Kloos Industries has projected the availability of capital over each of the next three years to be $$850,000,$ $$1,000,000,$ and $$1,200,000,$ respectively. It is considering four options for the disposition of the capital:

research and development of a promising new product

plant expansion

modernization of its current facilities

investment in a valuable piece of nearby real estate

Monies not invested in these projects in a given year will NOT be available for the following year's investment in the projects. The expected benefits three years hence from each of the four projects and the yearly capital outlays of the four options are summarized in the table below in millions of dollars.

Capital Outlay

Options Year $1$ Year $2$ Year $3$ Projected Benefits

New product R&D $0\mathrm{.}350\mathrm{.}550\mathrm{.}753\mathrm{.}2$

Plant expansion $0\mathrm{.}500\mathrm{.}500\mathrm{.}002\mathrm{.}8$

Modernization $0\mathrm{.}350\mathrm{.}400\mathrm{.}455\mathrm{.}2$

Real estate $0\mathrm{.}500\mathrm{.}000\mathrm{.}003\mathrm{.}6$

In addition, Kloos has decided to undertake exactly two of the projects. If plant expansion is selected, it will also modernize its current facilities.

Formulate this problem as a binary programming problem in order to maximize projected benefits $($in millions of dollars$).($Let xi $=1$ if project i is selected; $0$ otherwise.$)$

Max

s$.$t$.$

year $1$ available capital

year $2$ available capital

year $3$ available capital

exactly two projects

constraint on plant expansion and modernization

xi $=0,1$

Solve this problem.

$($x$1,$ x$2,$ x$3,$ x$4)=$