Problem $1$:

Homework #$1$Daimler Trucks North America $($DTNA$)$ is an automotive industry manufacturer of commercial vehicles headquartered in Portland, Oregon. DTNA also owns Detroit Diesel Corporation $($DDC$),$ a diesel engine manufacturing plant located in Detroit, Michigan, USA. DDC produces heavy$-$duty engines for the on$-$ highway and vocational commercial truck markets and has the capacity of producing $200$ engines per day. All engines are shipped from DDC to Portland by train. The train company operates trains daily on the national rail network. Each engine weighs on average $1000$ pounds. There is a limit on the weight that can be shipped across each link of the train network daily. The network below shows the daily carrying capacity $($in tons $=2000$ pounds$).$DNTA suspects that the current rail network may not have enough capacity to distribute all daily production to Portland. Therefore, it wants to know the maximum possible number of engines that can be shipped out of Detroit to Portland each day.

a$)$ Formulate this problem as a math program $($define all constraints, the objective function, and the variables algebraically$).$

b$)$ What type of math program is this? If you will use Excel Solver to solve it$,$ please also state the Solution Method $($e$.$g$.,$ GRG Nonlinear, Simplex LP$,$ or Evolutionary$).$ If you will use Pyomo to solve it$,$ please state what solver you will use $($e$.$g$.,$ CPLEX, Gurobi, KNITRO, GLPK$).$