Lecture 26: Quiz-II Discussion#
Sree Krishna Bricks operates two brick manufacturing plants in Medavakkam that cater to five client construction sites across Chennai. The table below presents supply capacities of the plants and demand requirements of the sites (in tonnes). Additionally, it details the cost of transporting a tonne of bricks between each plant and site (in ₹k). Determine the shipping plan for the company that minimises the transportation costs while satisifying the supply and demand constraints.
From/To
Site 1
Site 2
Site 3
Site 4
Site 5
Supply
Plant 1
15
11
18
12
14
80
Plant 2
16
17
21
10
13
70
Demand
45
20
30
30
25
1.1. Define the decision variables. (1)
\(x_{ij}\): brick flow from plant \(i \in [1,2]\) to site \(j \in [1,5]\) (in tonnes)
1.2. Formulate the objective function for this problem. (2)
Objective:
\[ \min_{\mathbf{x}} z = 15x_{11} + 11x_{12} + 18x_{13} + 12x_{14} + 14x_{15} + 16x_{21} + 17x_{22} + 21x_{23} + 10x_{24} + 13x_{25} \]1.3. Formulate the supply constraints. (2)
\[\begin{split} \begin{aligned} x_{11} + x_{12} + x_{13} + x_{14} + x_{15} & \leq 80 \\ x_{21} + x_{22} + x_{23} + x_{24} + x_{25} & \leq 70 \\ \end{aligned} \end{split}\]Note
each equation carries 1 mark
1.4. Formulate the demand constraints. (5)
\[\begin{split} \begin{aligned} x_{11} + x_{21} & \geq 45 \\ x_{12} + x_{22} & \geq 20 \\ x_{13} + x_{23} & \geq 30 \\ x_{14} + x_{24} & \geq 30 \\ x_{15} + x_{25} & \geq 25 \\ \end{aligned} \end{split}\]Note
each equation carries 1 mark
1.5. Formulate the domain constraint. (1)
\[ x_{ij} \geq 0 \ \ \ \ \ \forall \ i \in [1,2], \ j \in [1,5] \]Vehicle Routing Problem: We have a directed graph \(G=(N,A); N=\{{d} \cup C\}\), where d represents the depot node, C represents set of customer nodes, and A represents the set of arcs connecting these nodes. The depot has an associated set of electric delivery vehicles \(V\), wherein each vehicle has a capacity \(q_v\), range \(l_v\), and a traversal cost of \(π_o\) per km on arc \((i,j) \in A\) spanning a length of length \(d_{ij}\). Further, each customer node c∈C has a demand \(q_c\) that must be fulfilled from the depot nodes via one of its electric delivery vehicles. For the given problem,
2.1. Define the decision variables. (2)
\(x^v_{ij}\): traversal on arc \((i,j)\) for vehicle \(v \in V\)
\(y^v_k\): load of vehicle \(v \in V\) on arrival at node \(k \in N = \{C \cup {d}\}\)
Note
each decision variable varries one mark
2.2. Formulate the objective function (total cost of distribution) as the sum of fixed costs and net present value of operational cost. (2)
\[ \min_{\mathbf{x,y}} \pi_o \sum_{v \in V} \sum_{(i,j) \in A} d_{ij}x^v_{ij}\]2.3. Define head and tail set function. (2)
\[\begin{split} \begin{aligned} T(i) & = \{j; \ (j,i) \in A\} \\ H(i) & = \{j; \ (i,j) \in A\} \end{aligned} \end{split}\]Note
each function carries 1 mark
2.4. Formulate the following constraints
2.4.1. customer service constraint (1)
\[ \sum_{v \in V}\sum_{j \in H(c)} x^v_{cj} = 1 \ \ \ \ \ \forall \ c \in C \]2.4.2. flow conservation constraint (2)
\[ \sum_{i \in T(j)} x^v_{ij} = \sum_{k \in H(j)} x^v_{jk} \ \ \ \ \ \forall \ j \in N, \ v \in V \]2.4.3. vehicle use constraint (assume vehicle can perform at most one delivery tour) (1)
\[ \sum_{c \in H(d)} x^v_{dc} \leq 1 \ \ \ \ \ \forall \ v \in V \]2.4.4. fleet size constraint (1)
\[ \sum_{v \in V}\sum_{j \in H(d)} x^v_{dj} \leq |V| \]2.4.5. vehicle load balance constraint (3)
\[ y^v_k \leq x^v_{ck}(y^v_c - q_c) + (1 - x^v_{ck})\text{M} \ \ \ \ \ \forall \ c \in C, \ k \in H(c), \ v \in V\]2.4.6. vehicle load boundary condition (1)
\[ y^v_d = 0 \ \ \ \ \ \forall \ v \in V \]2.4.7. vehicle capacity constraint (1)
\[ y^v_c \leq q_v \ \ \ \ \ \forall \ c \in C, \ v \in V \]2.4.8. vehicle range constraint (1)
\[ \sum_{i \in N}\sum_{j \in H(i)} x^v_{ij}d_{ij} \leq l_v \ \ \ \ \ \forall \ v \in V_d, \ d \in D \]2.4.9. domain constraints (2)
\[\begin{split} \begin{aligned} x^v_{ij} & \in \{0,1\} & \ \forall \ (i,j) \in A, \ v \in V \\ y^v_c & \in \mathbb{R}_+ & \forall \ c \in C, \ v \in V \end{aligned} \end{split}\]