Problems 2454.38 Transform the following LP problem into the form required by Karmarkar’s method:
Minimizef= x 1 +x 2 +x 3subject tox 1 +x 2 −x 3 = 4
3 x 1 −x 2 = 0
xi≥ 0 , i= 1 , 2 , 34.39 A contractor has three sets of heavy construction equipment available at both New York
and Los Angeles. He has construction jobs in Seattle, Houston, and Detroit that require
two, three, and one set of equipment, respectively. The shipping costs per set between
citiesiandj(cij) are shown in Fig. 4.5. Formulate the problem of finding the shipping
pattern that minimizes the cost.
4.40
Minimizef (X)= 3 x^21 + 2 x^22 + 5 x^32 − 4 x 1 x 2 − 2 x 1 x 3 − 2 x 2 x 3
subject to3 x 1 + 5 x 2 + 2 x 3 ≥ 10
3 x 1 + 5 x 3 ≤ 15
xi≥ 0 , i= 1 , 2 , 3by quadratic programming.4.41 Find the solution of the quadratic programming problem stated in Example 1.5.
4.42 According to elastic–plastic theory, a frame structure fails (collapses) due to the formation
of a plastic hinge mechanism. The various possible mechanisms in which a portal frame
(Fig. 4.6) can fail are shown in Fig. 4.7. The reserve strengths of the frame in various
failure mechanisms (Zi) can be expressed in terms of the plastic moment capacities of the
hinges as indicated in Fig. 4.7. Assuming that the cost of the frame is proportional to 200
times each of the moment capacitiesM 1 , M 2 , M 6 , andM 7 , and 100 times each of the
moment capacitiesM 3 , M 4 , andM 5 , formulate the problem of minimizing the total cost
Figure 4.5 Shipping costs between cities.