116. For a device to work properly, three subsystems of the device must all function properly. To increasethe reliability of the device, spare units may be added to each system. It costs 100 TL to add a spareunit to system 1, 300 TL to system 2, and 200 TL to system 3. As a function of the number of addedspares (a maximum of two spares may be added to each system), the probability that each system willwork is given in Table. Use dynamic programming to maximize the probability that the device will workproperly, given that 600 TL is available for spare units. Identify and explain decision variables, stages,states and formulate a recursion to design the device with optimal reliability.Probability that a system workssystem 1Numberof sparessystem 3system 2.60.85.701.90.85.902.95.95.98117. You are planning on moving to a new house. You need to move n items of size aj , j-1,2,.,.n. You havebought m boxes (box i has size bi , i- 1,2,.,m). You need to rent a truck that will help fit all your items init. There are two sizes of trucks to choose from: the smaller truck has a size of Qi and the larger truckhas a size more than Qi. Since the rent of the smaller truck is cheaper, you would like to rent thesmaller truck if the move is possible with it. Formulate an integer (or mixed integer) programmingproblem in order to decide whether the move is possible with a single trip of the smaller truck. Notethat you can put multiple items in the same box and size is the only criterion determining if an item canbe put into a box.

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