Let 1,2, , 2 be the set of the first 2 natural numbers, where is a positive integer greater than 1 Suppose that all the numbers in will be arranged in a single file Once the numbers are arranged, we do the following operation for each number in the line that is not on either end, we compute its positive difference from the number before it and the number after it We then record each of these differences (thats two differences per number) For each number that is on an end of the line, we compute and record its positive difference from the single number that is adjacent to it (thats only one difference for each of these numbers) Formulate an ILP model (avoiding constraint deactivator variables) that will help you obtain a permutation that will maximize the sum of all the recorded differences Use an intuitive algorithm to obtain an optimal permutation for the case that 2, and briefly reflect on the uniqueness of your optimal arrangement (i e whether other optimal arrangements exist) Discuss how your optimal solution can be obtained by using the model you formulated in (a) (i e fully explain what the model does in order to get an optimal arrangement) State the values of the decision variables, as well as the optimal value Suppose that the numbers will instead be arranged in a by square array, with one number in each cell Then for each number in the array, we record its positive difference from each of its neighbors (right, up, left, and down only) Thus, for some numbers there are 4 differences to record while for some others, there are less If the goal is still to obtain a permutation that maximizes the sum of the recorded differences, what will be the ILP formulation (again, avoiding constraint deactivator variables) Do as in (b) but this time, the arrangement will be on a 2 by 2 array What are the values of your decision variables, and what is your value Do your decision variables satisfy all of your constraints Finally, suppose that we again arrange the 2 numbers in a single file, and the positive differences are recorded as described in (a) We note the highest among these positive differences, and then we count the number of times this highest difference appears Formulate an ILP model (again, avoiding constraint deactivator variables) that will help you obtain a permutation that maximizes this occurrence count Your model must be able to isolate the noted maximum difference and the occurrence count Find an optimal arrangement for the situation in (e) when 2 Do other optimal arrangements exist What are the values of your decision variables, and what is your value Do your decision variables satisfy all of your constraints

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Let = {1,2, ... , ^2 } be the set of the first ^2 natural numbers, where is a positive integer greater than 1. Suppose

Let = {1,2, ... , ^2 } be the set of the first ^2 natural numbers, where is a positive integer greater than 1.

Suppose that all the numbers in will be arranged in a single file. Once the numbers are arranged, we do the following operation: for each number in the line that is not on either end, we compute its positive difference from the number before it and the number after it. We then record each of these differences (thats two differences per number). For each number that is on an end of the line, we compute and record its positive difference from the single number that is adjacent to it (thats only one difference for each of these numbers). Formulate an ILP model (avoiding constraint deactivator variables) that will help you obtain a permutation that will maximize the sum of all the recorded differences.
Use an intuitive algorithm to obtain an optimal permutation for the case that = 2, and briefly reflect on the uniqueness of your optimal arrangement (i.e. whether other optimal arrangements exist). Discuss how your optimal solution can be obtained by using the model you formulated in (a) (i.e. fully explain what the model does in order to get an optimal arrangement). State the values of the decision variables, as well as the optimal value.
Suppose that the numbers will instead be arranged in a -by- square array, with one number in each cell. Then for each number in the array, we record its positive difference from each of its neighbors (right, up, left, and down only). Thus, for some numbers there are 4 differences to record; while for some others, there are less. If the goal is still to obtain a permutation that maximizes the sum of the recorded differences, what will be the ILP formulation (again, avoiding constraint deactivator variables)?
Do as in (b) but this time, the arrangement will be on a 2-by-2 array. What are the values of your decision variables, and what is your -value? Do your decision variables satisfy all of your constraints?
Finally, suppose that we again arrange the ^2 numbers in a single file, and the positive differences are recorded as described in (a). We note the highest among these positive differences, and then we count the number of times this highest difference appears. Formulate an ILP model (again, avoiding constraint deactivator variables) that will help you obtain a permutation that maximizes this occurrence count. Your model must be able to isolate the noted maximum difference and the occurrence count.
Find an optimal arrangement for the situation in (e) when = 2. Do other optimal arrangements exist? What are the values of your decision variables, and what is your -value? Do your decision variables satisfy all of your constraints?