An optimization of solid transportation problem with stochastic demand by Lagrangian function and KKT conditions

In this paper, a stochastic solid transportation problem (SSTP) is constructed where the demand of the item at the destinations are randomly distributed. Such SSTP is formulated with profit maximization form containing selling revenue, transportation cost and holding/shortage cost of the item. The proposed SSTP is framed as a nonlinear transportation problem which is optimized through Karush–Kuhn–Tucker (KKT) conditions of the Lagrangian function. The primary model is bifurcated into three different models for continuous and discrete demand patterns. The concavity of the objective functions is also presented here very carefully. Finally, a numerical example is illustrated to stabilize the models