cj  |  -1   |  -1    |   0  |   0  |  -M  |  -M  |
    CB | Xb  |  b   |  x1   |  x2    |  x3  |  x4  |  x5  |  x6  |
    ------------------------------------------------------------------------
    -M  | x5  |  4   |  2    |  1    |  -1  |  0   |  1   |  0   | (Row 1)
    -M  | x6  |  7   |  1    |  7    |  0   |  -1  |  0   |  1   | (Row 2)
    ------------------------------------------------------------------------
          Zj  |      | -3M   | -8M   |  M   |  M   |  -M  |  -M  |
          Cj  |      |  -1   |  -1   |  0   |  0   |  0   |  0   |
      Zj - Cj |      | -3M+1 | -8M+1 |  M   |  M   |  0   |  0   |
                                    
                                
    Since M is a large value, suppose it is 100, then the most negative value will be -8M+1. So in this column, x2 is the entering variable.
    Now, we will find the minimum ratio to identify the outgoing variable.
    Row 1 ratio of b/x2 = 4/1 = 4
    Row 2 ratio of b/x2 = 7/7 = 1 (this is the minimum ratio value).
    Thus, 7 becomes our key value, and we need to make it 1.
    x6 is the outgoing variable.
  • Remember, in the Big M method, we don't write the outgoing variable in the next table, so we will not write the x6 column.
    To make 7 into 1, we will perform the following operations:
    For row 2, divide row 2 by 7.
    For row 1, subtract the new row 2 from row 1.
                                    
             |  Cj   | -1        |  -1  |   0  |   0      |  -M  
    CB | Xb  |  b    |  x1       |  x2  |  x3  |  x4      |  x5  
    ---------------------------------------------------------------
    -M  | x5  |  3   | 13/7      |  0   |  -1  |  1/7     |  1   (Row 1)
    -1  | x2  |  1   |  1/7      |  1   |   0  | -1/7     |  0   (Row 2) new
    ---------------------------------------------------------------
          Zj  |      |(-13M-1)/7 |  -1  |  M    |(-M+1)/7 |  -M  
          Cj  |      |  -1       |  -1  |  0    |  0      |  0  
      Zj - Cj |      |(-13M-8)/7 |  0   |  M    |(-M+1)/7 |  0  
                                    
                                
    Here, (-13M-8)/7 is the most negative, which makes x1 the entering variable.
    Now, find the ratio of b and x1 to identify the outgoing variable.
    Row 1 ratio of b/x1 = 21/13
    Row 2 ratio of b/x1 = 7
    Thus, 21/13 is the smallest ratio, making x5 the outgoing variable, and 13/7 is the key value.
  • For row 1, we will multiply row 1 by 7/13.
    For row 2, we will perform the following operation: row 2 = row 2 - 1/7 * row 1 (new).
                                    
              |  Cj   |  -1  |  -1  |   0   |   0   
     CB | Xb  |  b    |  x1  |  x2  |  x3   |  x4  
    ---------------------------------------------------
    -1  | x1  | 21/13 |   1  |  0   | -7/13 | 1/13  (Row 1) new
    -1  | x2  | 10/13 |   0  |  1   |  1/13 | -2/13 (Row 2) 
    ----------------------------------------------------
          Zj  |       |  -1  |  -1  |  6/13 |  1/13  
          Cj  |       |  -1  |  -1  |  0    |  0    
      Zj - Cj |       |   0  |   0  |  6/13 |  1/13  
                                    
                                
    Now, as all Zj - Cj values are ≥ 0, we have found our optimal solution.
    x1 = 21/13
    x2 = 10/13
    Min Z = (21/13) + (10/13) = 31/13

Two Phase Method

The Two-Phase Method is used to solve Linear Programming Problems (LPP) when the problem involves artificial variables, typically in the case of equality constraints or greater-than-or-equal-to (≥) constraints. It ensures that a feasible solution is found even when such variables are introduced. The method is divided into two phases:

  1. Phase 1:
    • The goal is to minimize the sum of artificial variables, ensuring that they are driven to zero.
    • The original LPP is modified by adding artificial variables and creating an auxiliary objective function (minimizing the sum of these variables).
    • Solve the modified problem using the simplex method. If the optimal solution of the auxiliary problem has a value of zero for the artificial variables, it means a feasible solution is found.
  2. Phase 2:
    • After obtaining a feasible solution from Phase 1, the artificial variables are removed from the problem.
    • The original objective function is restored, and the feasible solution is used as the starting point for solving the original LPP using the simplex method.

Min Z = x1 + x2
Constraints:
2x1 + x2 ≥ 4
x1 + 7x2 ≥ 7
x1 ≥ 0

  • First, we have to convert it into a maximization function.
    For that, we will multiply it by -1.
    Max z' = -x1 - x2
    Since we have ≥, we need to subtract a surplus variable.
    2x1 + x2 - x3 + 0x4 = 4
    x1 + 7x2 + 0x3 - x4 = 7
    Now, the problem is that we don't have a unit matrix, so we add 2 more variables called artificial variables.
    Max z' = -x1 - x2 + 0x3 + 0x4 - x5 - x6
    Phase 1:
    for phase 1 our max z' function becomes 0x1 + 0x2 + 0x3 + 0x4 - x5 - x6
    Max z' = -x5 - x6
    Following are the constraints
    2x1 + x2 - x3 + 0x4 + x5 + 0x6 = 4
    x1 + 7x2 + 0x3 - x4 + 0x5 + x6 = 7
  •                                 
             |  Cj  |   0   |   0    |   0  |   0  |  -1  |  -1  |
    CB  | Xb |  b   |  x1   |  x2    |  x3  |  x4  |  x5  |  x6  |
    ------------------------------------------------------------------------
    -1  | x5  |  4   |  2   |  1    |  -1  |  0   |  1   |  0   | (Row 1)
    -1  | x6  |  7   |  1   |  7    |  0   |  -1  |  0   |  1   | (Row 2)
    ------------------------------------------------------------------------
          Zj  |      | -3   |  -8   |  1   |  1   |  -1  |  -1  |
          Cj  |      |  0   |   0   |  0   |  0   | -1   |  -1  |
      Zj - Cj |      | -3   | -8    |  1   |  1   |  0   |  0   |
                                    
                                
    The most negative value will be -8. So in this column, x2 is the entering variable.
    Now, we will find the minimum ratio to identify the outgoing variable.
    Row 1 ratio of b/x2 = 4/1 = 4
    Row 2 ratio of b/x2 = 7/7 = 1 (this is the minimum ratio value).
    Thus, 7 becomes our key value, and we need to make it 1.
    x6 is the outgoing variable.
  • Remember, in the 2 phase method also, we don't write the outgoing variable in the next table, so we will not write the x6 column.
    To make 7 into 1, we will perform the following operations:
    For row 2, divide row 2 by 7.
    For row 1, subtract the new row 2 from row 1.
                                    
             |  Cj   |  0        |   0  |   0  |   0      |  -1  
    CB | Xb  |  b    |  x1       |  x2  |  x3  |  x4      |  x5  
    ---------------------------------------------------------------
    -1  | x5  |  3   | 13/7      |  0   |  -1  |  1/7     |  1   (Row 1)
     0  | x2  |  1   |  1/7      |  1   |   0  | -1/7     |  0   (Row 2) new
    ---------------------------------------------------------------
          Zj  |      | -13/7     |  -1  |  1   | -1/7     |  -1  
          Cj  |      |   0       |   0  |  0   |    0     |  -1  
      Zj - Cj |      | -13/7     |  -1  |  1   | -1/7     |   0  
                                    
                                
    Here, -13/7 is the most negative, which makes x1 the entering variable.
    Now, find the ratio of b and x1 to identify the outgoing variable.
    Row 1 ratio of b/x1 = 21/13
    Row 2 ratio of b/x1 = 7
    Thus, 21/13 is the smallest ratio, making x5 the outgoing variable, and 13/7 is the key value.
  • For row 1, we will multiply row 1 by 7/13.
    For row 2, we will perform the following operation: row 2 = row 2 - 1/7 * row 1 (new).
                                    
              |  Cj  |   0  |   0    |   0     |   0   
     CB | Xb  |  b   |  x1  |  x2    |  x3     |  x4  
    ---------------------------------------------------
     0  | x1  | 21/13 |   1  |  0    | -7/13   | 1/13  (Row 1) new
     0  | x2  | 10/13 |   0  |  1    |  1/13   | -2/13 (Row 2) 
    ----------------------------------------------------
          Zj  |       |   0  |  0    |    0   |  0  
          Cj  |       |   0  |   0   |  0     |  0    
      Zj - Cj |       |   0  |   0   |  0     |  0 
                                    
                                
    Now, as all Zj - Cj values are ≥ 0, so our phase 1 is ended
  • Phase 2:
    now for phase 2 our objective function becomes:
    Max z' = -x1 - x2 + 0x3 + 0x4
  •                             
              |  Cj  |  -1  |  -1    |   0     |   0   
     CB | Xb  |  b   |  x1  |  x2    |  x3     |  x4  
    ---------------------------------------------------
    -1  | x1  | 21/13 |   1  |  0    | -7/13   | 1/13  (Row 1) new
    -1  | x2  | 10/13 |   0  |  1    |  1/13   | -2/13 (Row 2) 
    ---------------------------------------------------- 
      Zj - Cj |       |   0  |   0   | 6/13    |  1/13
                                
                            
    Now, as all Zj-Cj values are >= 0, so we have found our optimal solution
    x1 = 21/13 and x2 = 10/13
    min z = 31/13

Transportation Problem

Methods of finding the inital feasible solution

  1. Nort-west corner rule
  2. Least cost method
  3. Vogel's Approximation Method

Basic Feasible Solution (BFS):

  • The Transportation Problem is a special case of Linear Programming Problems (LPP), so the definition of a Basic Feasible Solution (BFS) is similar to that in LPP.
  • In the case of the Transportation Problem, there are only (m + n - 1) basic variables out of a total of m * n variables. Here, m represents the number of rows (suppliers) and n represents the number of columns (destinations) in the transportation problem. Therefore, the BFS of a transportation problem consists of at most (m + n - 1) variables, with the remaining variables being zero.

Example Problem - Find the Initial BFS (Basic Feasible Solution) of the Following Transportation Problem:

                    
                                        |  Warehouse
    Factory |  w1  |  w2  |  w3  |  w4  | Capacity
----------------------------------------------------
        F1  |  19  |  30  |  50  |  10  |   7
        F2  |  70  |  30  |  40  |  60  |   9
        F3  |  40  |   8  |  70  |  20  |  18
----------------------------------------------------
Warehouse   |   5  |   8  |   7  |  14  |
Requirement |
                    
                

North-West Corner Rule

  • We will solve this using the North-West Corner Rule.
  • First, we check if the supply and demand are balanced. This means the total supply should equal the total demand. In this case, the total capacity (7 + 9 + 18 = 34) is equal to the total requirement (5 + 8 + 7 + 14 = 34), so the problem is balanced.
  • In the North-West Corner method, we start at the top-left (north-west) corner of the cost matrix and assign the supply or demand, whichever is smaller, to that cell. We then subtract the smaller value from the larger one and move to the next cell, following the north-west direction until all supplies and demands are met.
  • Start with top-left and assign smallest supply or demand.
                                
        Factory |  w1      |  w2  |  w3  |  w4  | Capacity
    ----------------------------------------------------
            F1  |  19 (5)  |  30  |  50  |  10  |   7-5=2
            F2  |  70      |  30  |  40  |  60  |   9
            F3  |  40      |   8  |  70  |  20  |  18
    ----------------------------------------------------
    Requirement | 5-5=0    |   8  |   7  |  14  |
                                
                            
    now we will remove the column 1
    again we assign
                                
        Factory |  w2    |  w3  |  w4  | Capacity
    ----------------------------------------------------
            F1  |  30(2) |  50  |  10  |   2-2=0
            F2  |  30    |  40  |  60  |   9
            F3  |   8    |  70  |  20  |  18
    ----------------------------------------------------
    Requirement | 8-2=6  |   7  |  14  |
                                
                            
    now we remove row 1
                                
        Factory |  w2     |  w3  |  w4  | Capacity
    ----------------------------------------------------
            F2  |  30(6)  |  40  |  60  |   9-6=3
            F3  |   8     |  70  |  20  |  18
    ----------------------------------------------------
    Requirement |  6-6=0  |   7  |  14  |
                                
                            
    now we remove column 1
                                
        Factory |  w3     |  w4  | Capacity
    --------------------------------------------
            F2  |  40(3)  |  60  |   3-3=0
            F3  |  70     |  20  |  18
    --------------------------------------------
    Requirement |  7-3=4  |  14  |
                                
                            
    now we remove row 1
    Now only 2 values are left so we can directly assign.
                                
        Factory |  w3     |  w4      | Capacity
    -----------------------------------------------
            F3  |  70(4)  |  20(14)  |  18-4-14=0
    -----------------------------------------------
    Requirement |   4-4=0 |  14-14=0 |
                                
                            
    so finaly we have:
                                
        Factory |  w1      |    w2   |     w3  |     w4   | Capacity
    ----------------------------------------------------------------
            F1  |  19 (5)  |  30(2)  |  50     |  10      |   7
            F2  |  70      |  30(6)  |  40(3)  |  60      |   9
            F3  |  40      |   8     |  70(4)  |  20(14)  |  18
    ----------------------------------------------------------------
    Requirement |   5      |   8  |   7  |  14  |
                                
                            
    Now we will find its transportation cost:
    (19 * 5) + (30 * 2) + (30 * 6) + (40 * 3) + (70 * 4) + (20 * 14)
    = 1015

Least Cost Method

Solving the Same Question Using the Least Cost Method

  • In this method, we find the minimum cost and assign the minimum value from either the supply or demand.
    Here, the minimum cost is 8, so we allocate accordingly.
                
     Factory |  w1  |  w2     |  w3  |  w4  |   D
    -------------------------------------------------
         F1  |  19  |  30     |  50  |  10  |   7
         F2  |  70  |  30     |  40  |  60  |   9
         F3  |  40  |   8(8)  |  70  |  20  |  18-8=10
    -------------------------------------------------
          S  |   5  |   8-8=0 |   7  |  14  |  34
                
            
    Now, we remove the w2 column.
    Then, we assign the next smallest cost.
                
     Factory |  w1  |  w3  |  w4     |   D
    -------------------------------------------------
         F1  |  19  |  50  |  10(7)  |   7-7=0
         F2  |  70  |  40  |  60     |   9
         F3  |  40  |  70  |  20     |  10
    -------------------------------------------------
          S  |   5  |   7  |  14-7=7 |  34
                
            
    Now, we remove row 1.
    Then, we assign the next smallest cost.
                
     Factory |  w1  |  w3  |  w4     |   D
    -----------------------------------------
         F2  |  70  |  40  |  60     |   9
         F3  |  40  |  70  |  20(7)  |  10-7=3
    -----------------------------------------
          S  |   5  |   7  |   7-7=0 |  34
                
            
    Now, we remove the w4 column.
    Since we have two minimum values, we can allocate to either one of them.
                
     Factory |  w1  |  w3     |  D
    -------------------------------
         F2  |  70  |  40(7)  |  9-7=2
         F3  |  40  |  70     |  3
    -------------------------------
          S  |   5  |   7-7=0 |  
                
            
    Now, we remove the w3 column.
                
     Factory |  w1     |  D
    -----------------------
         F2  |  70(2)  |  2-2=0
         F3  |  40(3)  |  3-3=0
    -----------------------
          S  | 5-2-3=0 |  
                
            
    So finally, we have:
                                
     Factory |  w1     |  w2    |  w3    |  w4  
    ----------------------------------------------
         F1  |  19     |  30    |  50    |  10(7)  
         F2  |  70(2)  |  30    |  40(7) |  60  
         F3  |  40(3)  |  8(8)  |  70    |  20(7)  
                                
                            
    The number of allocations should be m + n - 1:
    3 + 4 - 1 = 6
    And we also have 6 allocations, so we did it correctly.
    Now, we will find the transportation cost using the Least Cost Method:
    (10 * 7) + (70 * 2) + (40 * 7) + (40 * 3) + (8 * 8) + (20 * 7)
    70 + 140 + 280 + 120 + 64 + 140 = 814
    So, we know that using the North-West Corner Rule, the transportation cost was 1015, and using the Least Cost Method, the cost is 814. When we solve this same question using Vogel's Approximation Method, the cost will decrease further, making it the best method for solving transportation problems.

Vogel's Approximation Method

Again Solving this above question using Vogel's Approximation Method

                    
 Factory |  w1  |  w2  |  w3  |  w4  |   D
---------------------------------------------
     F1  |  19  |  30  |  50  |  10  |   7
     F2  |  70  |  30  |  40  |  60  |   9
     F3  |  40  |   8  |  70  |  20  |  18
---------------------------------------------
      S  |   5  |   8  |   7  |  14  |  34
                    
                
  • Here we first mention the penalties for each row and column.
    We find 2 smallest cost in a row or column and write down there difference.
    Then we look for biggest penalty and look for smallest cost in that row or column and simply assign.
                                
     Factory |  w1  |  w2   |  w3  |  w4  |   D     |  P
    ---------------------------------------------------------
         F1  |  19  |  30   |  50  |  10  |   7     |  (9)
         F2  |  70  |  30   |  40  |  60  |   9     |  (10)
         F3  |  40  | 8(8)  |  70  |  20  | 18-8=10 |  (12)
    ---------------------------------------------------------
          S  |   5  | 8-8=0 |   7  |  14  |  
          P  | (21) | (22)  | (10) | (10) |  
                        ↓
                 Biggest Penalty
                                
                            
    Now we will remove w2 column
                                
     Factory |  w1   |  w3  |  w4  |   D    |  P
    ------------------------------------------------------
         F1  | 19(5) |  50  |  10  | 7-5=2  |  (9)
         F2  |  70   |  40  |  60  |   9    |  (20)
         F3  |  40   |  70  |  20  |  10    |  (20)
    ------------------------------------------------------
          S  | 5-5=0 |   7  |  14  |  
          P  | (21)  | (10) | (10) |  
                 ↓
          Biggest Penalty
                                
                            
    Now we remove the w1 column
                                
     Factory |  w3  |  w4    |  D      |  P
    -------------------------------------------
         F1  |  50  |  10    |  2      |  (40)
         F2  |  40  |  60    |  9      |  (20)
         F3  |  70  | 20(10) | 10-10=0 |  (50) → Biggest Penalty
    -------------------------------------------
          S  |   7  | 14-10=4|  
          P  | (10) | (10)   |  
                                
                            
    Now we remove row 3
                                
     Factory |  w3  |  w4   |  D    |  P
    -------------------------------------------
         F1  |  50  | 10(2) | 2-2=0 |  (40)
         F2  |  40  |  60   |  9    |  (20)
    -------------------------------------------
          S  |   7  | 4-2=2 |  
          P  | (10) | (50)  |  
                       ↓
                Biggest Penalty
                                
                            
    Now we will remove row 1
                                
     Factory |  w3   |  w4   |  D      |  P
    -------------------------------------------
         F2  | 40(7) | 60(2) | 9-7-2=0 |  (20)
    -------------------------------------------
          S  | 7-7=0 | 2-2=0 |
                                
                            
    Finally we have:
                                
     Factory |  w1   |  w2  |  w3   |  w4  
    -------------------------------------
         F1  | 19(5) |  30  |  50   |  10(2)  
         F2  |  70   |  30  | 40(7) |  60(2)  
         F3  |  40   | 8(8) |  70   |  20(10)  
                                
                            
    Transportation cost:
    (19 * 5) + (10 * 2) + (40 * 7) + (60 * 2) + (8 * 8) + (20 * 10)
    =779

What if supply and demand are not equal??

Example Problem - Find the Initial BFS (Basic Feasible Solution) of the Following Transportation Problem:

    
            |  w1  |  w2  |  w3  | Capacity
------------------------------------------------------------
        F1  |  10  |  0   |  0   |   10
        F2  |  0   |  14  |  1   |   15
        F3  |  2   |  0   |  9   |   20
------------------------------------------------------------
Warehouse   |  12  |  14  |  10  |
Requirement |
    
  • Checking if the supply and demand are balanced. This means the total supply should equal the total demand. In this case:
    • Total supply = 10 (F1) + 15 (F2) + 20 (F3) = 45
    • Total demand = 12 (w1) + 14 (w2) + 10 (w3) = 36
    Since the total supply (45) is greater than the total demand (36), the problem is unbalanced.
  • When the supply is greater than the demand, we need to add a dummy warehouse with a demand equal to the difference between the total supply and total demand. In this case, the difference is 9 (45 - 36 = 9), so we add a dummy warehouse (w4) with a demand of 9.
  • The new transportation table will include this dummy warehouse (w4) with a demand of 9, and we add a column of 0s for this dummy warehouse (w4) since it's not actually supplying any products. The table now looks like this:
    
            |  w1  |  w2  |  w3  |  w4  | Capacity
------------------------------------------------------------
        F1  |  10  |  0   |  0   |  0   |   10
        F2  |  0   |  14  |  1   |  0   |   15
        F3  |  2   |  0   |  9   |  0   |   20
------------------------------------------------------------
Warehouse   |  12  |  14  |  10  |  9   |
Requirement |
    
  • Now the total supply (45) and total demand (45) are balanced, and we can proceed to solve the problem using any method.

Optimality Test

This test helps in determining whether the current solution to an optimization problem is optimal. It evaluates if any further improvements can be made to the solution. If the solution satisfies the optimality criteria, it is declared the best possible solution; otherwise, adjustments are made to improve the results, and the process is repeated.

  • First, we have to find the total transportation cost then we can apply optimality test.
  • Finding transportation cost for the following problem using Vogel's Approximation Method:
                                
        |  D1 |  D2 |  D3 |  D4 | Avail
    -------------------------------------
     O1 |  1  |  2  |  1  |  4  |  30
     O2 |  3  |  3  |  2  |  1  |  50
     O3 |  4  |  2  |  5  |  9  |  20
    -------------------------------------
    Req | 20  | 40  | 30  | 10  | 100
                                
                            
    Writing penalities
    Note: if we have smallest cost which are same like 2, 2 then we don't subtract them instead we find another just bigger then them.
                                
        |  D1 |  D2 |  D3 |  D4    | Avail    |  P
    ----------------------------------------------------
     O1 |  1  |  2  |  1  |  4     |  30      |  (1)
     O2 |  3  |  3  |  2  | 1(10)  | 50-10=40 |  (1)
     O3 |  4  |  2  |  5  |  9     |  20      |  (2)
    ---------------------------------------------------
    Req | 20  | 40  | 30  | 10-10=0   
      P | (2) | (1) | (1) | (3)  
                             ↓
                      Biggest Penalty
                                
                            
    Remove D4 column
                                
        |  D1     |  D2 |  D3 |   Avail  |  P
    -----------------------------------------------
     O1 | 1(20)   |  2  |  1  | 30-20=10 |  (1)
     O2 |  3      |  3  |  2  |  40      |  (1)
     O3 |  4      |  2  |  5  |  20      |  (2)
    -----------------------------------------------
    Req | 20-20=0 | 40  | 30  |    
      P |  (2)    | (1) | (1) |   
            ↓
     Biggest Penalty
                                
                            
    Remove D1 column
                                
        |  D2      |  D3 |  Avail  |  P
    ----------------------------------------------------
     O1 |  2       |  1  |  10     | (1)
     O2 |  3       |  2  |  40     | (1)
     O3 | 2(20)    |  5  | 20-20=0 | (2)  → Biggest Penalty
    ---------------------------------------------------
    Req | 40-20=20 | 30  |    
      P |   (1)    | (1) |
                                
                            
    Remove row 3
    Now as we have same penalties then we can take the row or column with smallest supply or demand
                                
        |  D2  |  D3      |  Avail  |  P
    ------------------------------------------
     O1 |  2   | 1(10)    | 10-10=0 | (1)
     O2 |  3   |  2       |  40     | (1)
    ------------------------------------------
    Req |  20  | 30-10=20 |    
      P | (1)  | (1)      |
                                
                            
    remove row 1
                                
        |  D2     |  D3     | Avail      |  P
    ------------------------------------------
     O2 | 3(20)   | 2(20)   | 40-20-20=0 | (1)
    ------------------------------------------
    Req | 20-20=0 | 20-20=0 |    
      P | (1)   | (1)   
                                
                            
    Finally we have
                                
        |    D1   |    D2   |    D3   |  D4 
    -------------------------------------------
     O1 |  1(20)  |  2      |  1(10)  |  4  
     O2 |  3      |  3(20)  |  2(20)  |  1(10) 
     O3 |  4      |  2(20)  |  5      |  9  
                                
                            
    Total transportation cost = 20 + 10 + 60 + 40 + 10 + 40 = 180
  • Now we have found transportation cost now we will perform the optimality test using MODI (Modification) method:
  • For allocated cell Cij = ui + vj
                                
        |    D1   |    D2   |    D3   |   D4   |  
    -------------------------------------------------
     O1 |  1(20)  |    .    |  1(10)  |   .    |  u1 = -1
     O2 |    .    |  3(20)  |  2(20)  |  1(10) |  u2 = 0
     O3 |    .    |  2(20)  |   .     |   .    |  u3 = -1
     -------------------------------------------
             v1       v2        v3        v4
              2        3         2         1
                                
                            
    We put vi or ui as zero where there are most allocated cost in a row or column. so in the above table at row 3 we have the most allocated cost so we will put u2 = 0
    Now for non allocated cell cij - (ui + vj) >= 0
                                
        |    D1   |    D2   |    D3   |  D4   |
    -------------------------------------------
     O1 |    .    |   2(2)  |    .    |  4(0) |  u1 = -1
     O2 |  3(2)   |   .     |    .    |   .   |  u2 = 0
     O3 |  4(1)   |   .     |  5(1)   |  9(0) |  u3 = -1
     ------------------------------------------ 
           v1        v2         v3       v4
            2        3          2        1
                                
                            
    Now here for all cij - (ui + vj) the difference is >= 0
    so transportation cost 180 is optimal. which means we can't decrease it cost less than 180.

Assignment Problem

Application of Assignment Problem:

  1. To minimize the total cost/time required to complete task.
  2. To assign a best job to best person.
  3. To assign vehicles to the routes.
  4. To assign sales representative to the sales teritories.
  5. In minimizing the time of arrival and departure of airlines.

Example problem:
Q- Solve the following assignment problem using the Hungarian Method:

    
    | J1 | J2 | J3 |
--------------------
 P1 | 10 | 20 | 30 |  
 P2 | 20 | 10 | 40 | 
 P3 | 50 | 30 | 20 |
--------------------  
    
  • First, we perform row reduction, followed by column reduction, and then we allocate the tasks.
    Row reduction: We find the smallest cost in each row and subtract it from every cost in that particular row.
                
        | J1 | J2 | J3 |
    --------------------
     P1 | 10 | 20 | 30 |  row 1's minimum cost = 10 
     P2 | 20 | 10 | 40 |  row 2's minimum cost = 10 
     P3 | 50 | 30 | 20 |  row 3's minimum cost = 20
    --------------------
    
        |    J1    |    J2    |    J3    |
    --------------------------------------
     P1 | 10-10=0  | 20-10=10 | 30-10=20 |  
     P2 | 20-10=10 | 10-10=0  | 40-10=30 | 
     P3 | 50-20=30 | 30-20=10 | 20-20=0  |
    ---------------------------------------
    
        | J1 | J2 | J3 |
    --------------------
     P1 | 0  | 10 | 20 |  
     P2 | 10 | 0  | 30 | 
     P3 | 30 | 10 | 0  |
    --------------------
                
            
  • Next, we perform column reduction. Since the minimum value in each column is 0, the matrix remains unchanged.
                
        | J1 | J2 | J3 |
    --------------------
     P1 | 0  | 10 | 20 |  
     P2 | 10 | 0  | 30 | 
     P3 | 30 | 10 | 0  |
    --------------------
                
            
  • Now we proceed with the allocation:
    Wherever there is a 0, that cost will be assigned to the corresponding job-person pair.
                
                         | Cost |
    -----------------------------
                   P1-J1 |  10  |  
                   P2-J2 |  10  | 
                   P3-J3 |  20  |
    -----------------------------
    Total Assignment Cost: 40
                
            

Another Example:

Q- Solve the following problem using the Hungarian Assignment Method:

    
    | J1 | J2 | J3 | J4 |
----+----+----+----+----+
 P1 | 12 | 30 | 21 | 15 |
 P2 | 18 | 33 |  9 | 31 |
 P3 | 44 | 25 | 24 | 21 |
 P4 | 23 | 30 | 28 | 14 |
----+----+----+----+----+
    
  • Row reduction:
                
        |    J1    |    J2    |    J3    |    J4    |
    ----+----------+----------+----------+----------+
     P1 | 12-12=0  | 30-12=18 | 21-12=9  | 15-12=3  |
     P2 | 18-9=9   | 33-9=24  |  9-9=0   | 31-9=22  |
     P3 | 44-21=23 | 25-21=4  | 24-21=3  | 21-21=0  |
     P4 | 23-14=9  | 30-14=16 | 28-14=14 | 14-14=0  |
    ----+----------+----------+----------+----------+
    
    It becomes:
    
        | J1 | J2 | J3 | J4 |
    ----+----+----+----+----+
     P1 |  0 | 18 |  9 |  3 |
     P2 |  9 | 24 |  0 | 22 |
     P3 | 23 |  4 |  3 |  0 |
     P4 |  9 | 16 | 14 |  0 |
    ----+----+----+----+----+
                
            
  • Column reduction:
                
        | J1 |    J2   | J3 | J4 |
    ----+----+---------+----+----+
     P1 |  0 | 18-4=14 |  9 |  3 |
     P2 |  9 | 24-4=20 |  0 | 22 |
     P3 | 23 |  4-4=0  |  3 |  0 |
     P4 |  9 | 16-4=12 | 14 |  0 |
    ----+----+----+----+---------+
    
    It becomes:
    
        | J1 | J2 | J3 | J4 |
    ----+----+----+----+----+
     P1 |  0 | 14 |  9 |  3 |
     P2 |  9 | 20 |  0 | 22 |
     P3 | 23 |  0 |  3 |  0 |
     P4 |  9 | 12 | 14 |  0 |
    ----+----+----+----+----+
                
            
  • Assignment:
    Select the rows where 0 appears only once.
    Once a job is assigned to a person, it cannot be reassigned, so the remaining 0s will be assigned to the unassigned person.
                
        |   J1 |   J2 |  J3  |  J4   |
    ----+------+------+------+-------+
     P1 |  0 A | 14   |  9   |    3  |
     P2 |  9   | 20   |  0 A |  22   |
     P3 | 23   |  0 A |  3   |  0 C |
     P4 |  9   |  12  | 14   |  0 A  |
    ----+------+------+------+-------+
    A - assigned 
    C - can't assign
                
            
                
              | Cost |
    ----------+-------
        P1-J1 |  12
        P2-J3 |   9
        P3-J2 |  25
        P4-J4 |  14
    ----------+-------
      Total Assignment Cost: 60 /-
                
            

Q- Four professors are each capable of teaching any one of the four different subects. Class preparation time (in hrs) for different topics varies from professor to professor and is given in the table. Each professor should be assigned only one subject. Find the schedule so as to minimize the total subject preparation time for all subjects/professor.

                    
    | S1 | S2 | S3 | S4 |
----+----+----+----+----+
 P1 |  2 | 10 |  9 |  7 |
 P2 | 15 |  4 | 14 |  8 |
 P3 | 13 | 14 | 16 | 11 |
 P4 |  3 | 15 | 13 |  8 |
----+----+----+----+----+
                    
                

Unbalanced Assignment Problem

In an unbalanced assignment problem, the number of rows (tasks) is not equal to the number of columns (jobs). This happens when the number of tasks to be assigned differs from the number of resources or workers available. To handle such cases, we convert the unbalanced problem into a balanced one by adding dummy rows or columns. These dummy rows or columns are introduced with a cost of zero for all their elements, ensuring that the problem becomes balanced while keeping the integrity of the assignment process intact. This way, each task can still be assigned to a worker, or a job can be assigned to a task, without any imbalance. For example:

    
    Original Matrix (Unbalanced):
       | J1 | J2 | J3 |
-------+----+----+----+
    P1 | 15 | 20 | 35 |  
    P2 | 25 | 30 | 25 |  
-------+----+----+----+

    After adding a dummy row to balance the matrix:

    | J1 | J2 | J3 |
----+----+----+----+
 P1 | 15 | 20 | 35 |  
 P2 | 25 | 30 | 25 |  
 D1 |  0 |  0 |  0 |  
----+----+----+----+

    Now the problem is balanced, and you can apply the usual steps of row reduction, column reduction, and assignment using the Hungarian Method.
    

Maximization Problem:

In assignment problems, the objective is usually to minimize the total cost. However, in some cases, the goal is to maximize the total profit or benefit. These are called maximization problems. The Hungarian method, which is typically used for minimization problems, can also be applied to maximization problems with a slight modification. To solve a maximization problem, we convert it into a minimization problem. This is done by subtracting each element of the given matrix from the highest element in the matrix. This transformation effectively inverts the problem, allowing us to use the same Hungarian Method steps of row reduction, column reduction, and allocation for maximization problems. For example:

                    
Original Profit Matrix (Maximization):
   | J1 | J2 | J3 |
---+----+----+----+
P1 | 50 | 60 | 40 |  
P2 | 70 | 80 | 60 |  
P3 | 65 | 55 | 45 |  
---+----+----+----+

Step 1: Find the maximum value in the matrix (which is 80 here) and subtract each element from this value:

| J1     |   J2     |   J3     |
---------+----------+----------+
80-50=30 | 80-60=20 | 80-40=40 |
80-70=10 | 80-80=0  | 80-60=20 |
80-65=15 | 80-55=25 | 80-45=35 |
---------+----------+----------+

Resulting Cost Matrix (for minimization):
     | J1 | J2 | J3 |
-----+----+----+----+
 P1  | 30 | 20 | 40 |  
 P2  | 10 |  0 | 20 |  
 P3  | 15 | 25 | 35 |  
-----+----+----+----+
                    
                

Now, you can apply the usual steps of the Hungarian Method to this transformed matrix to find the optimal assignment.

Travelling Salesman Problem

Conditions:

  1. No place is to be visited twice before the tour of all places is completed. This means that each location must be visited exactly once during the tour, ensuring a complete path through all designated points without repetition.
  2. Going from one place to itself is not allowed. This condition emphasizes that the salesman must travel to different locations, and returning to the starting point without visiting other places is not an option during the tour.

Application of TSP:

  • Postal Deliveries: In postal delivery systems, TSP can be used to optimize the routes that delivery vehicles take to minimize travel distance and time while ensuring all delivery points are reached.
  • Inspection: TSP is applicable in scenarios where inspections are required at multiple sites (e.g., infrastructure inspections), allowing inspectors to plan efficient routes that cover all sites with minimal travel.
  • Cable Connection: In telecommunications and networking, TSP can help design optimal routes for laying cables between multiple points, ensuring the least distance is covered, which can reduce costs and time.
  • School Bus Routes: TSP can be utilized to determine the most efficient routes for school buses, ensuring that all student pickups and drop-offs are conducted with minimal travel time and distance.

Q- Solve the TSP given by:

    
     |  A  |  B  |  C  |  D  |  E
-----------------------------------                        
  A  |  ∞  |  2  |  5  |  7  |  1
  B  |  6  |  ∞  |  3  |  8  |  2
  C  |  8  |  7  |  ∞  |  4  |  7
  D  | 12  |  4  |  6  |  ∞  |  5
  E  |  1  |  3  |  2  |  8  |  ∞
    
  • ∞ is there to show that we can't go from A to A, B to B, and so on.
    Note: In the Traveling Salesman Problem (TSP), you must visit all places exactly once, and the tour must return to the starting point. The path cannot repeat the same place until the tour is complete.
  • Row Reduction:
    We will perform row reduction by subtracting the smallest value in each row from all elements in that row:
                
         |  A  |  B  |  C  |  D  |  E
    -----------------------------------                        
      A  |  ∞  |  1  |  4  |  6  |  0
      B  |  4  |  ∞  |  1  |  6  |  0
      C  |  4  |  3  |  ∞  |  0  |  3
      D  |  8  |  0  |  2  |  ∞  |  1
      E  |  0  |  2  |  1  |  7  |  ∞
                
            
  • Column Reduction:
    We now perform column reduction by subtracting the smallest value in each column from all elements in that column:
                
         |  A   |  B  |  C   |  D   |  E
    -----------------------------------                        
      A  |  ∞   |  1  |  3   |  6   |  0A
      B  |  4   |  ∞  |  0A  |  6   |  0C
      C  |  4   |  3  |  ∞   |  0A  |  3
      D  |  8   |  0A |  1   |  ∞   |  1
      E  |  0A  |  2  |  0C  |  7   |  ∞
                
            
    Now, we try to assign paths, but there will be no complete cycle. This means the current solution is not optimal.
  • Assigning Greater Than Zero:
    We now look for the next smallest value greater than zero, which is 1, and attempt to make assignments:
                
         |  A  |  B  |  C   |  D  |  E
    -----------------------------------                        
      A  |  ∞  |  1A  |  3  |  6  |  0C
      B  |  4  |  ∞   |  0C |  6  |  0A
      C  |  4  |  3   |  ∞  |  0A |  3
      D  |  8  |  0C  |  1A |  ∞  |  1C
      E  |  0A |  2   |  0C |  7  |  ∞
                
            
    There is still no complete cycle, so we adjust the assignments.
  • Adjusting Assignments:
    We adjust the assignments further to ensure a complete cycle is formed:
                
         |  A  |  B  |  C  |  D  |  E
    -----------------------------------                        
      A  |  ∞  |  1A |  3  |  6  |  0C
      B  |  4  |  ∞  |  0A |  6  |  0C
      C  |  4  |  3  |  ∞  |  0A |  3
      D  |  8  |  0C |  1C |  ∞  |  1A
      E  |  0A |  2  |  0C |  7  |  ∞
                
            
    The cycle is now complete: A - B - C - D - E - A, which forms the optimal solution.
  • Calculating the Total Cost:
    The total cost of the optimal tour is the sum of the selected paths:
    • From A to B: 2
    • From B to C: 3
    • From C to D: 4
    • From D to E: 5
    • From E to A: 1
    So, the total TSP cost = 2 + 3 + 4 + 5 + 1 = 15.

Reference