Assignment # 9: Sensitivity analysis in linear programming

• 50

a) No effect the short dowel constraint is not binding and has 360 slack units;
b)nothing, again 360 extra short dowels can absorb this shortfall;
c) 1240 and up ( allowable increase is infinity allowable decrease 360, the slack.

• 51
  • a) This is a binding constraint and the 10 unit reduction is within the allowable decrease thus: SP* change in RHS= Change in OV  or . 6* (-10) = -60
  • b) Can be tightened by 90, the amount of surplus, i .e, can tighten the constraint to x1 + x2 >= 190.
  • c) Shadow price of legs is  $6.0 and  is valid for 50 additional legs , hence OV will increase by $6 * 50 = $300

• 52

Optimal solution is degenerate; in general when the allowable increase or decrease of a RHS is zero the solution is degenerate.
Also if the allowable increase or decrease of an objective function coefficient is zero then we know there are alternative optima.

• 53

Degenerate, because there are 6 constraints and only 5 non-zero variables: Qty of Captains and Mates, Surplus variable of chair production constraint, Slacks of short dowels and light seats.

• 54
  • a) Since the reduction of $10 is within the allowable decrease of $12 the solution (130,60) remains the same.
  • b) However OV will be reduced by Change in coeff * value of the variable = (-10)*60 = -$600; i.e., they will be making $10 less for each of 60 Mates they are making and selling.

• 55
  • a) Since the increase equals the max allowable increase, the solution will remain optimal albeit an alternative optimum will exist.
  • b) OV increases by  change in coeff * value of the variable = $24 * 130 = 3120 i.e.,  they will be making $24 more for each of the 130 captains  they are making and selling.

• 56
  • a) T4=0; reduced cost is $91.11, thus it must become cheaper by at least 91.11 before it becomes attractive enough to use ore 4.
  • b) No change in solution (allowable increase is 120), however the OV will be reduced by $80*0.259=$20.72.
  • c) No change in solution ( allowable increase is 223.64), however the OV increases by $100 * 0.259 = $25.90.

 

• 57
  • a) No change in solution (allowable increase is 85.71), however the OV will increase by change in coeff * value of the variable = $50 * 0.037 =$1.85/per ton.
  • b) Since the change is exactly equal to the allowable decrease, the solution will remain optimal; however there will be an alternative optimum that uses more T3 and less of the other ores. 
  • c) The new OV is less by $118.269 * 0.037 = $4.38.