To be coded in Matlab.

The task is to find the minimum cost of shipping sets of items, and also the positioning and allocation of items on craft for each of these sets. Determine the optimal solution for the Sample Mixes listed in Table 2, but also give consideration to your solution being applied to generic, and larger problems (such as more items, different item types, and different craft sizes.

Assume there are 4 different types of item (Table 1), and the mixes of these items needing to be shipped are in Table 2.

A standard craft is width 3 units and length 7 units, where the number of craft will be limited cost is expensive so this needs to be minimised also.

Formulate the problem as an ILP (Integer Linear Program) (a) Solving the relaxed problem, what (if anything) does this tell you about the actual solution? How does the running time change when the sample Mix is changed? (b) Solve the exact problem. Consider the running time as you change the Sample Mix. Compare to your results from (a). (c) Consider re-formulating so some items might be rotatable.

Reformulate the problem where a priority is given to certain items (there may be a benefit from some items arriving before others) (d) Formulate your problem as an ILP where item Ii has priority value vi associated; discuss the priority value and number of craft used trade-off when determining an objective function. (e) Let vi = i for i = 1, . . . , 4. Solve the ILP. How does this solution change from parts a, b and c? (f) To find an approximate (or exact) solution to the general problem, propose a heuristic method and implement and compare with previous results.

Item Width Length 2 13 3 TABLE 1. Items types and sizes. Item Width Length 2 13 3 TABLE 1. Items types and sizes