Question

14. An automobile manufacturer has five outdated plants: one each in Michigan, Ohio, and California and two in New York. Management is considering modernizing these plants to manufacture engine blocks and transmissions for a new model car the cost to modernize each plant and the manufacturing capacity after modernization are as follows:

Plant	Cost ($ millions)	Engine Blocks (1000s)	Transmissions (1000s)
Michigan	25	500	300
New York	35	800	400
New York	35	400	800
Ohio	40	900	600
California	20	200	300

The projected needs are for total capacities of 900,000 engine blocks and 900,000 transmissions. Management wants to determine which plants to modernize to meet projected manufacturing needs and, at the same time, minimize the total cost of modernization.

a. Develop a table that lists every possible option available to management. As part of your table, indicate the total engine block capacity and transmission capacity for each possible option, whether the option is feasible based on the projected needs, and the total modernization cost for each option.

b. Based on your analysis in part (a), what recommendation would you provide management?

c. Formulate a 0-1 integer programming model that could be used to determine the optimal solution to the modernization question facing management.

d. Solve the model formulated in part (c) to provide a recommendation for management.

Answer 1

Lets define some binary variables with name M,NY1, NY2, O, C. Which indicated if that plant will be open or not.
So the Z function will be our cost function which we have to minimise.
Z= 25*M + NY1*35 + NY2*35 + O*40 + C*20
There are 2 more constraint apart from Binary variables
1. Engine Equation
500*M+ 800*NY1 + 400*NY2 + 900*O + 200*C >= 900
2. Transmission Equation
300*M + 400*NY1 + 800*NY2 + 600*O + 1100*C>=900

As shown in the excel solver
M and NY2 turns out to be 1 rest variables are 0.

Apart from this the management also have option to open every other plant as well