Optimization Techniques — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

Semester VIII retail stores A, B and C. The number of units available at factories X and Y are 200 and 300 1.(a) A firm has two factories X and Y. It wishes to ship its products from its two factories to three respectively, while the demands at retail stores A, B and C are 100, 150 and 250 respectively. Rather transshipment. The unit transportation cost (in rupees) is given in the following table. Find the optimal than shipping directly from factories to retail stores, it is added to investigate possibility of shipping schedule. Source Destination x A B weight w; in tons and the unit revenue in thousands of dollars ri, for item. How many units of each (b) A 4-ton vessel can be loaded with one or more three items. The following table gives the unit item be loaded to maximize the value? Item Weight Value 2. (a) Suppose we want to select the shortest highway route between two cities. The network in given figure provides the possible route between the starting city node "l'and the destination city at node '7'. Solve by the method of forward recursive equation and use it to find the optimal solution for the following data: (b) In a cargo loading problem, there are 4 items of different weights/unit and different values/unit as given below: Item Weight Value The maximum cargo load is restricted to 17. How many units of each item be loaded to maximize the value? 3. (2) Solve the following ILPP by using Gomory's Cutting Plane Method Max Z = 2xq†*z subject to 1+x2=5 ×,, Xz ≥ 0 and both are integers. (b) Consider the following ILLP 1, Xz ≥ 0 and both are integers (i) Convert the above defined problem into the standard Linear programming problem. (ii) Write the two LPP's by branching from z and find the next branching variable, if the initial solution is xy = 0 and xz = 3, the maximum value of Z = 7, with the last simplex table as follows: 4. (a) Consider the following ILPP Xy, X2 ≥ 0 and both are integers. (i) Find the optimal solution of the LP by using the simplex method. (ii) Solve the given ILPP graphically. (b) Consider the following optimal simplex tableay of the associated Linear Programming Problem for a given AILP (with maximization from and Xz, X4 as slack variables) with Max Z = 18.75. Generate the Gomory's cut through Z and hence solve the given AILP. 5. (a) (i) Define a convex function on a convex set S £ R". Is the function f(x) = x?,x € R, convex? Justify your answer using definition. (ii) Suppose that S C R" be a convex set and f: S → R. Prove that f is a convex function on S if and only if its epigraph Ef is a convex set. (b) (i) Is every convex function continuous? Explain with an example. (i) Using Charnes and Cooper algorithm, solve the following linear fractional programming problem: Max Z = -x1+X2+1 -subject to X1+ x2 ≤1 6. (a) (i) For the following problem, check whether it is a convex programming problem with justification: (ii) Define a quasilinear function on a convex set S C R". Give an example of a quasilinear function.