6. (total 15 points) A simple replacement problem for one kind of equipment having a 3-year lifetime can be represented by the following network diagram: 18 Each node represents a year in your planning horizon. An arc from node i to node j represents the possibility of buying equipment in year i and holding it until year j. Thus the decision variable corresponding to the arc is: Dij pieces of equipment bought in year i, and used until year j The associated cost is: Cij net cost of buying one piece of equipment in year i, operating and maintaining it through year j - 1, and selling it in year In the example represented by the diagram, the costs are shown adjacent to the arcs; for example, C12 is 10, and Cas is 26. All of the following questions pertain to the example. Where part of a constraint is given, fill in the rest of it. a). (3 points) Exactly one piece of equipment is to be purchased in the first year. What is the linear constraint corresponding to this requirement? b). (3 points) In years k = 2,3,4, the number of pieces purchased must equal the number sold so that you continue to have exactly one machine available. What is the linear constraint corresponding to this requirement in year 3? c). (3 points) The objective is to minimize the total four-year cost of operation. What is the linear expression for this total? d). (3 points) This equipment replacement problem is a linear program, which could be solved by the simplex method. However, it is a special kind of network op- timization problem, which has an even more efficient solution algorithm. What kind of network optimization problem is this? e). (3 points) What is the minimum total four-year cost in this case? In what years can you replace the equipment to achieve this minimum cost