x 1 x 2 x 3 x 4 x 5 x 6 b (essence of widget) (enhancement time) (production time)

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1 (Widgets International, LTD.) A company makes three types of widgets. Type widget requires lbs. of essence of widget to produce while types 2 and 3 require and 2 lbs., respectively. Type and type 3 widgets are also put through a widget enhancement process to increase their value. This requires 2 hours for a batch of type widgets and 3 hours for a batch of type 3 widgets. There are only 5 hours of enhancement processing available per day. The total number of daily production hours, excluding the widget enhancement time, is limited to 7. The widgets take 2,, and 3 hours to produce widget batches of the type, type 2, and type 3 widgets, respectively. If there are only 4 lbs. of essence of widget available per day and the per widget profits are \$3, \$2, and \$4 for types, 2, and 3 widgets, respectively, then what is the daily production schedule that maximizes profit? The plant manager has formulated this problem as an LP with initial tableau (a) It is possible to purchase more essence of widget to increase productivity. (a) Should more be purchased? (b) If we purchase more, how much are we willing to pay for it beyond what we currently pay? (c) How much more should we purchase at the higher price?

2 (a) It is possible to purchase more essence of widget to increase productivity. (a) Should more be purchased? (b) If we purchase more, how much are we willing to pay for it beyond what we currently pay? (c) How much more should we purchase at the higher price? Solution: (a) Yes, essense of widget has a shadow price of \$.2 a pound. That is, every extra pound, up to some limit, increases profit by \$.2. (b) We are willing to pay up to \$.2 beyond what we are currently paying. (c) The range analysis on essense of widget gives 5 +.θ 5 θ.5.θ θ 5 Hence 5 θ 5, which implies that we can add up to 5 pounds more essense of widget without changing the currently optimal basis. If we do add more than 5 lbs, then we will need to perform a dual simplex pivot on the third row.

3 (b) A new style widget is being considered for production. It requires 3 lbs of essence of widget and it takes 5 hours to produce a batch of. In addition, the new widget will require 6 hours of enhancement time per widget batch. What is the minimum profit level is required for this new widget in order to make it efficient to produce?

4 (b) A new style widget is being considered for production. It requires 3 lbs of essence of widget and it takes 5 hours to produce a batch of. In addition, the new widget will require 6 hours of enhancement time per widget batch. What is the minimum profit level is required for this new widget in order to make it efficient to produce? Solution: Price out the new widget: T.2 5 = = 9

5 (c) Do not consider questions (a) and (b) above in answering this question. Production capacity at Widgets International has been increased. This increase gives us an additional hour of widget enhancement time and 2 additional hours of widget production time. Moreover, they have arranged to receive and additional 5 lbs pounds of essence of widget each day at the usual price. Do these changes alter the optimal production schedule? and, if so, what is the new schedule and the new optimal value?

6 (c) Do not consider questions (a) and (b) above in answering this question. Production capacity at Widgets International has been increased. This increase gives us an additional hour of widget enhancement time and 2 additional hours of widget production time. Moreover, they have arranged to receive and additional 5 lbs pounds of essence of widget each day at the usual price. Do these changes alter the optimal production schedule? and, if so, what is the new schedule and the new optimal value? Solution: The new information can be represented as a change to the rhs b with b = 5 2 The new rhs is Rb+R b = = = This gives a new production schedule of x x 2 x 3 = 3 5 The new profit is y T b + y T b = T 5 2 = = 2.

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