Lecture 25: Assignment 4 Discussion

Lecture 25: Assignment 4 Discussion#

  1. Point-to-Point Routing

Consider a 5×5 grid-like network (Table 1.)

From

To

dij (km)

tij (mins)

fij (l)

1

2

1.30

1.08

0.99

1

6

1.38

1.39

1.33

2

3

1.80

1.48

1.17

2

7

1.07

2.23

1.41

3

4

1.83

1.43

1.28

3

8

1.39

1.37

1.03

4

5

1.77

1.20

1.12

4

9

1.54

1.54

1.24

5

10

1.75

1.29

1.15

6

7

1.40

1.36

1.07

6

11

1.45

1.52

1.19

7

8

1.23

1.83

1.36

7

12

1.60

1.46

1.21

8

9

1.72

1.50

1.10

8

13

1.12

2.01

1.37

9

10

1.65

1.35

1.08

9

14

1.35

1.25

0.98

10

15

1.95

1.34

1.30

11

12

1.70

1.29

1.11

11

16

1.55

1.38

1.16

12

13

1.47

1.49

1.09

12

17

1.72

1.28

1.18

13

14

1.27

1.75

1.40

13

18

1.61

1.40

1.22

14

15

1.81

1.33

1.14

14

19

1.19

1.78

1.39

15

20

1.85

1.31

1.25

16

17

1.66

1.22

1.06

16

21

1.58

1.37

1.13

17

18

1.49

1.27

1.04

17

22

1.67

1.41

1.15

18

19

1.42

1.45

1.10

18

23

1.64

1.32

1.17

19

20

1.36

1.52

1.35

19

24

1.78

1.42

1.20

20

25

1.99

1.37

1.28

21

22

1.53

1.39

1.12

22

23

1.61

1.35

1.09

23

24

1.50

1.31

1.08

24

25

1.70

1.42

1.15

Assuming node 1 as the origin node 25 as the destination, answer the questions below.

a. Formulate the objective functions for the shortest route, fastest route, and eco-route. (1)

Shortest Route:

minxD=1.3x1,2+1.38x1,6+1.8x2,3+1.07x2,7+1.83x3,4+1.39x3,8+1.77x4,5+1.54x4,9+1.75x5,10+1.4x6,7+1.45x6,11+1.23x7,8+1.6x7,12+1.72x8,9+1.12x8,13+1.65x9,10+1.35x9,14+1.95x10,15+1.7x11,12+1.55x11,16+1.47x12,13+1.72x12,17+1.27x13,14+1.61x13,18+1.81x14,15+1.19x14,19+1.85x15,20+1.66x16,17+1.58x16,21+1.49x17,18+1.67x17,22+1.42x18,19+1.64x18,23+1.36x19,20+1.78x19,24+1.99x20,25+1.53x21,22+1.61x22,23+1.5x23,24+1.7x24,25

Fastest Route:

minxT=1.08x1,2+1.39x1,6+1.48x2,3+2.23x2,7+1.43x3,4+1.37x3,8+1.20x4,5+1.54x4,9+1.29x5,10+1.36x6,7+1.52x6,11+1.83x7,8+1.46x7,12+1.50x8,9+2.01x8,13+1.35x9,10+1.25x9,14+1.34x10,15+1.29x11,12+1.38x11,16+1.49x12,13+1.28x12,17+1.75x13,14+1.40x13,18+1.33x14,15+1.78x14,19+1.31x15,20+1.22x16,17+1.37x16,21+1.27x17,18+1.41x17,22+1.45x18,19+1.32x18,23+1.52x19,20+1.42x19,24+1.37x20,25+1.39x21,22+1.35x22,23+1.31x23,24+1.42x24,25

Eco-Route:

minxF=0.99x1,2+1.33x1,6+1.17x2,3+1.41x2,7+1.28x3,4+1.03x3,8+1.12x4,5+1.24x4,9+1.15x5,10+1.07x6,7+1.19x6,11+1.36x7,8+1.21x7,12+1.10x8,9+1.37x8,13+1.08x9,10+0.98x9,14+1.30x10,15+1.11x11,12+1.16x11,16+1.09x12,13+1.18x12,17+1.40x13,14+1.22x13,18+1.14x14,15+1.39x14,19+1.25x15,20+1.06x16,17+1.13x16,21+1.04x17,18+1.15x17,22+1.10x18,19+1.17x18,23+1.35x19,20+1.20x19,24+1.28x20,25+1.12x21,22+1.09x22,23+1.08x23,24+1.15x24,25

b. Formulate all constraints. (5)

1=x1,2+x1,6x1,2=x2,3+x2,7x2,3=x3,4+x3,8x3,4=x4,5+x4,9x4,5=x5,10x1,6=x6,7+x6,11x2,7+x6,7=x7,8+x7,12x3,8+x7,8=x8,9+x8,13x4,9+x8,9=x9,10+x9,14x5,10+x9,10=x10,15x6,11=x11,12+x11,16x7,12+x11,12=x12,13+x12,17x8,13+x12,13=x13,14+x13,18x9,14+x13,14=x14,15+x14,19x10,15+x14,15=x15,20x11,16=x16,17+x16,21x12,17+x16,17=x17,18+x17,22x13,18+x17,18=x18,19+x18,23x14,19+x18,19=x19,20+x19,24x15,20+x19,20=x20,25x16,21=x21,22x17,22+x21,22=x22,23x18,23+x22,23=x23,24x19,24+x23,24=x24,25x20,25+x24,25=1xi,j{0,1}  (i,j)A

c. Formulate the above optimisation problems (each) in a spreadsheet. (9)

Note

Each sheet carries 3 marks

d. Report the optimal solution (total distance, total time, and total fuel consumed in each shortest, fastest, and eco-route). (1)

Shortest Route: 1 - 2 - 7 - 8 - 13 - 14 - 19 - 20 - 25

Fastest Route: 1 - 2 - 3 - 8 - 9 - 14 - 19 - 20 - 25

Eco-Route: 1 - 2 - 3 - 4 - 5 - 10 - 15 - 20 - 25

Path

Distance

Time

Fuel

SR

9.64

13.47

10.26

FR

14.05

10.01

9.32

ER

11.53

12.47

8.16

Note

Each solution carries 1/3 marks

  1. Location Routing Problem

Amazon plans to serve 10000 customers in a service region of size 307.78 km2 from the following potential distribution facilities (Table 2.) using a fleet of diesel and electric vans (Table 3.). Cosnidering a planning horizon of 7 years, each with 330 working days, which facilities should Amazon choose to operate from?

Table 2. Potential Distribution Facility Locations

Location

Fixed Cost (in ₹cr)

Distance from Service Region (in km)

Capacity (in customers)

Location #1

75

1

3000

Location #2

50

5

10000

Location #3

10

20

30000

Table 3. Fleet Characteristics

Vehicle Type

Purchase Cost (₹)

Operational Cost (₹ per km)

Maximum Fleet Size

Maximum Tours

Maximum Customers

#1 Diesel Van

6,00,000

₹35

20

3

200

#2 Electric Van

9,00,000

₹28

-

2

150

Using the following notations, answer the questions below,

Notations:

  • number of type v delivery vehicles purchased at depot node d: fvd  v[1,2], d[1,3]

  • number of tours per type v delivery vehicle at depot node d: mvd  v[1,2], d[1,3]

  • number of customer per delivery tour per type v delivery vehicle at depot node d: cvd  v[1,2], d[1,3]: ci  i[1,3]

a. Formulate the objective function. (1)

η=((11.037)/0.03)330=2056δ=10000/307.78=5.7
minz=y1(75×107+6×105f11+9×105f21+2056×(35×(2×1+0.57c11/5.7)m11f11+28×(2×1+0.57c21/5.7)m21f21))+y2(50×107+6×105f12+9×105f22+2056×(35×(2×5+0.57c12/5.7)m12f12+28×(2×5+0.57c22/5.7)m22f22))+y3(10×107+6×105f13+9×105f23+2056×(35×(2×20+0.57c13/5.7)m13f13+28×(2×20+0.57c23/5.7)m23f23))

Rendering,

minz=y1(75×107+6×105f11+9×105f21+(143920+7196c11)m11f11+(115136+5756.8c21)m21f21))+y2(50×107+6×105f12+9×105f22+(719600+7196c12)m12f12+(575680+5756.8c22)m22f22))+y3(10×107+6×105f13+9×105f23+(2878400+7196c13)m13f13+(2302720+5756.8c23)m23f23))

Note

Each row of the equation carries 1/3 marks

b. Formulate the constraints. (3)

f11+f12+f1320m113m212m123m222m133m232c11200c21150c12200c22150c13200c23150c11m11f11+c21m21f213000c12m12f12+c22m22f2210000c13m13f13+c23m23f2330000y1(c11m11f11+c21m21f21)+y2(c12m12f12+c22m22f22)+y3(c13m13f13+c23m23f23)10000cvd,mvd,fvdZ+ vV, dDyd{0,1}

Note

Each constraint carries 1/6 marks

c. Formulate the above optimisation problem in a spreadsheet. (3)

d. Report the optimal solution. (2)

Table 4. Decision Variable Values - yd

Location

yd

1

0

2

0

3

1

Table 5. Decision Variable Values - fv3,mv3,cv3

Vehicle Fleet

fv3

mv3

cv3

#1 Diesel Van

17

3

197

#2 Electric Van

0

0

0

Note

Each decision variable carries 1/2 marks