Lecture 17: Transshipment Problem

General Description

The Transshipment Problem is a generalized form of the Transportation Problem, where intermediate nodes, called transshipment nodes, are introduced between supply nodes and demand nodes. These nodes can serve as transfer nodes, where goods can be temporarily stored or redistributed. These transshipment nodes could include distribution centers, warehouses, or modal interchange hubs in a logistics network.

Hence, the problem involves shipping goods from \(m\) supply nodes to \(n\) demand nodes via \(p\) transshipment nodes. Here, each supply node has a maximum supply capacity of \(s_i \ \forall \ i \in [1,m]\), and each demand node has a minimum demand requirement of \(d_j \ \forall \ j \in [1,n]\). Likewise, each transshipment node has a maximum handling volume of \(q_k \ \forall \ k \in [1,p]\). Further, goods shipped from supply node \(i\) to demand node \(j\) through a trans-shipment node incur a cost \(c_{ik}\) in the first leg and \(c_{kj}\) in the second. Thus, the objective of the Transshipment Problem is to minimize the total transportation cost while satisfying the supply, demand, and transshipment constraints.

Objective:

\[ \min_{\mathbf{x}} z = \sum_{i=1}^{m} \sum_{k=1}^{p} c_{ik}x_{ik} + \sum_{j=1}^{n} \sum_{k=1}^{p} c_{kj}x_{kj} \]

Subject to:

\[\begin{split} \begin{aligned} \sum_{k=1}^{p} x_{ik} & \leq s_i & \ \forall \ i \in [1,m] \\ \sum_{k=1}^{p} x_{kj} & \geq d_j & \ \forall \ j \in [1,n] \\ \sum_{i=1}^{m} x_{ik} & \leq q_k & \ \forall \ k \in [1,p] \\ \sum_{i=1}^{m} x_{ik} & = \sum_{j=1}^{n} x_{kj} & \ \forall \ k \in [1,p] \\ x_{ik}, \ x_{kj} & \geq 0 & \ \forall \ i \in [1,m], \ j \in [1,n], \ k \in [1,p] \end{aligned} \end{split}\]

Here \(x_{ik}\) is the amount of goods flow between supply node \(i\) and transhipment node \(k\), while \(x_{kj}\) is the amount of goods flow between transhipment node \(k\) and demand node \(j\).

Caution

The volume constraint for transshipment nodes \(\sum_{i=1}^{m} x_{ik} \leq q_k \ \forall \ k \in [1,p]\) assumes that all inflows occur in a single time-period, while all outflows occur in a seperate time-period. For instance, the goods may arrive at the transshipment node on a particular day, stored overnight, and then dispatched the next day.

Further, flow balance at transhipment nodes \(\sum_{i=1}^{m} x_{ik} = \sum_{j=1}^{n} x_{kj} \ \forall \ k \in [1,p]\) assumes no exchange between transhipment nodes.

Example

Sree Krishna Bricks operates three brick manufacturing plants in Medavakkam that cater to four client constuction sites across Chennai via two warehouses. Table 1 below presents supply capacities of the plants, handling volume of warehouses, and demand requirements of the sites (in tonnes). Further, Table 2 and 3 detail the cost of transporting a tonne of bricks (in ₹k) from each plant to each warehouse (first leg), and from each warehouse to each site (second leg), respectively. Determine the shipping plan for the company that minimises the transportation costs while satsifying the supply, demand and volume constraints.

Table 1. Threshold quantity of bricks

Facility	Quantity
Plant 1	35
Plant 2	50
Plant 3	40
Warehouse 1	70
Warehouse 2	55
Site 1	45
Site 2	20
Site 3	30
Site 4	30

Table 2. First leg cost

From/To	Warehouse 1	Warehouse 2
Plant 1	4	5
Plant 2	3	4
Plant 3	6	2

Table 3. Second leg cost

From/To	Site 1	Site 2	Site 3	Site 4
Warehouse 1	5	3	6	4
Warehouse 2	7	2	5	3

1. Formulate a linear optimisation model for this problem.

   Objective:

   \[ \min_{\mathbf{u,v}} z = (4u_{11} + 5u_{12} + 3u_{21} + 4u_{22} + 6u_{31} + 2u_{32}) + (5v_{11} + 3v_{12} + 6v_{13} + 4v_{14} + 7v_{21} + 2v_{22} + 5v_{23} + 3v_{24}) \]

   Subject to:

   \[\begin{split} \begin{aligned} u_{11} + u_{12} & \leq 35 \\ u_{21} + u_{22} & \leq 50 \\ u_{31} + u_{32} & \leq 40 \\ v_{11} + v_{21} & \geq 45 \\ v_{12} + v_{22} & \geq 20 \\ v_{13} + v_{23} & \geq 30 \\ v_{14} + v_{24} & \geq 30 \\ u_{11} + u_{21} + u_{31} & \leq 70 \\ u_{12} + u_{22} + u_{32} & \leq 55 \\ u_{11} + u_{21} + u_{31} & = v_{11} + v_{12} + v_{13} + v_{14} \\ u_{12} + u_{22} + u_{32} & = v_{21} + v_{22} + v_{23} + v_{24} \\ u_{ik}, \ v_{kj} & \geq 0 & \ \forall \ i \in [1,m], \ j \in [1,n], \ k \in [1,p] \end{aligned} \end{split}\]

2. Solve the above linear optimisation model using a spreadsheet to find the optimal solution.

   Table 4. First leg flow

   From/To	Warehouse 1	Warehouse 2
   Plant 1	20	15
   Plant 2	50	0
   Plant 3	0	40

   Table 5. Second leg flow

   From/To	Site 1	Site 2	Site 3	Site 4
   Warehouse 1	45	20	5	0
   Warehouse 2	0	0	25	30

   The total cost of shipping is ₹0.915m (₹9.15L).

3. Transform the above optimisation model into standard primal form.

   \[ \min_{\mathbf{u,v}} z = (4u_{11} + 5u_{12} + 3u_{21} + 4u_{22} + 6u_{31} + 2u_{32}) + (5v_{11} + 3v_{12} + 6v_{13} + 4v_{14} + 7v_{21} + 2v_{22} + 5v_{23} + 3v_{24}) \]

   Subject to:

   \[\begin{split} \begin{aligned} - u_{11} - u_{12} & \geq - 35 \\ - u_{21} - u_{22} & \geq - 50 \\ - u_{31} - u_{32} & \geq - 40 \\ v_{11} + v_{21} & \geq 45 \\ v_{12} + v_{22} & \geq 20 \\ v_{13} + v_{23} & \geq 30 \\ v_{14} + v_{24} & \geq 30 \\ - u_{11} - u_{21} - u_{31} & \geq - 70 \\ - u_{12} - u_{22} - u_{32} & \geq - 55 \\ u_{11} + u_{21} + u_{31} & \geq v_{11} + v_{12} + v_{13} + v_{14} \\ - u_{11} - u_{21} - u_{31} & \geq - v_{11} - v_{12} - v_{13} - v_{14} \\ u_{12} + u_{22} + u_{32} & \geq v_{21} + v_{22} + v_{23} + v_{24} \\ - u_{12} - u_{22} - u_{32} & \geq - v_{21} - v_{22} - v_{23} - v_{24} \\ u_{ik}, \ v_{kj} & \geq 0 & \ \forall \ i \in [1,m], \ j \in [1,n], \ k \in [1,p] \end{aligned} \end{split}\]

4. Infer shadow price by relaxing each constraints for each plant supply capacity constraint, warehouse handling volume constraint, and site demand requirement constraint.

   Table 6. Shadow Price

   Facility	Shadow Price
   Plant 1	0
   Plant 2	1
   Plant 3	3
   Warehouse 1	0
   Warehouse 2	0
   Site 1	9
   Site 2	7
   Site 3	10
   Site 4	8