inf π ∈ Π( µ$_{0}$,µ$_{1}$ ) ∫ c ( y$_{0}$, y$_{1}$ ) dπ. (2) 12 The objective function in the minimization problem is linear in the joint distribution π and the constraint is that the joint distribution π should have fixed marginal distributions µ$_{0}$ and µ$_{1}$ . c ( y$_{0}$, y$_{1}$ ) and ∫ c ( y$_{0}$, y$_{1}$ ) dπ are called the cost function and the total cost , respectively. Kantorovich (1942) developed a dual formulation for the problem (2), which is a key feature of the optimal transportation approach. An optimal transportation problem was first formulated by Monge (1781) who studied the most efficient way to move a given distribution of mass to another distribution in a different location. Much later Monge’s problem was rediscovered and developed by Kantorovich. The optimal transportation problem of Monge- Kantorovich type is posed as follows. Let c ( y$_{0}$, y$_{1}$ ) be a nonnegative lower semicontinuous function on R 2 and define Π ( µ$_{0}$, µ$_{1}$ ) to be the set of joint distributions on$_{R}$ 2 that have µ$_{0}$ and µ$_{1}$ as marginal distributions. The optimal transportation problem solves Methods of establishing sharp bounds under this class of restrictions and fixed marginal distributions have remained unanswered in the literature. The central difficulty lies in finding out the particular joint distributions achieving sharp bounds among all joint distributions that have the given marginal distribu- tions and satisfy support restrictions. The next subsection shows that an optimal transportation approach circumvents this difficulty through its dual formulation. improved under MTR because MTR allows multiple mutually exclusive rectangles to be placed below the straight line Y$_{1}$ = Y$_{0}$ + δ . Figure 5: Makarov bounds are not best possible under MTR 2.3 Optimal Transportation Approach