Algorithms for minimizing total cost, bottleneck time and bottleneck shipment in transportation problems

Abstract

We consider the transportation problem of determining nonnegative shipments from a set of m warehouses with given avnilabilities to a set of n markets with given requirements. Three objectives are defined for each solution: (i) total cost, TC, (ii) bottleneck time, BT (i.e., maximum transportation time for a positive shipment), and (iii) bottleneck shipment, S B (i.e., total shipment over routes with bottleneck time). An algorithm is given for determining all efficient (pareto-optimal or nondominated) ( T C , B T ) solution pairs. The special case of this algorithm when all the unit cost coefficients are zero is shown to be the same as the algorithms for minimizing BT provided by Szwarc and Hammer. This algorithm for minimizing BT is shown to be computationally superior. Transportation or assignment prohlems with m=n=100 average about a second on the UNIVAC 1108 computer ( F O R T R A N T)) t o the threshold algorithm for minimizing BT. The algorithm is then extended, t o provide not only all the efficient ( T C , BT) solution pairs but also, for each such BT, all the efficient (TC, S B ) solution pairs. The algorithms are based on the cost operator theory of parametric programming for the transportation problem developed by the authors.

An efficient primal approach to bottleneck transportation problems

“…For example, the bottleneck solution required about 40% less CPU time than standard assignment problems. The same kind of conclusion has been drawn by Srinivasan and Thompson [20]. The data in Table 1 indicate that solution times are sensitive to the change in tii for problems which are highly dense.…”

“…The finiteness and convergence of this algorithm has been previously argued by others [20,21]. The validity of step 4 follows directly from the poly-w technique of Charnes and Cooper [4].…”

“…The algorithms of Hammer, Szwarc, and Srinivasan and Thompson are essentially equivalent. Srinivasan and Thompson [20] coded the Szwarc-Hammer primal approach exploiting some methods and rules found effective on the standard transportation problem. Their computational results indicate that the specialized primal approach is roughly three times as fast as the threshold algorithm in solving bottleneck transportation problems.…”

“…A thorough review of eastern European literature on the problem can be found in the paper by Szwarc [21]. More recently Hammer [15], Szwarc [21], and Srinivasan and Thompson [20] have developed primal simplex approaches; Garfinkel and Rao [9] have developed a "threshold algorithm" utilizing the FordFulkerson [7] algorithm. Garfinkel and Rao found that their threshold algorithm was computationally superior to the primal approach of Algorithm I above when applied to bottleneck transportation problems.…”

“…Areas of application for linear bottleneck problems include military operations and the transportation of perishable goods [ 151 as well as assembly line balancing problems [20].…”