the total value shipped net of total costs. To solve this problem, the manager decides to set up another MIP and he redefines the following decision variables: Xij = 1 if item j is loaded on vani, otherwise X = 0. i = 1,.,4 and j = 1,...,5. Call X's "X variables". S; = 1 if item j is shipped, otherwise S; = 0. j = 1,...,5. Call S,'s "S variables". U = 1 if van i is used, otherwise U = 0. i = 1,.,4. Call U's "U variables". 25) (4 points) The objective of this MIP is: 26) (4 points) To express the capacity limit of van 2, the constraint is: 27) (4 points) For item 3, we need a constraint to establish the linkage among the relevant X variables, S variables and U variables (whichever variables are relevant to this item). This constraint is: Answer question 25)~27) based on the problem description below. Consider the problem setting of 21)-24) (Luckman Shipping Co.). After an internal meeting, the following changes are made compared to the situation in part A). First, the firm decides that both requirements ((i) and (ii)) are waived due to rearrangement of the shipping routes. Second, in addition to the existing van (indexed by 1) there are three other vans (indexed by 2,3,4) are available. Now, all four vans can be used, with each subject to a cost if the van is used. The information on the four vans are summarized below. Van Capacity (pound) 1 2000 2 1000 3 750 4 500 Cost (in $1000) 500 400 300 200 The firm needs to decide on a loading plan (in other words, which item to be shipped by which van). Note that it is not necessary that all items are shipped. The goal of the firm is to maximize Answer question 21) ~24) based on the problem description below. Luckman Shipping Co. is deciding on which items to load on a van of 2000 pounds capacity. The items awaiting shipment have values and weights specified by the table below. Value (in $1000) Weight (pound) 1000 Item 1 500 2 625 1200 345 3 300 400 4 350 600 5 450 850 The goal of the firm is to maximize the total value shipped. Also, the following requirements are stipulated by the logistics optimization department: (i) Item 1 can be shipped only if item 4 is shipped. (ii) Item 2 and item 3 cannot be shipped together. To solve this problem, the manager decides to set up a mixed integer program (MIP) and he defines the following decision variables: X = 1 if item j is chosen, otherwise X = 0; j = 1,... 5.