1. Linear Programming I: Simplex method
Linear programming is an optimization method applicable for the solution of problems in which the objective function and the constraints appear as linear functions of the decision variables.
Simplex method is the most efficient and popular method for solving general linear programming problems.
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2. Linear Programming I: Simplex method-Applications
Petroleum refineries
choice of buying crude oil from several different sources with differing compositions and at differing prices
manufacturing different products such as aviation fuel, diesel fuel, and gasoline, in varying quantities
Constraints due to the restrictions on the quantity of the crude oil from a particular source, the capacity of the refinery to produce a particular product
A mix of the purchased crude oil and the manufactured products is sought that gives the maximum profit
Optimal production plan in a manufacturing firm
Pay overtime rates to achieve higher production during periods of higher demand
The routing of aircraft and ships can also be decided using LP
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3. Standard Form of a Linear Programming Problem
Scalar form
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4. Standard Form of a Linear Programming Problem
Matrix form
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5. Characteristic of a Linear Programming Problem
The objective function is of the minimization type
All the constraints are of the equality type
All the decision variables are nonnegative
The number of the variables in the problem is n. This includes the slack and surplus variables.
The number of constraints is m (m < n).
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6. Characteristic of a Linear Programming Problem
The number of basic variables is m (same as the number of constraints).
The number of nonbasic variables is n-m.
The column of the right hand side b is positive and greater than or equal to zero.
The calculations are organized in a table.
Only the values of the coefficients are necessary for the calculations. The table therefore contains only coefficient values, the matrix A in previous discussions. These are the coefficients in the constraint equations.
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7. Characteristic of a Linear Programming Problem
The objective function is the last row in the table. The constraint coefficients are written first.
Row operations consist of adding (subtracting)a definite multiple of the pivot row from other rows of the table.
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8. Transformation of LP Problems into Standard Form
The maximization of a function f(x1,x2,…,xn ) is equivalent to the minimization of the negative of the same function. For example, the objective function
Consequently, the objective function can be stated in the minimization form in any linear programming problem.
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9. Transformation of LP Problems into Standard Form
A variable may be unrestricted in sign in some problems. In such cases, an unrestricted variable (which can take a positive, negative or zero value) can be written as the difference of two nonnegative variables.
Thus if xj is unrestricted in sign, it can be written as xj=xj'-xj", where
It can be seen that xj will be negative, zero or positive, depending on whether xj" is greater than, equal to, or less than xj’ .
In most engineering optimization problems, the decision variables represent some physical dimensions and hence the variables xj will be nonnegative.
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10. Transformation of LP Problems into Standard Form
If a constraint appears in the form of a “less than or equal to” type of inequality as:
it can be converted into the equality form by adding a nonnegative slack variable xn+1 as follows:
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11. Transformation of LP Problems into Standard Form
If a constraint appears in the form of a “greater than or equal to” type of inequality as:
it can be converted into the equality form by subtracting a variable as:
where xn+1 is a nonnegative variable known as a surplus variable.
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12. Geometry of LP Problems
Example: A manufacturing firm produces two machine parts using lathes, milling machines, and grinding machines. The different machining times required for each part, the machining times available for different machines, and the profit on each machine part are given in the following table. Determine the number of parts I and II to be manufactured per week to maximize the profit.
Type of machine
Machine time required (min)
Machine Part I
Machine Part II
Maximum time available per week (min)
Lathes 10 5
2500
Milling machines 4
2000
Grinding machines 1
1.5
450
Profit per unit
$50
$100
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13. Geometry of LP Problems
Solution: Let the machine parts I and II manufactured per week be denoted by x and y, respectively. The constraints due to the maximum time limitations on the various machines are given by:
Since the variables x and y can not take negative values, we have
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15. Geometry of LP Problems
Solution: The total profit is given by:
Thus the problem is to determine the nonnegative values of x and y that satisfy the constraints stated in Eqs.(E1) to (E3) and maximize the objective function given by (E5). The inequalities (E1) to (E4) can be plotted in the xy plane and the feasible region identified as shown in the figure. Our objective is to find at least one point out of the infinite points in the shaded region in figure which maximizes the profit function (E5).
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16. Geometry of LP Problems
Solution: The contours of the objective function, f, are defined by the linear equation:
As k is varied, the objective function line is moved parallel to itself. The maximum value of f is the largest k whose objective function line has at least one point in common with the feasible region). Such a point can be identified as point G in the figure. The optimum solution corresponds to a value of x*=187.5, y*=125, and a profit of $21,
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17. Geometry of LP Problems
Solution: In some cases, the optimum solution may not be unique. For example, if the profit rates for the machine parts I and II are $40 and $100 instead of $50 and $100 respectively, the contours of the profit function will be parallel to side CG of the feasible region as shown in the figure. In this case, line P”Q” which coincides with the boundary line CG will correspond to the maximum feasible profit. Thus there is no unique optimal solution to the problem and any point between C and G online P”Q” can be taken as optimum solution with a profit value of $
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18. Geometry of LP Problems
Solution: There are three other possibilities. In some problems. In some prolems, the feasible region may not be a closed convex polygon. In such a case, it may happen that the profit level can be increased to an infinitely large value without leaving the feasible region as shown in the figure. In this case, the solution of the linear programming problem is said to be unbounded.
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19. Geometry of LP problems
On the other extreme, the constraint set may be empty in some problems. This could be due to the inconsistency of the constraints; or, sometimes, even though the constraints may be consistent, no point satisfying the constraints may also satisfy the nonnegativity restrictions.
The last possible case is when the feasible region consists of a single point. This can occur only if the number of constraints is at least equal to the number of variables. A problem of this kind is of no interest to us since there is only one feasible point and there is nothing to be optimized.
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20. Solution of LP Problems
A linear programming problem may have
A unique and finite optimum solution
An infinite number of optimal solutions
An unbounded solution
No solution
A unique feasible point
Example 3.2. on page 135
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21. Solution of LP Problems
A linear programming problem may have a unique and finite optimum solution:
The condition necessary for this to occur is:
The objective function and the constraints have dissimilar slopes
The feasible region is bounded/closed.
Example 3.2. on page 135
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22. Solution of LP Problems
A linear programming problem may have infinite solution.
The condition necessary for this to occur is:
The objective function must be parallel to one of the constraints.
Example 3.2. on page 135
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23. Solution of LP Problems
A linear programming problem may have no solution.
The condition necessary for this to occur is:
There is no point that is feasible. There is no solution to the problem.
Example 3.2. on page 135
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24. Solution of LP Problems
A linear programming problem may have unbounded solution.
In this case, the feasible region is not bounded.
The presence of an unbounded solution also suggests that the formulation of the problem may be lacking. Additional meaningful constraints can be accomodated to define the solution.
Example 3.2. on page 135
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