this is a $$fixed cost$$ problem Hart Manufacturing makes three products. Each product requires manufacturing operations in

B $2\mathrm{.}001\mathrm{.}002\mathrm{.}50$ C $0\mathrm{.}250\mathrm{.}250\mathrm{.}25$ During the next production period, the labor$-$hours available are $450$ in department A$,350$ in department B$,$ and $50$ in department C$.$ The profit contributions per unit are $$25$ for product $1,$ $$28$ for product $2,$ and $$30$ for product $3.($a$)$ Formulate a linear programming model for maximizing total profit contribution. $($b$)$ Use Excel solver to solve the linear program formulated in part $($a$).$ How much of each product should be produced, and what is the projected total profit contribution? $($c$)$ After evaluating the solution obtained in part $($b$),$ one of the production supervisors noted that production setup costs had not been taken into account. She noted that setup costs are $$400$ for product $1,$ $$550$ for product $2,$ and $$600$ for product $3.$ If the solution developed in part $($b$)$ is to be used, what is the total profit contribution after taking into account the setup costs? $($d$)$ Management realized that the optimal product mix, taking setup costs into account, might be different from the one recommended in part $($b$).$ Formulate a linear program that takes setup costs into account. Management also stated that we should not consider making more than $175$ units of product $1,150$ units of product $2,$ or $140$ units of product $3($hint: you need to introduce $3$ binary variables, say Y$1,$ Y$2$ and Y$3.$ Each Yi variable will get a value of $1$ if the respective product $($Pi$)$ is produced and a value of $0$ if the product $($Pi$)$ is not produced$)($e$)$ Solve the model you formulated in part $($d$).$ How much of each product should be produced, and what is the projected total profit contribution? Compare this profit contribution to that obtained in part $($c$).$ See Problem #$2$ on next page OIM $310-$ HW$8$ OP $($OPTIONAL$)$ Fixed Costs & Asset Selection Due: Fri, Nov $29,11$:$59$pm Page $2$ Problem #$214$ pts $($not from text$)($this is an asset selection problem that takes into consideration fixed costs$)$ Utopia Company operates a plant in Avalon with a monthly capacity of $35,000$ units and a plant in Bridgetown with a monthly capacity of $40,000$ units. Product is shipped to regional distribution centers located in Riverdale, Seaside and Teatown. The long$-$range planning group at Utopia reevaluated their forecasts and determined that due to an anticipated increase in demand in the next $5$ years, the company should consider increasing capacity by constructing a new plant in one or more of the following cities: Cedar, Daphne, Elm or Florence. The estimated monthly fixed cost and the monthly capacity for the four proposed plants are as follows: Proposed Plant Location of Plant Monthly Fixed Cost Monthly Capacity C Cedar $$20,00010,000$ D Daphne $$30,00020,000$ E Elm $$40,00030,000$ F Florence $$60,00040,000$ The anticipated monthly demands at the distribution centers are as follows: Distribution Center $($DC$)$ Location of DC Monthly Demand $1$ Riverdale $45,0002$ Seaside $30,0003$ Teatown $35,000$ The shipping cost per unit from each plant to each distribution center is shown below Distribution Center Plant Site $123$ A $523$ B $843$ C $975$ D $1042$ E $654$ F $789$ A diagram of the network associated with the above problem is shown on the next page Let Xij $=$ the units shipped from plant i to distribution center j$,$ with i $=$ A$,$ B$,$ C$,$ D$,$ E or F and j $=1,2,3.$ For example, XD$1$ designates the # of units shipped from Plant D to Distribution Center $1.$ OIM $310-$ HW$8$ OP $($OPTIONAL$)$ Fixed Costs & Asset Selection Due: Fri, Nov $29,11$:$59$pm Page $3($a$)$ Why is the company considering opening at least one new plant? $($b$)$ How many shipment variables will the LP formulation have? $($That is$,$ how many Xij variables are included in the formulation?$)($c$)$ Why do we need to introduce binary variables YC$,$ YD$,$ YE and YF$?$ Do we need a binary variables YA and YB$?$ Why? $($d$)$ Formulate a linear programming model for minimizing total costs $($shipping costs$+$fixed costs$)$; you should have a total of $9$ constraints. You can use steps $($i$)-($iv$)$ below as a guide for writing the LP i$.$ Use suitable variables to express the objective function $($minimize shipping costs $+$ fixed costs$).$ ii$.$ Write the demand constraints for Distribution Centers $1,2$ and $3.$ iii. Write the supply constraints for plants A and B$.$ iv$.$ Use suitable variables to write the supply constraints for plants C$,$ D$,$ E and F$.($e$)$ Write a constraint to express a requirement that the company will open only one new plant $($f$)$ If the company will open only one plant, which plant do you think it will be C$,$ D$,$ E or F$?$ Why? $($g$)$ Write a constraint to express a requirement that the company will either open both plants C and D or neither of them $($that is$,$ if plant C opens then plant D has to open as well and if plant D is open then Plant C has to open too$)$