**7.2 ATSRASSR algorithm**

For small number of nodes, ATSRAorg algorithm performs well. But as the number of nodes increases the time taken by the algorithm to run becomes considerable. ATSRASSR is proposed to improve performance for large number of nodes. It is based on finding a lower bound for the cost minimization problem formulated in Equation (7). The basic idea is to replace a "difficult" minimization problem by a simpler minimization problem whose optimal value is at least as small as cost4-tier.

For the "relaxed" problem to have this property, there are two possibilities.

1. Enlarge the set of feasible solutions so that one optimizes. If the set of feasible solutions is represented by P, then it means to find �� such that ���� .

OR

2. Replace the minimum objective function of Equation (7) by a function that has the same or a smaller value everywhere.

For the ATSRASSR, we have chosen the first approach. We formulate a new optimization problem called the relaxation of the original problem, using the same objective function but a larger feasible region �� that includes P as a subset. Because �� contains P, any solution which belongs to P, also belongs to �� as well. This relaxed cost is denoted by cost4-tier-relaxed.. Tenlarge the set of feasible solutions, constraint relaxation technique is used. In ATSRASSR, the constraint relaxation technique is used at the Architectural Template selection stage only. Once an Architectural Template has been chosen, exact LP formulation is used for resource allocation. It is thus named as ATSRA Single Stage Relaxation (SSR) or ATSRASSR, as the constraint relaxation is applied only to first stage of the algorithm.


The ATSRASSR starts by calculating the costs associated with 2- tier Architectural Templates 2a and 2b, using Equations (3) and (4). The minimum of these two is called as costmin. For 4 tier Architectural Templates, instead of calculating exact cost4-tier, cost4-tier-relaxed is calculated. For constraint relaxation, the fifth constraint (i.e. ��� ≤ ��) is dropped.
