Abstract
In this paper, we consider a multi-index constrained transportation problem (CTP) of axial constraints with bounds on destination requirements, source availabilities, and multiple types of commodities. The specified problem is converted into a related transportation problem by adding a source, a destination, and a commodity, making it equivalent to a standard axial sum problem. This related problem is transformed into a multi-index transportation problem that can be solved easily. The provided solution method is very useful for transporting heterogeneous commodities. A transportation model may sometimes have various capacity constraints on the flow between pairs of origins and destinations. Moreover, budgetary, political, and emergency situations may impair or enhance the flow between origins and destinations, making it critical for a manager to reevaluate allocations. These considerations have motivated us to explore the multi-index CTP with impaired and enhanced flow. We present several numerical examples to demonstrate the proposed algorithms.
Highlights
In the classical transportation problem (TP), a commodity is transported from each of m sources to each of n destinations
We have provided a solution method for a constrained transportation problem (CTP) with axial constraints with bounds on source availabilities, destination demands, and various commodities
We discussed solution methods for the case of impaired and enhanced flow in a CTP, which involves the addition of a flow constraint to a given CTP
Summary
In the classical transportation problem (TP), a commodity is transported from each of m sources to each of n destinations. We provide a solution method for a constrained transportation problem (CTP) with axial constraints with bounds on source availabilities, destination demands and various commodities. Let {xijk}I × J × K be corresponding feasible solutions of P1 and {yijk }I ×J ×K be corresponding feasible solution of problem P2 with objective function value as Z. These situations may require a government/country/company to reserve stock of some goods (e.g., weapons, medicines, grains etc.) This situation leads an impaired flow 3-dimensional CTP with bounds on source availabilities, destination demands and various types of commodities. The objective function values of P4 at a feasible solution {xijk}i ∈ I, j ∈ J, k ∈ Kand P5 at its corresponding corner feasible solution {yijk }i I , j J , k K are equal. Optimizing P4 is equivalent to optimizing P5, provided P4 has a feasible solution
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