Q3. Graves Lighting is planning this weeks schedule for a metal stamping machine. It has been decided that they will make several models of exterior lanterns. All necessary accessories and glass inserts are in ample supply. Model Stamping time/unit Assembly time/unit Packaging time/unit Profit/unit MX142 1.2 minutes 2.4 minutes 1 minute $2.56 QX156 1.3 minutes 1.9 minute 1 minute $2.34 RX188 2.0 minutes 1.8 minute 1 minute $3.09 SZ203 1.5 minutes 2.0 minute 1 minute $2.74 Time available 40 hours 53 hours and 20 minutes 30 hours Graves is committed to producing at least 500 each of models MX142 and SZ203. The number of RX188 should not exceed 40% of the total number of units produced. A Linear Programming (LP) model for this situation is given below: M = # of MX142 models made Q = # of QX156 models made R = # of RX188 models made S = # of SZ203 models made Maximize Profit = 2.56M + 2.34 Q + 3.09R + 2.74S Subject to: 1.2M + 1.3Q + 2.0R + 1.5S 2400 minutes of stamping time 2.4M + 1.9Q + 1.8R + 2.0S 3200 minutes of assembly time 1.0M + 1.0Q + 1.0R + 1.0S 1800 minutes of packaging time M 500 minimum output of MX142 S 500 minimum output of SZ203 R 0.40(M + Q + R + S) RX188 no more than 40% of total -0.4M 0.4Q + 0.6R 0.4S 0 All variables are non-negative. The Sensitivity Report for the optimal solution to this model is: Variable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$4 # units MX142 500 0 2.56 0.152 1E+30 $C$4 # units QX156 0 -0.143 2.34 0.143 1E+30 $D$4 # units RX188 461.54 0 3.09 0.165 0.624 $E$4 # units SZ203 584.62 0 2.74 0.693 0.0748 Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $F$11 R<=40% -156.92 0 0 1E+30 156.92 $F$6 Stamping 2400 0.96 2400 61.11 300 $F$7 Assembly 3200 0.65 3200 400 55 $F$8 Packaging 1546.15 0 1800 1E+30 253.85 $F$9 min MX142 500 -0.152 500 41.67 500 $F$10 min SZ203 584.62 0 500 84.62 1E+30

a) What is the optimal production plan?

b) What is the maximum profit?

c) Which constraints are binding?

d) If the profit per unit of MX142 increased to $2.70 would the production plan change? Would the total profit change? If so, by how much?

e) If the profit per unit of the SZ203 dropped to $2.50, would the plan change? Explain.

f) Management has decided that some of every product should be made. What will be the impact on profit if at least one unit of every item is made?

g) Suppose we can get employees in stamping to work an additional hour. What is the maximum that we would pay these workers?

h) What if they offered to work an extra 3 hours? Can you evaluate the impact now? Explain.

i) The packaging line will have to shut down for 3 hours for maintenance. What will be the impact on profits?

j) The minimum requirements for MX142 has been reduced to 400 units. How will this affect profits?

k) Management has asked that the NP111 model be added to the production plan. The NP111 requires 1.5 minutes of stamping, 1.7 minutes of assembly and 1.0 minutes of packaging and generates a profit of $2.81 per unit. How will each unit of NP111 affect total profits if it is made?