Lecture 25: Assignment 4 Discussion

Lecture 25: Assignment 4 Discussion#

  1. Point-to-Point Routing

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

From

To

\(d_{ij}\) (km)

\(t_{ij}\) (mins)

\(f_{ij}\) (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:

\[\begin{split} \begin{aligned} \min_\mathbf{x} D = & 1.3x_{1,2} + 1.38x_{1,6} + 1.8x_{2,3} + 1.07x_{2,7} + 1.83x_{3,4} + 1.39x_{3,8} + 1.77x_{4,5} + 1.54x_{4,9} + 1.75x_{5,10} + \\ & 1.4x_{6,7} + 1.45x_{6,11} + 1.23x_{7,8} + 1.6x_{7,12} + 1.72x_{8,9} + 1.12x_{8,13} + 1.65x_{9,10} + 1.35x_{9,14} + 1.95x_{10,15} + \\ & 1.7x_{11,12} + 1.55x_{11,16} + 1.47x_{12,13} + 1.72x_{12,17} + 1.27x_{13,14} + 1.61x_{13,18} + 1.81x_{14,15} + 1.19x_{14,19} + 1.85x_{15,20} + \\ & 1.66x_{16,17} + 1.58x_{16,21} + 1.49x_{17,18} + 1.67x_{17,22} + 1.42x_{18,19} + 1.64x_{18,23} + 1.36x_{19,20} + 1.78x_{19,24} + 1.99x_{20,25} + \\ & 1.53x_{21,22} + 1.61x_{22,23} + 1.5x_{23,24} + 1.7x_{24,25} \end{aligned} \end{split}\]

Fastest Route:

\[\begin{split} \begin{aligned} \min_\mathbf{x} T = & 1.08x_{1,2} + 1.39x_{1,6} + 1.48x_{2,3} + 2.23x_{2,7} + 1.43x_{3,4} + 1.37x_{3,8} + 1.20x_{4,5} + 1.54x_{4,9} + 1.29x_{5,10} + \\ & 1.36x_{6,7} + 1.52x_{6,11} + 1.83x_{7,8} + 1.46x_{7,12} + 1.50x_{8,9} + 2.01x_{8,13} + 1.35x_{9,10} + 1.25x_{9,14} + 1.34x_{10,15} + \\ & 1.29x_{11,12} + 1.38x_{11,16} + 1.49x_{12,13} + 1.28x_{12,17} + 1.75x_{13,14} + 1.40x_{13,18} + 1.33x_{14,15} + 1.78x_{14,19} + 1.31x_{15,20} + \\ & 1.22x_{16,17} + 1.37x_{16,21} + 1.27x_{17,18} + 1.41x_{17,22} + 1.45x_{18,19} + 1.32x_{18,23} + 1.52x_{19,20} + 1.42x_{19,24} + 1.37x_{20,25} + \\ & 1.39x_{21,22} + 1.35x_{22,23} + 1.31x_{23,24} + 1.42x_{24,25} \end{aligned} \end{split}\]

Eco-Route:

\[\begin{split} \begin{aligned} \min_\mathbf{x} F = & 0.99x_{1,2} + 1.33x_{1,6} + 1.17x_{2,3} + 1.41x_{2,7} + 1.28x_{3,4} + 1.03x_{3,8} + 1.12x_{4,5} + 1.24x_{4,9} + 1.15x_{5,10} + \\ & 1.07x_{6,7} + 1.19x_{6,11} + 1.36x_{7,8} + 1.21x_{7,12} + 1.10x_{8,9} + 1.37x_{8,13} + 1.08x_{9,10} + 0.98x_{9,14} + 1.30x_{10,15} + \\ & 1.11x_{11,12} + 1.16x_{11,16} + 1.09x_{12,13} + 1.18x_{12,17} + 1.40x_{13,14} + 1.22x_{13,18} + 1.14x_{14,15} + 1.39x_{14,19} + 1.25x_{15,20} + \\ & 1.06x_{16,17} + 1.13x_{16,21} + 1.04x_{17,18} + 1.15x_{17,22} + 1.10x_{18,19} + 1.17x_{18,23} + 1.35x_{19,20} + 1.20x_{19,24} + 1.28x_{20,25} + \\ & 1.12x_{21,22} + 1.09x_{22,23} + 1.08x_{23,24} + 1.15x_{24,25} \end{aligned} \end{split}\]

b. Formulate all constraints. (5)

\[\begin{split} \begin{aligned} 1 & = x_{1,2} + x_{1,6} \\ x_{1,2} & = x_{2,3} + x_{2,7} \\ x_{2,3} & = x_{3,4} + x_{3,8} \\ x_{3,4} & = x_{4,5} + x_{4,9} \\ x_{4,5} & = x_{5,10} \\ x_{1,6} & = x_{6,7} + x_{6,11} \\ x_{2,7} + x_{6,7} & = x_{7,8} + x_{7,12} \\ x_{3,8} + x_{7,8} & = x_{8,9} + x_{8,13} \\ x_{4,9} + x_{8,9} & = x_{9,10} + x_{9,14} \\ x_{5,10} + x_{9,10} & = x_{10,15} \\ x_{6,11} & = x_{11,12} + x_{11,16} \\ x_{7,12} + x_{11,12} & = x_{12,13} + x_{12,17} \\ x_{8,13} + x_{12,13} & = x_{13,14} + x_{13,18} \\ x_{9,14} + x_{13,14} & = x_{14,15} + x_{14,19} \\ x_{10,15} + x_{14,15} & = x_{15,20} \\ x_{11,16} & = x_{16,17} + x_{16,21} \\ x_{12,17} + x_{16,17} & = x_{17,18} + x_{17,22} \\ x_{13,18} + x_{17,18} & = x_{18,19} + x_{18,23} \\ x_{14,19} + x_{18,19} & = x_{19,20} + x_{19,24} \\ x_{15,20} + x_{19,20} & = x_{20,25} \\ x_{16,21} & = x_{21,22} \\ x_{17,22} + x_{21,22} & = x_{22,23} \\ x_{18,23} + x_{22,23} & = x_{23,24} \\ x_{19,24} + x_{23,24} & = x_{24,25} \\ x_{20,25} + x_{24,25} & = 1 \\ x_{i,j} & \in \{0,1\} \ \forall \ (i,j) \in A \end{aligned} \end{split}\]

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 \(\text{km}^2\) 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 \(d\): \(f^d_v \ \forall \ v \in [1,2], \ d \in [1,3]\)

  • number of tours per type \(v\) delivery vehicle at depot node \(d\): \(m^d_v \ \forall \ v \in [1,2], \ d \in [1,3]\)

  • number of customer per delivery tour per type \(v\) delivery vehicle at depot node \(d\): \(c^d_v \ \forall \ v \in [1,2], \ d \in [1,3]\): \(c_i \ \forall \ i \in [1,3]\)

a. Formulate the objective function. (1)

\[\begin{split} \begin{aligned} \eta & = ((1 - 1.03 ^ {-7})/ 0.03) * 330 = 2056 \\ \sqrt\delta & = \sqrt{10000/ 307.78} = 5.7 \end{aligned} \end{split}\]
\[\begin{split} \begin{aligned} \min z = & y_1(75 \times 10^7 + 6 \times 10^5f^1_1 + 9 \times 10^5f^1_2 + 2056 \times (35 \times (2 \times 1 + 0.57 * c^1_1/5.7)m^1_1f^1_1 + 28 \times (2 \times 1 + 0.57 * c^1_2/5.7)m^1_2f^1_2)) + \\ & y_2(50 \times 10^7 + 6 \times 10^5f^2_1 + 9 \times 10^5f^2_2 + 2056 \times (35 \times (2 \times 5 + 0.57 * c^2_1/5.7)m^2_1f^2_1 + 28 \times (2 \times 5 + 0.57 * c^2_2/5.7)m^2_2f^2_2)) + \\ & y_3(10 \times 10^7 + 6 \times 10^5f^3_1 + 9 \times 10^5f^3_2 + 2056 \times (35 \times (2 \times 20 + 0.57 * c^3_1/5.7)m^3_1f^3_1 + 28 \times (2 \times 20 + 0.57 * c^3_2/5.7)m^3_2f^3_2)) \end{aligned} \end{split}\]

Rendering,

\[\begin{split} \begin{aligned} \min z = & y_1(75 \times 10^7 + 6 \times 10^5f^1_1 + 9 \times 10^5f^1_2 + (143920 + 7196c^1_1)m^1_1f^1_1 + (115136 + 5756.8c^1_2)m^1_2f^1_2)) + \\ & y_2(50 \times 10^7 + 6 \times 10^5f^2_1 + 9 \times 10^5f^2_2 + (719600 + 7196c^2_1)m^2_1f^2_1 + (575680 + 5756.8c^2_2)m^2_2f^2_2)) + \\ & y_3(10 \times 10^7 + 6 \times 10^5f^3_1 + 9 \times 10^5f^3_2 + (2878400 + 7196c^3_1)m^3_1f^3_1 + (2302720 + 5756.8c^3_2)m^3_2f^3_2)) \end{aligned} \end{split}\]

Note

Each row of the equation carries 1/3 marks

b. Formulate the constraints. (3)

\[\begin{split} \begin{aligned} f^1_1 + f^2_1 + f^3_1 & \leq 20 \\ m^1_1 & \leq 3 \\ m^1_2 & \leq 2 \\ m^2_1 & \leq 3 \\ m^2_2 & \leq 2 \\ m^3_1 & \leq 3 \\ m^3_2 & \leq 2 \\ c^1_1 & \leq 200 \\ c^1_2 & \leq 150 \\ c^2_1 & \leq 200 \\ c^2_2 & \leq 150 \\ c^3_1 & \leq 200 \\ c^3_2 & \leq 150 \\ c^1_1m^1_1f^1_1 + c^1_2m^1_2f^1_2 & \leq 3000 \\ c^2_1m^2_1f^2_1 + c^2_2m^2_2f^2_2 & \leq 10000 \\ c^3_1m^3_1f^3_1 + c^3_2m^3_2f^3_2 & \leq 30000 \\ y_1(c^1_1m^1_1f^1_1 + c^1_2m^1_2f^1_2) + y_2(c^2_1m^2_1f^2_1 + c^2_2m^2_2f^2_2) + y_3(c^3_1m^3_1f^3_1 + c^3_2m^3_2f^3_2) & \geq 10000 \\ c^d_v, m^d_v, f^d_v & \in \mathbf{Z}_+ \ \forall v \in V, \ d \in D \\ y_d & \in \{0,1\} \end{aligned} \end{split}\]

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 - \(y_d\)

Location

\(y_d\)

1

0

2

0

3

1

Table 5. Decision Variable Values - \(f^3_v, m^3_v, c^3_v\)

Vehicle Fleet

\(f^3_v\)

\(m^3_v\)

\(c^3_v\)

#1 Diesel Van

17

3

197

#2 Electric Van

0

0

0

Note

Each decision variable carries 1/2 marks