Sensitivity Analysis of Costs in a Transpostation Problem

Let’s consider m providers (distributors, manufacturers, etc.) F1 − Fm of an identical product that they possess in quantities a1 − am as well as n consummers or beneficiaries (users) B1 − Bn whose needs are b1 − bn . Let us denote by xij the quantity that is transported (transferred, allocated) from Fi to Bj and with cij as unit transportation costs from Fi to Bj. The cost of transportation for the quantity xij from Fi to Bj will equal cij ·xij and for all the providers and all the benficiaries, the overall cost will equal f = ∑m i=1 ∑n j=1 cij · xij which must be minimised. Obviously, at the level of provider Fi the total quantity provided will equal ∑n j=1 xij ≤ ai, 1 ≤ i ≤ m , and everything that the beneficiary Bj can get is . We search for that transportation (or allocation) option which satisfies all partners Fi şi Bj and leads to a minimum total cost. Keeping in mind the considerations and example presented, wecan sum up the fact that, in a general form, a linear model of transportation type may be written as follows:  [opt]f = ∑m i=1 ∑n j=1 cij · xij ∑n j=1 xij ≤ ai, 1 ≤ i ≤ m ∑m i=1 xij ≥ bj, 1 ≤ j ≤ n xij, i = 1, m, j = 1, n