Explore sensitivity to the marginal rates. For example, the marginal rate for "economy" $($expressed as $$/$lb$)$ seems really high$$perhaps the marginal rate of $$20$ per pound is too large.

Make a plot, with the MRS for $$/$lb on the x$-$axis and the score for each alternative on the y$-$axis. Start at MRS $$2/$lb and plot the scores at steps of $$2/$lb up to the value of $$20/$lb$.$

At what value of the marginal rate for $$/$lb $($currently at $$20/$lb$)$ does the initially preferred camper no longer rank the highest? $($You can estimate this from the graph$).$ How would your recommendation change at this value?

Another interesting way to visualize the sensitivity of your findings is to take two MRS values at a time$$the ones you feel you are the least sure about and$/$or the ones that are most critical to the client. Take the MRS of $$/$cubic foot of interior space and $$/$lb$.$ Make an x$-$y plot with each of these MRS values on one axis $($put $$/$cubic foot on the y$-$axis, $$/$lb on the x$-$axis$).$

For $$/$cubic foot, start at $$5/$cubic foot and move at $$5/$cubic foot increments up to $$30/$cubic foot interior space. For $$/$lb$,$ start at $$2/$lb and move at $$2/$lb increments up to $$20/$lb overall weight.

On the plot itself $($see below$),$ indicate the most preferred alternative at each x$-$y point on the graph. In the end, the plot will show the regions of MRS values where each alternative is most preferred. You may want to use a color code or shading pattern to show the regions where different campers are the preferred alternative.

Given

Weight: $$20/$lb$.$

Convenience: $$2,000$ for $1$ rank improvement.

Safety Rating: $$1,500$ per rating point.

Interior Space: $$15/$ft$^3.$

Length: $$500/$ft$.$

Data in Feet $($Decimal$)$ Format

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Mobility $($Length: Shorter is Better$)$ Economy $($Weight: Lighter is Better$)($lb$)$ Safety Rating $($Higher is Better$)$ Comfort $($Interior Dimensions: Larger is Better$)$ Convenience $($Ranking; $1=$Best$)$ Price $($Retail Purchase Price: Lower is Better$)($$$)$

Casita $13188060427\mathrm{.}71414000$

Sport $16\mathrm{.}42860100692\mathrm{.}74142000$

rPob $18\mathrm{.}4220580797\mathrm{.}16217000$

Scamp $13135065406\mathrm{.}25311200$

Maximum $18\mathrm{.}42860100797\mathrm{.}16442000$

Minimum $13135060406\mathrm{.}25111200$

Delta $($Max$-$Min$)5\mathrm{.}4151040390\mathrm{.}91330800$

Normalization

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Mobility Economy Safety Comfort Convenience Price

Casita $00\mathrm{.}3509933800\mathrm{.}05489754710\mathrm{.}090909091$

Sport $0\mathrm{.}62962963110\mathrm{.}73287969101$

rPob $10\mathrm{.}566225170\mathrm{.}510\mathrm{.}3333333330\mathrm{.}188311688$

Scamp $000\mathrm{.}12500\mathrm{.}6666666670$

Normalization $($With Safety and Comfort Adjusted$-$Lower is Better$)$

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Mobility Economy Safety Comfort Convenience Price

Casita $00\mathrm{.}3509933810\mathrm{.}94510245310\mathrm{.}090909091$

Sport $0\mathrm{.}62962963100\mathrm{.}26712030901$

rPob $10\mathrm{.}566225170\mathrm{.}500\mathrm{.}3333333330\mathrm{.}188311688$

Scamp $000\mathrm{.}87510\mathrm{.}6666666670$

MRS Calculations

MRSxy $1/$MRSxy

Economy$/$Convenience MRSbe $100/11000\mathrm{.}01$

Comfort$/$Mobility MRSda $33\mathrm{.}33/133\mathrm{.}330\mathrm{.}030003$

Comfort$/$Convenience MRSde $133\mathrm{.}33/1133\mathrm{.}330\mathrm{.}007500188$

Price$/$Safety MRSfc $1500/115000\mathrm{.}000666667$

Economy$/$Price MRSbf $1/200\mathrm{.}0520$

Score $($Lower is Better$)$ and Rank

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Mobility Economy Safety Comfort Convenience Price Total Score Rank

Relative Weight $111\mathrm{.}186024211\mathrm{.}63492872\mathrm{.}1719394162\mathrm{.}22238891111\mathrm{.}40826867$

Normalized Weight $0\mathrm{.}025890940\mathrm{.}289616680\mathrm{.}301239240\mathrm{.}0562335520\mathrm{.}0575397370\mathrm{.}295370796$

Casita $00\mathrm{.}3509933810\mathrm{.}94510245310\mathrm{.}0909090910\mathrm{.}540430873$

Sport $0\mathrm{.}62962963100\mathrm{.}267120309010\mathrm{.}61631034$

rPob $10\mathrm{.}566225170\mathrm{.}500\mathrm{.}3333333330\mathrm{.}1883116880\mathrm{.}415300492$

Scamp $000\mathrm{.}87510\mathrm{.}66666666700\mathrm{.}358177711$

Please also explain the steps for creating the plot