SENSITIVITY ANALYSIS

2 Sensitivity Analysis Sensitivity analysis is carried out after optimal solution is found. Hence called as post optimality analysis. Goal is to find out how changes in model coefficients affect the solution. Following can happen  The current solution remains unchanged.  The current solution becomes infeasible.  The current solution becomes non-optimal.  Current solution becomes infeasible and non-optimal.

SENSITIVITY ANALYSIS3 Solution Alternatives If the solution remains unchanged then we need not do any further calculations. If solution becomes infeasible then use dual simplex to restore feasibility. If solution becomes non-optimal then use primal simplex method to find new optimality. If the solution becomes both infeasible and non- optimal then we use both primal and dual simplex method to solve.

SENSITIVITY ANALYSIS4 What Changes can be made? Changes in objective coefficients  For non-basic variables  For basic variables Changes in availability of resources Changes in input/output coefficients Addition of new variables (column) Addition of new constraint (row)

SENSITIVITY ANALYSIS5 A little bit of Algebra Basicxjxj Starting X B Solution zc j - z j XBXB B –1 P j B -1 B –1 b LP form  Max z =  c j x j (j = 1 to n) subject to  P j x j = b  P j and b are column vectors

SENSITIVITY ANALYSIS6 Algebra continued.. The table on last page shows  Current basis matrix, starting X’s and Solution of BV. Let Y = (y1, y2… ym) be the m row vector for dual variables. The dual variables are calculated as  Y = C B B –1  where C B is the m row vector comprised of the original objective coefficients c j with the basis vector X B Thus c j – z j is computed as  c j – z j = c j -  a ij y i = c j – YP j Pj = aj (a matrix)  The above equation shows that cj – zj is the difference between the LHS and RHS of the dual constraints.

SENSITIVITY ANALYSIS7 Factors Affecting Feasibility The feasibility of a current solution can be changed if  the right side of constraints b is changed.  a new constraint is added. In both cases infeasibility occurs if one of the solution values (B –1 b) becomes negative, if one of the current basic variables becomes negative. Refer example.

SENSITIVITY ANALYSIS8 Adding New Constraint The addition of constraint can lead to two cases  The added constraint is redundant and hence solution remains same.  The new constraint is violated and hence we need to use dual simplex method for recovering feasibility. Refer excel sheet for example

SENSITIVITY ANALYSIS9 EXAMPLE A company wants to produce products A,B &C. the unit profit on these products are RS 4, Rs 6 & Rs. 2 respectively. The products require two resources namely manpower and raw material. Maximise Z = 4x1 + 6x2 + 2x3 Raw material constraint  x1 + 4x2 + 7x3 < 9 Manpower constraint  x1 + x2 + x3 < 3  x1, x2 & x3 > 0

SENSITIVITY ANALYSIS10 INITIAL SOLUTION

SENSITIVITY ANALYSIS11 CHANGE IN RHS OF THE CONSTRAINTS

SENSITIVITY ANALYSIS12 RANGE OF RHS CONSTRAINTS

SENSITIVITY ANALYSIS13 ADDING NEW CONSTRAINT

SENSITIVITY ANALYSIS14 ADDING NEW CONSTRAINT

SENSITIVITY ANALYSIS15 DUAL SIMPLEX

SENSITIVITY ANALYSIS16 Changes Affecting Optimality The current solution will cease to be optimal when cj – zj violate the optimality condition. This can change only when we change either the objective coefficients cj (C B ) or the unit resource usage vector Pj or aj. We know that  c j - z j = c j – Ya j where aj is unit resource usage vector.

SENSITIVITY ANALYSIS17 Changes in Objective Coefficients The effect of making changes in cj entails recomputing cj-zj for the NBV only. BV will remain zero (why?) Compute the dual prices (shadow) Y=C B B -1 using the new vector C B if it has changed Compute cj – zj = cj – Yaj for all current NBV. Following cases may arise  If the optimality condition is satisfied then current solution will remain.  If the optimality is changed we apply primal simplex to recover optimality.

SENSITIVITY ANALYSIS18 ADDITION OF NEW ACTIVITY This is akin to adding new variable in the objective function. Desirable only if the new economic activity is profitable. Check cj-zj = cj – YPj for the new activity where Y is current dual values, Pj (aj) is the resource for new variable and cj the contribution.

SENSITIVITY ANALYSIS19 Change of cj (NBV)

SENSITIVITY ANALYSIS20 Change of cj for Basic Variable