Q&A Hero

Question

Answer: Attached is the graph of the total cost with respect to units produced.

To solve for the crossover points students should set the total cost functions equal for the various options. This gives the following equations

50000+2x=100000+x, x=50000 units

50000+2x=60000+1.4x, x=16667 units

100000+x=60000+1.4x, x=100000 units

To check which function is the lowest cost for a given range students should test each function for its total cost. For example

A(0)=50000+2(0)=50000

B(0)=100000+1(0)=100000

C(0)=60000+1.4(0)=60000

Therefore for the range 0 to 16667 option A is cheapest.

A( 20000)=90000

B(20000)= 120000

C(20000)=88000

Therefore for the range 16667 to 50000 option C is cheapest.

A(60000)=170000

B(60000)=160000

C(60000)=144000

Therefore for the range 50000 to 100000 option C is cheapest

A(110000)=270000

B(110000)=210000

C(110000)=214000

Therefore for the range 100000 or more option B is cheapest

Combining these ranges shows that for production of under 16667 units A is cheapest, for between 16667 and 100000 C is cheapest, and for 100000 and above B is cheapest.

22) A grocery chain is deciding on where to locate its new distribution center. The new DC will serve four grocery stores, each with a demand of 10,000 units. If the coordinates of the stores are (112,108), (110,50), (40, 85), and (10, 25) where should the DC be located? Suppose now that each store instead had demand of 20,000 units. Where should the DC go in this case?

Answer: Since each store has the same demand the x and y coordinates can simply be averaged and the DC will be located in the same spot for each case.

X=(112+110+40+10)/4 =68

Y=(108+50+85+25)=67

Thus the DC should be located at (68,67) for both cases.

23) Suppose that a bus company is deciding where to locate its central hub. There are 6 possible destinations for the buses. Suppose that the center of town will be used as the reference for describing the possible destinations. A is located 5 miles South and 3 miles West. B is located 3 miles North and 2 miles East. C is located 1 mile South and 5 miles East. D is located 2 miles North and 3 miles West. E is located exactly in the center of town. F is located 10 miles North and 5 miles East. Assume that traffic to each destination will be equal. Where should the hub go so that travel time is minimized?

Answer: Since each site has the same demand the coordinates can simply be averaged. Converting the center of town to (0,0) will mean that coordinates that are labeled South become
-y and those labeled West become -x, while North becomes +y and East +x. Thus the location is found to be:

X= (-3+2+5-3+0+5)/6 =1

Y= (-5+3-1+2+0+10)/6=1.5

Converting back into direction yields 1 mile East and 1.5 miles North of the center of town should be the location for the hub.