Question

Bavarian Autoworks is an automobile repair company specializing in German cars such as BMW, Audi, Mercedes, and Volkswagen. The company does two kinds of services most frequently: oil changes and tune-ups. The company has three kinds of capacity limitations: labor time, garage space, and hoist time. Doing an oil change requires 4 units of labor, 2 units of garage space, and 8 units of hoist time. A tune-up requires 5 units of labor, 1 unit of garage space, and 2 units of hoist time. Bavarian has 800 units of labor, 300 units of garage space, and 400 units of hoist time available each week.

Part1:

Suppose that labor is the only input with a capacity limitation and all labor is used to produce oil changes. How many oil changes can Bavarian do in a week?    Now suppose that instead of oil changes, Bavarian devotes all of its units of labor to producing tune-ups and faces no other capacity constraints. How many tune-ups can Bavarian do in a week?

Part2:

Now suppose that garage space is the capacity limitation, and consider devoting all of the available garage space to producing oil changes. How many oil changes can Bavarian do in this case?    How many tune-ups can Bavarian do if it devotes all its garage space to tune-ups?   

Part 3:

Finally, consider the case where hoist time becomes the binding constraint. If Bavarian devotes all of its hoist time to doing oil changes, how many can it do in a week?    If all of the hoist time is devoted to doing tune-ups, how many can Bavarian do in a week?     

Part 4:

Now consider the range of possible outcomes if all three constraints must be satisfied simultaneously. In this case, one of the constraints doesn't matter. Which one?( garage space,labor time or hoist time?)

Part 5:

Suppose Bavarian can charge $120 for an oil change and $600 for a tune-up. How many oil changes should the company do in a week if it wants to maximize its revenue?  

    What about tune-ups?   

Answer 1

capacity limitations	oil change	tune-ups
labor time	4 units	5 units
garage space	2 units	1 unit
hoist time	8 units	2 units

The company has 800 units of labor, 300 units of garage space, and 400 units of hoist time.

Part 1. if the company devote all its labor in oil change then the oil changed in a week is = 800/4 = 200 units. this happens because no other inputs have been used.

And if the company devotes all its units of labor in tune-ups then the total tune-ups in a week will be = 800/5 = 160 units.

part 2. now we have 300 units of garage space for a week, and we use 2 units of garage space in oil change, so if we use all units of garage space in oil change then we get total units that used in oil changed = 300/2 = 150 units of the garage are produced in oil change.

And if the company tune-ups by using all units of the garage then = 300/ 1 = 300 units are produced in garage use.

part 3. The company has 400 units of hoist time. if all used in oil changed then = 400/8 = 50 units produced in a week.  

And if the company uses all its units in tune-ups then the units will be = 400/2 = 200 units of hoist used for tune-ups per week.

part 4. All the constraints ( labor, garage, and hoist time ) satisfied at the same time and this condition can be fulfilled when any one of the constraints does not work and others can be used instead of that. here the garage can be replaced by hoists. as the replacement can fill the gap. so the garage doesn't matter for the company.

part 5. if com[any charges $120 for an oil change then the maximum revenue will be,

now the company uses 4 units of labor + 2 units of garage + 8 units of hoist time. so the total units used by the company are 4+2+8 = 14 units per car oiling.

And the company has a total of 800 units of labor + 300 units of garage + 400 units of hoists, so the total units are 800+300+400 = 1500 units for a week. so if divide total units by per unit then we get the total work done by the company, so

1500 14 = 107.14 units of work

company charge $120 for oil change and now thr maximum profit company can earn is,

$120 107.14 = $ 12856.8 per week