Bridge Investment Decision$-$Making at the Network Level

Consider your homecity's bridge network shown below. Assume that project$-$level optimizations for each of the $12$ bridges on the network yielded the optimal actions for Year $2012$ as indicated by each bridge in the figure. Assume a bridge preservation budget of $\backslash (\backslash$$ $1\mathrm{.}5\backslash )$ million. It is sought to choose projects that maximize total network benefits within the given constraints. Costs of the projects are as follows: Superstructure replacement: $\backslash$$$200/$ft$^2$; Deck replacement: $\backslash$$$80/$ft$2$; Deck rehabilitation: $\backslash (\backslash$$ $20/\backslash$mathrm$\{$ft$\}^\{2\}\backslash ).$ The benefit of each project is the product of the traffic volume $($in thousands$)$ and the remaining service life given in Equation $1.$

$80-20(0\mathrm{.}992)\backslash (\{\}^\{$F$\}\backslash )$

$(1)$

Where $\backslash ($ P$=\backslash )$ total rehab. and maintenance expenditure in constant dollars per sq$.$ ft $.$

For example, for a remaining service life of $50$ years and average daily traffic of $70$ K vehicles per day, the benefit is: $3,500$ ADT$-$years.

Write the complete mathematical formulation for this optimization problem using appropriate symbols. Use a knapsack spreadsheet to find the optimal solution. Copy and paste your spreadsheet $($showing the solution$)$ on your HW submission sheet.