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SOLUTION(USING(SIMPLEX(METHOD(
(
If#you#would#like#to#further#talk#about#the#solution#methods#in#the#classroom,#you#could#show#how#
to#solve#this#problem#using#simplex#method#as#follows:#
#
In#our#diet#problem,#we#are#looking#at#a#“standard”#minimization#problem.##What#does#the#
“standard#minimization”#mean?##It#means#that#the#constraints#are#all#linear;#the#constants#are#nonC
negative;#the#inequality#symbols#are#greater#or#equal#to;#the#nonCnegativity#constraints#are#placed#
on#our#variables;#and,#we’re#minimizing.##After#recognizing#that#it’s#a#standard#minimization#
problem,#in#order#to#apply#the#simplex#method,#we#need#to#convert#it#to#a#standard#maximization#
problem.##To#do#that,#we#will#start#writing#a#matrix,#which#comes#from#the#coefficients#and#
constants#of#the#constraints#as#well#as#the#coefficients#and#constant#of#the#objective.##
#
A#=# #
#
Then,#we#will#find#the#transpose#of#this#matrix#by#interchanging#its#rows#and#columns.#
#
AT#=# #
#
We#can#see#that#the#rows#of#this#matrix#are#the#columns#of#the#first#matrix.##This#new#matrix#is#a#
maximization#problem#so#that#the#corresponding#maximization#problem#is:#
#
Maximize#w#=#135.5#y1#+#45.5#y2#+#32.5#y3#+#1,000#y4##
#
subject#to#the#constraints#
#
12#y1#+#3#y2#+#9#y3#+#171#y4#<=###2#
33#y1#+#6#y2#+####y3#+#150#y4#<=#1.5#
#
y1#>#0,#y2#>=#0,#y3#>=#0,#and#y4#>=#0#
##
This#formulation#is#called#the#dual#of#the#original#problem.##Since#this#is#the#standard#form#for#a#
maximization#problem,#we#can#now#apply#the#simplex#method.##To#do#that,#the#first#step#is#to#
#
Step#1:##Rewrite#the#objective#formula#as#an#equation:##w##C#135.5#y1#–#45.5#y2#–#32.5#y3#C#1,000#y4#=#0.##
We#will#call#this#the#objective#equation.###
#
Step#2:##Remove#the#inequalities#from#the#constraints.##To#do#that,#we#will#introduce#two#slack#
variables#as#m#and#n#and#add#them#to#the#constraints.##We#will#call#these#the#constraint#equations:#
