Answer Happy

A company who works with car manufacturers produces and sells engines of various types and sizes. There is one particular style of engine that comes in three different sizes (call them "size 1," "size 2," and "size 3"). The contractor processes raw materials to procude the engines. Processing a pallet of raw materials costs \$700 per pallet and produces 2 size 1 engines, 1 size 2 engine, and 1 size 3 engine. Size 1 engines sell for \$160 each, size 2 engines sell for \$210 each, and size 3 engines sell for \$300 each. Engines (of any size) can be stored from month to month. Keeping engines in good working condition costs $5×(engine size) per engine per month (i.e., \$5 for size 1,\$10 for size 2, \$15 for size 3). The table below shows the maximum amount of each engine the contractor can sell in the next three months as well as the maximum number of pallets of raw materials it can process in each month. Demand that is not met in the month it occurs cannot be carried over to a later month (i.e., no backlogging is allowed - so the demand amounts in the table represent the maximum that can be sell - it may not be possible to sell this much). The contractor currently has 10 size 1 engines and 2 size 3 engines (no size 2 engines) in stock, and wishes to have at least that much - plus at least one size 2 engine - in stock at the end of month 3. Max raw Month (t) materials (RM) Max size 1 (M1) Max size 2 (M2) Max size 3 (M3) 1 2 3 15 20 12 30 15 20 12 18 13 15 17 21 Formulate a linear programming model to help the contractor maximize its profits over the next three months. Solve the model in Julia/JuMP and display the production plan (how many of each engine size to produce, store, and sell in each month). (b) Now suppose the contractor would like to allow backlogging of demand. Demand not met in a month may be met in a future month (backlogged) or may be lost. The contractor pays a fee of \$19 per engine per month backlogged, regardless of engine size. Modify your code from part (a) to include the ability to backlog. The same ending stock requirements should be enforced in part (b) and no engines should be backlogged in the final month. How, if at all, does this change the optimal objective value and solution? Speculate as to the reason for the change (or lack of change).