Suppose you have one machine and a set of n tasks a1; a2; : : : ; an. each task aj has a processing time pj , a pro t pj , and a deadline dj . if you complete a task aj by deadline dj , you receive a pro t pj (otherwise, you receive no pro t). you wish to nd a schedule that completes all the tasks and derives the maximum pro t. (a) state the decision version of this problem. (b) show that it is np-complete by reducing the knapsack problem to it. (c) give a polynomial-time algorithm for the decision problem, assuming that all processing times are integers from 1 to n

Acommercial grade cubical freezer, 4 m on a side, has a composite wall consisting of an exterior sheet of 5.0-mm thick plain carbon steel (kst= 60.5 w/m k), an intermediate layer of 100-mm thick polyurethane insulation (kins 0.02 w/m k), and an inner sheet of 5.0- mm thick aluminium alloy (kal polyurethane insulation and both metallic sheets are each characterized by a thermal contact resistance of r 2.5 x 104 m2 k/w. (a) what is the steady-state cooling load that must be maintained by the refrigerator under conditions for which the outer and inner surface temperatures are 25°c and -5°c, respectively? (b) for power saving purpose, which wall material should be increased/reduced in. thickness in order to reduce 50% of the cooling load found in part (a)? redesign the thickness of the proposed material. 177 w/m-k). adhesive interfaces between the q=575.93 w