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Simplex Algorithm

Simplex Algorithm. Solving linear programming problems algebraically. Initial Example : Maximise P = 10x + 12y subject to the constraints: x + y ≤ 40 (i) x + 2y ≤ 75 (ii) x ≥ 0, y ≥ 0.

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Simplex Algorithm

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  1. Simplex Algorithm Solving linear programming problems algebraically

  2. Initial Example: Maximise P = 10x + 12y subject to the constraints: x + y ≤ 40 (i) x + 2y ≤ 75 (ii) x ≥ 0, y ≥ 0. Step 1: Introduce slack variables to convert the non-trivial inequalities into equalities: Equation (i): x + y + s = 40 s ≥ 0 Equation (ii): x + 2y + t = 75 t ≥ 0 s, t are slack variables.

  3. Step 2: Rewrite the objective function so that the RHS is a number: P = 10x + 12y → P – 10x – 12y = 0. Step 3: Write the objective function and the non-trivial constraints in tableau format:

  4. Tableau Format Last column P-10x-12y=0 Pivotal column x+y+s=40 x+2y+t=75 The aim is to solve the equations by combining rows together. The solution is reached when all entries in the first row (except possibly the value in the last column) are non-negative. The shaded cells should be non-negative We begin by identifying the most negative entry in the objective function row, here -12 in the y column.

  5. We highlight the pivotal column. We then divide every entry in the l column by the corresponding value in the highlighted column. Pick the least positive of these. This is the pivotal row. Pivotal column pivot

  6. Equation Ratio P x y s t l 450/ - 4= - 112.5 (4)=(1)+12×(6) 1 - 4 0 0 6 450 2.5/0.5 = 5 0 0 0 - ½ 2.5 (5) = (2) – (6) ½ 37.5/0.5 = 75 (6) = (3)/2 0 ½ 1 0 ½ 37.5 Divide the pivotal row by the pivot value. The aim is to now get 0 entries elsewhere in the pivotal column. We now repeat the process, first selecting the new pivotal column, i.e. the one with the most negative value in the objective function row.

  7. Having identified the pivotal row and the pivot value, we now divide every entry in the pivotal row by the pivot value. The process is now finished as every entry on the objective function row is non-negative.

  8. The values of x, y and P can be read from the table: x = 5, y = 35, P = 470. This is the optimal solution.

  9. Interpretation

  10. Examination Question A clocksmith makes 3 types of luxury watches. The mechanism for each watch is assembled by hand by a skilled watchmaker and then the complete watch is formed, weatherproofed and packaged for sale by a fitter. The table below shows the times (in mins) for each stage of the process. It also gives the profits to be made on each watch. The watchmaker works for a maximum of 30 hours per week; the fitter for 25 hours per week. Let x, y, z represent the number of type A, B, C watches to be made (respectively).

  11. Setting up the problem Profit function: P = Constraint 1: Constraint 2:

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