An engineering firm won a $$2,000,000$ contract to develop a blade spar for helicopters. The blade spar is a metal tube that runs the length of and provides strength to the helicopter blade.$$ Due to the special length and size of the blade, the firm is unable to produce a single$-$piece blade spar of the required dimensions using existing extrusion equipment and material. The engineering department has prepared two alternatives for developing the blade spar: $(1)$ sectioning or $(2)$ an improved extrusion process.$$ The firm must decide which process to use.$($Backing out of the contract at any point is not an option.$)$ The risk report has been prepared by the engineering department.$$ The information from this report is explained next.$$

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The sectioning option involves joining several shorter lengths of extruded metal into a blade spar of sufficient length.$$ This work will require extensive testing and rework over $12$ months at a total cost of $$1,800,000.$ Although this process will definitely produce an adequate blade spar, it merely represents an extension of existing technology.$$

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To improve the extrusion process, on the other hand, it will be necessary to perform two steps: $(1)$ improve the material used, at a cost of $$300,000$ and $(2)$ modify the extrusion press, at a cost of $$960,000.$ The first step will require six months of work, and if this first step is successful, the second step will require another six months of work.$$ If both steps are successful, the blade spar will be available at that time, that is$,$ a year from now.$$ The engineers estimate that the probabilities of succeeding in Steps $1$ and $2$ are $0\mathrm{.}90$ and $0\mathrm{.}75,$ respectively.$$ However, if either step is unsuccessful $($which will be known only in six months for Step $1$ and in a year for Step $2),$ The firm will have no alternative but to switch to the sectioning process $$ and incur the sectioning cost on top of any costs already incurred.$$

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Development of the blade spar must be completed within $18$ months to avoid holding up the rest of the contract.$$ If necessary, the sectioning work can be done on an accelerated basis in six months, but the cost of sectioning will then increase from $$1\mathrm{.}8$ million to $$2\mathrm{.}4$ million.$$ The Chief Engineer, Dr$.$ Whitmann, wants to try developing an improved extrusion process.$$ Dr$.$ Whitmann reasons that this is not only cheaper $($if successful$)$ for the current project, but its expected side benefits for future projects could be sizable.$$ Although these side benefits are difficult to gauge, Dr$.$ Whitmann$$s best guess is an additional $$2$ million.$($These side benefits are obtained only if both steps of the modified extrusion process are completed successfully.$)$

$1.$ How much would the firm be willing to pay, right now, for perfect information about both steps of the improved extrusion process? This information would tell the firm, right now, the ultimate success or failure outcomes of both steps.