Challenge $1$: Operations Research and Simulation

Total Points: $25$ Help asap

The Jordan Phosphate Mines Company $($JPMC$)$ operates three mines in Jordan: Eshidiya, AI Hassa, and Wadi Al$-$Abiad mines. It supplies phosphate to four different plants. The cost of shipping phosphate from each mine to each plant, the capacity at each of the three mines, the demand at each plant and the percentage of Fluoride and sulfur content per ton of phosphate at each mine are shown in the following table: Plant $($$$/$ton$)$

$\backslash$table$[[$Mine$,,,$Plant $($$$/$ton$)],[$Fluoride $\%,$Sulfur $\%,1,2,3,4,\backslash$table$[[$Capacity$],[($tons$)]]],[$Eshidiya$,9,6,7,9,10,12,220],[$Al Hassa,$5,4,9,7,8,12,170],[\backslash$table$[[$WadiAl$-],[$Abiad$]],4,3,11,14,5,7,280],[$Demand $(,(,110,160,90,180,,]]$

The cost of mining and processing phosphate is $$62$ per ton at Eshidiya, $$67$ per ton at Al Hassa, and $$75$ per ton at Wadi Al$-$Abiad.

Each plant has different cleaning equipment. Plant $1$ requires that the phosphate it receives have no more than $6\%$ Fluoride and $5\%$ sulfur; plant $2$ phosphate can have no more than $5\%$ Fluoride and sulfur combined; plant $3$ can have no more than $5\%$ Fluoride and $7\%$ sulfur; and plant $4$ can have no more than $6\%$ Fluoride and sulfur combined. JPMC wants to determine the amount of phosphate to produce at each mine and ship to its customers that will minimize its total cost.

Required:

A$.$ Formulate a linear programming model for this problem. Define you decision variables clearly! $(12$ points$)$

B$.$ If the minimum and maximum Truck loads are $0\mathrm{.}5$ and $5$ tons respectively, include this condition into your model. $($Define new variables if necessary!$)(6$ points$)$

C$.$ Solve the model, compare your results in part $1$ and $2.(7$ points$)$P