Part $2$: Goal Programming for Multi$-$Objective Decision MakingBuilding on the previous linear programming model, Green Tech Solutions now wants to incorporate multiple objectives into their decision$-$making process. They've identified several goals, some of which may conflict with each other. The company wants to use goal programming to find a satisfactory solution that balances these objectives.Goals:$1.$Profit Goal: Achieve a total profit of at least $$20,0002.$Production Volume Goal: Produce at least $1000$ units in total across all product lines$3.$Sustainability Goal: Limit raw material usage to $500$ kg or less$4.$Labor Utilization Goal: Use at least $350$ labor hours to maintain workforce stability$5.$High$-$Tech Focus Goal: Ensure that the number of energy$-$efficient laptops and smart thermostats produced is at least $30\%$ of the total productionConstraints:$($Keep all constraints from Part $1)$Goal Programming Formulation:For each goal, introduce positive and negative deviational variables $($d$+$ and d$-)$ to represent over$-$ and under$-$achievement of the goal$\mathrm{.}1.$Profit Goal: $30$x$1+150$x$2+80$x$3+10$x$4+200$x$5+$ d$1$n $-$ d$1$p $=200002.$Production Volume Goal: x$1+$ x$2+$ x$3+$ x$4+$ x$5+$ d$2$n $-$ d$2$p $=10003.$Sustainability Goal: $3$x$1+10$x$2+4$x$3+$ x$4+8$x$5+$ d$3$n $-$ d$3$p $=5004.$Labor Utilization Goal: $2$x$1+5$x$2+3$x$3+$ x$4+4$x$5+$ d$4$n $-$ d$4+=3505.$High$-$Tech Focus Goal: $0\mathrm{.}3$x$1+0\mathrm{.}7$x$2+0\mathrm{.}7$x$3+0\mathrm{.}3$x$4+0\mathrm{.}3$x$5+$ d$5$n $-$ d$5$p $=0$Objective Function:Minimize Z $=$ w$1($d$1$n $+$ d$1$p$)+$ w$2($d$2$n $+$ d$2$p$)+$ w$3($d$3$p $+$ d$3$n$)+$ w$4($d$4$n$)+$ w$5($d$5$n$)$Where w$1,$ w$2,$ w$3,$ w$4,$ and w$5$ are weights assigned to each goal based on their relative importance to the company.Tasks for Part $2$:$1.$Discuss the potential conflicts between the different goals and how goal programming can help address these conflicts$\mathrm{.}2.$Formulate the complete goal programming model, including all constraints from Part $1$ and the new goal constraints$\mathrm{.}3.$Assign weights to each goal based on a hypothetical priority structure for Green Tech Solutions. Justify your choice of weights$\mathrm{.}4.$Solve the goal programming model using appropriate software$\mathrm{.}5.$Interpret the results, focusing on: a$.$ Which goals were fully achieved, partially achieved, or not achieved? b$.$ What trade$-$offs were made between different goals? c$.$ How does the solution compare to the original linear programming solution?

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