One requirement of the transportation problem is advance knowledge of the method of distribution of units from each source i to each destination j, so that the corresponding cost per unit (xij) can be determined. Sometimes, however, the best method of distribution is not clear because of the possibility of transshipments, whereby shipments would go through intermediate transfer points (which might be other sources or destinations).
Transshipments could be investigated in advance to determine the cheapest route from each source to each destination. However, this might be a very complicated and time-consuming task if there are many possible intermediate transfer points. Therefore, it may be much more convenient to let a computer algorithm solve simultaneously for the amount to ship from each source to each destination and the route to follow for each shipment so as to minimize the total shipping cost.
This extension of the transportation problem to include the routing decisions is referred to as the transshipment problem. This problem is the special case of the minimum cost flow problem where there are no restrictions on the amount that can be shipped through each shipping lane (unlimited arc capacities). The network representation of such a problem, where each two-sided arrow indicates that a shipment can be sent in either direction between the corresponding pair of locations.
The objective of the transshipment problem is to determine how many units should be shipped over each node so that all the demand requirements are met with the minimum transportation cost.
Considering a company with its manufacturing facilities situated at two places, Coimbatore and Pune. The units produced at each facility are shipped to either of the company’s warehouse hubs located at Chennai and Mumbai. The company has its own retail outlets in Delhi, Hyderabad, Bangalore and Thiruvananthapuram. The network diagram representing the nodes and transportation per unit cost is shown in Figure. The supply and demand requirements are also given.
Manufacturing Warehouses Retail Outlets Demand facility (Origin nodes) (Transshipment nodes) (Destination nodes)
Network Representation of Transshipment Problem
Solving Transshipment Problem using Linear Programming
xij be the number of units shipped from node i to node j,
x13 be the number of units shipped from Coimbatore to Chennai,
x24 be the number of units shipped from Pune to Mumbai, and so on
The following table shows the unit transportation cost from sources to destination.
TP of the Shipment
The objective is to minimize the total cost
Z = 4x13+ 7x14+ 6x23+ 3x24+ 7x35+ 4x36+ 3x37+ 5x38+ 5x45+6x46+ 7x47+ 8x48
Constraints: The number of units shipped from Coimbatore must be less than or equal to 800. Because the supply from Coimbatore facility is 800 units. Therefore, the constraints equation is as follows:
x13+ x14< 800 …………………….. (i)
Similarly, for Pune facility
x23+ x24< 600 ……………………...(ii)
Now, considering the node 3,
Number of units shipped out from node 1 and 2 are,
Number of units shipped out from node 3 is,
x35 + x36 + x37 + x38
The number of units shipped in must be equal to number of units shipped out, therefore
x13 + x23 = x35 + x36 + x37 + x38
Bringing all the variables to one side, we get
– x13– x23 + x35 + x36 + x37 + x38 = 0 ………….(iii)
Similarly for node 4
– x14– x24 + x45+x46 + x47 + x48 =0 …………..(iv)
Now considering the retail outlet nodes, the demand requirements of each outlet must be
satisfied. Therefore for retail node 5, the constraint equation is
x35 + x45 = 350 ................(v)
Similarly for nodes 6, 7, and 8, we get,
x36 + x46 = 200 ……...........(vi)
x37 + x47 = 400 ……...........(vii)
x38 + x48 = 450 ……...........(viii)
Linear Programming formulation,
Minimize Z = 4x13+7x14+6X22+3x24+7x35+4x36+3x37+5x38+5x45+6x46+7x47+8x48
Subject to constraints
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