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. PROF. SWETA SHAH

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 PROF. SWETA SHAH

Standard Form of a Linear Programming Problem Scalar form PROF. SWETA SHAH

Standard Form of a Linear Programming Problem Matrix form PROF. SWETA SHAH

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). PROF. SWETA SHAH

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. PROF. SWETA SHAH

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. PROF. SWETA SHAH

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. PROF. SWETA SHAH

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. PROF. SWETA SHAH

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: PROF. SWETA SHAH

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. PROF. SWETA SHAH

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 PROF. SWETA SHAH

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 PROF. SWETA SHAH

PROF. SWETA SHAH

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). PROF. SWETA SHAH

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 $21,875.00. PROF. SWETA SHAH

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 $ 20000. PROF. SWETA SHAH

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. PROF. SWETA SHAH

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. PROF. SWETA SHAH

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 PROF. SWETA SHAH

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 PROF. SWETA SHAH

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 PROF. SWETA SHAH

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 PROF. SWETA SHAH

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 PROF. SWETA SHAH