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In this paper, we consider the fully fuzzy unbalanced transportation problem in which the total availability/production is more than the total demand and propose a method to solve it. Such problems are usually solved by adding a dummy destination. Since the dummy destination has no existence in reality, the excess availability is not transported at all and is held back at one or more origins. The method proposed in this paper gives the additional information that to which of the destination(s) the excess availability be transported for future demand at minimum cost. The advantage of the proposed method over the existing method is that the fuzzy optimal solution obtained does not involve the dummy destination. The method has been illustrated with the help of an example.

In the present research, the AZ31 alloy is machined by wire-cut electric discharge machining (WEDM). The experiments were designed according to the Box-Behnken design (BBD) of response surface methodology (RSM). The input process variables, namely servo feed (SF), pulse on-time (Ton), servo voltage (SV), and pulse off-time (Toff), were planned by BBD, and experiments were performed to investigate the cutting rate (CR) and recast layer thickness (RCL). The analysis of variance (ANOVA) was performed to determine the influence of machining variables on response characteristics. The empirical models developed for CR and RCL were solved using Multi-Objective Particle Swarm Optimization (MOPSO). Pareto optimal front is used for the collective optimization of CR and RCL. The optimal solution suggested by the hybrid approach of RSM-MOPSO is further verified using a confirmation test on the random setting indicated by the hybrid algorithm. It is found that the minimum RCL (6.34 µm) is obtained at SF: 1700; SV: 51 V; Toff: 10.5 µs; and Ton: 0.5 µs. However, maximum CR (3.18 m/min) is predicted at SF: 1900; SV: 40 V; Toff: 7 µs; and Ton: 0.9 µs. The error percentage of ±5.3% between the experimental results and predicted solutions confirms the suitability of the proposed hybrid approach for WEDM of AZ31.

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The solid transportation problem is an important generalization of the classical transportation problem as it also considers the conveyance constraints along with the source and destination constraints. The problem can be made more effective by incorporating some other factors, which make it useful in real life situations. In this paper, we consider a fully fuzzy multi-objective multi-item solid transportation problem and present a method to find its fuzzy optimal-compromise solution using the fuzzy programming technique. To take into account the imprecision in finding the exact values of parameters, all the parameters are taken as trapezoidal fuzzy numbers. A numerical example is solved to illustrate the methodology.