A firm wants to choose the site of a new warehouse that will simultaneously serv

Question

A firm wants to choose the site of a new warehouse that will simultaneously serve four retail branches. The average daily transportation need of the four retail branches from the warehouse are: Retail branch Coordinates Ashmouth Bedsbottom3,10) Cadillac Danhunt Number of daily deliveries (8,2) 6 7 (8,15) (14,13) Suppose that travel costs are $1 per mile. Roads are in a grid, running vertically or horizontally every mile. (so there is a vertical road at 1-1, another one at r = 1.25, another at x 1.50, etc) The firm would like to determine where to locate the new warehouse in a way that minimizes the daily transportation cost (a). Formulate an optimization model to determine the new warehouse's location. Remember that a delivery is a round-trip (b). Suppose that zoning laws restrict the location of the new warehouse to lie in the four-sided shaded region in the figure below. Formulate a new optimization model to determine the new warehouse's location, taking these additional restrictions into consideration. Cadillac (8.15) Danhunt (14,13) sbottom (3,10) (10,9) (13,9) (10,5) (17,5) Ashmouth (8,2) Formulate this problem as a linear program (LP). Clearly define in words what your decision variables represent. In the LP, label each constraint with a keyword for what the constraint corresponds o. You do not need to solve the LP. Do not worry about decision variable integrality.

Explanation / Answer

a ) For any general point (x,y) as warehouse location we have

(D2,C2) = (x,y)

for x co-ordinate in D2, y co-ordinate in C2

C3=CEILING(C2,0.25)

C4=FLOOR(C2,0.25)

For y direction traverse cost C6 = |y-2|*1*2 + |y-10|*1*6 + |y-15|*1*7 + |y-13|*1*8 where y=C3  

For y direction traverse cost C7 = |y-2|*1*2 + |y-10|*1*6 + |y-15|*1*7 + |y-13|*1*8 where y=C4  

C5=MIN(C6,C7)

For x dir traverse cost D6= |x-8|*1*2 + |x-3|*1*6 + |x-8|*1*7  + |x-14|*1*8 where x=D3

For x dir traverse cost D7= |x-8|*1*2 + |x-3|*1*6 + |x-8|*1*7  + |x-14|*1*8 where x=D4

x co-ordinate suppose in D2 so D3=CEILING(D2,0.25)

D4=FLOOR(D2,0.25) D5=MIN(D6,D7)

here mod | | represents abs() function abs(cell number)

Objective cell = E7=SUM(C5:D5)

b) Additional constraint for (x,y)

y>=5 and y<=9

x>=10

y+x-22<=0 here x,y denote respectively cell number

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