Lecture 16: Transportation Problem#
General Description#
A Transportation Problem involves shipping goods from \(m\) supply nodes to \(n\) demand nodes. Here, each supply node has a maximum supply capacity of \(s_i \ \forall \ i \in [1,m]\), and every demand node has a minimum demand requirement of \(d_j \ \forall \ i \in [1,n]\). Further, each unit of goods shipped from a supply node \(i\) to a demand node \(j\) incurs a cost \(c_ij\). Hence, the objective of a Transportation Problem is to minimise the cost of transporting goods from supply to demand nodes while satisfying for the supply and demand constraints.
Objective:
Subject to:
Here \(x_{ij}\) is the amount of goods flow between supply node \(i\) and demand node \(j\).
Example#
Sree Krishna Bricks operates three brick manufacturing plants in Medavakkam that cater to four client constuction 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 satsifying the supply and demand constraints.
From/To |
Site 1 |
Site 2 |
Site 3 |
Site 4 |
Supply |
|---|---|---|---|---|---|
Plant 1 |
8 |
6 |
10 |
9 |
35 |
Plant 2 |
9 |
12 |
13 |
7 |
50 |
Plant 3 |
14 |
9 |
16 |
5 |
40 |
Demand |
45 |
20 |
30 |
30 |
Formulate a linear optimisation model for this problem.
Objective:
\[ \min_{\mathbf{x}} z = 8x_{11} + 6x_{12} + 10x_{13} + 9x_{14} + 9x_{21} + 12x_{22} + 13x_{23} + 7x_{24} + 14x_{31} + 9x_{32} + 16x_{33} + 5x_{34} \]Subject to:
\[\begin{split} \begin{aligned} x_{11} + x_{12} + x_{13} + x_{14} & \leq 35 \\ x_{21} + x_{22} + x_{23} + x_{24} & \leq 50 \\ x_{31} + x_{32} + x_{33} + x_{34} & \leq 40 \\ x_{11} + x_{21} + x_{31} & \geq 45 \\ x_{12} + x_{22} + x_{32} & \geq 20 \\ x_{13} + x_{23} + x_{33} & \geq 30 \\ x_{14} + x_{24} + x_{34} & \geq 30 \\ x_{ij} & \geq 0 \ \forall \ i \in [1,3], \ j \in [1,4] \end{aligned} \end{split}\]Solve the above linear optimisation model using a spreadsheet to find the optimal solution.
From/To
Site 1
Site 2
Site 3
Site 4
Plant 1
0
10
25
0
Plant 2
45
0
5
0
Plant 3
0
10
0
30
The total cost of shipping is ₹1.02m (₹10.2L).
Introduce slack into each technical constraint and transform the above linear optimisation model.
Objective:
\[ \min_{\mathbf{x}} z = 8x_{11} + 6x_{12} + 10x_{13} + 9x_{14} + 9x_{21} + 12x_{22} + 13x_{23} + 7x_{24} + 14x_{31} + 9x_{32} + 16x_{33} + 5x_{34} \]Subject to:
\[\begin{split} \begin{aligned} - x_{11} - x_{12} - x_{13} - x_{14} & = - 35 + s_1 \\ - x_{21} - x_{22} - x_{23} - x_{24} & = - 50 + s_2 \\ - x_{31} - x_{32} - x_{33} - x_{34} & = - 40 + s_3 \\ x_{11} + x_{21} + x_{31} & = 45 + s_4 \\ x_{12} + x_{22} + x_{32} & = 20 + s_5 \\ x_{13} + x_{23} + x_{33} & = 30 + s_6 \\ x_{14} + x_{24} + x_{34} & = 30 + s_7 \\ x_{ij} & \geq 0 \ \forall \ i \in [1,3], \ j \in [1,4] \\ s_k & \geq 0 \ \forall \ k \in [1,7] \end{aligned} \end{split}\]Evaluate slack at the optimal.
From/To
Site 1
Site 2
Site 3
Site 4
Flow from plant
Supply
Plant 1
0
10
25
0
35
35
Plant 2
45
0
5
0
50
50
Plant 3
0
10
0
30
40
40
Flow to site
45
20
30
30
Demand
45
20
30
30
For the optimal flow from brick manufacturing plants to client construction sites, the slacks turn out to be zero.
For the above developed primal optimisation problem, formulate the dual problem.
Primal
Objective:
\[ \min_{\mathbf{x}} z = 8x_{11} + 6x_{12} + 10x_{13} + 9x_{14} + 9x_{21} + 12x_{22} + 13x_{23} + 7x_{24} + 14x_{31} + 9x_{32} + 16x_{33} + 5x_{34} \]Subject to:
\[\begin{split} \begin{aligned} - x_{11} - x_{12} - x_{13} - x_{14} & \geq - 35 \\ - x_{21} - x_{22} - x_{23} - x_{24} & \geq - 50 \\ - x_{31} - x_{32} - x_{33} - x_{34} & \geq - 40 \\ x_{11} + x_{21} + x_{31} & \geq 45 \\ x_{12} + x_{22} + x_{32} & \geq 20 \\ x_{13} + x_{23} + x_{33} & \geq 30 \\ x_{14} + x_{24} + x_{34} & \geq 30 \\ x_{ij} & \geq 0 \ \forall \ i \in [1,3], \ j \in [1,4] \end{aligned} \end{split}\]Dual
Objective:
\[ \max_{\mathbf{u,v}} z = - 35u_1 - 50u_2 - 40u_3 + 45v_1 + 20v_2 + 30v_3 + 30v_4 \]Subject to:
\[\begin{split} \begin{aligned} - u_1 + v_1 & \leq 8 \\ - u_1 + v_2 & \leq 6 \\ - u_1 + v_3 & \leq 10 \\ - u_1 + v_4 & \leq 9 \\ - u_2 + v_1 & \leq 9 \\ - u_2 + v_2 & \leq 12 \\ - u_2 + v_3 & \leq 13 \\ - u_2 + v_4 & \leq 7 \\ - u_3 + v_1 & \leq 14 \\ - u_3 + v_2 & \leq 9 \\ - u_3 + v_3 & \leq 16 \\ - u_3 + v_4 & \leq 5 \\ u_i & \geq 0 \ \forall \ i \in [1,3] \\ v_j & \geq 0 \ \forall \ i \in [1,4] \end{aligned} \end{split}\]Here, \(\{u_i; i \in [1,3]\}\) are the slack terms associated with each supply capacity constraint, while \(\{v_j; j \in [1,4]\}\) are the slack terms associated iwth each demand requirement constraint.
Solve the above dual optimisation problem and infer the results.
Solving the above dual optimisation problem yeilds a shadow price of ₹3k, ₹0k, and ₹0k for the supply capacities of the three brick manufacturing plants, respectively, and a shadow price of ₹9k, ₹9k, ₹13k, and ₹5k for the demand requirements of the four client construction sites. The former indicate the reduction in total shipping costs resulting from a unit increase in supply, while the latter reflect the cost savings achieved through a unit decrease in demand.