LP case study

Case  Study:  Cutting  Cafeteria  Costs   A  cafeteria   at  All-­‐State  University  has  one   special  dish   it   serves   like   clockwork   every  Thursday  at  noon.  This  supposedly  tasty  dish  is  a  casserole  that  contains   sautéed   onions,   boiled   sliced   potatoes,   green   beans,   and   cream   of   mushroom   soup.  Unfortunately,  students  fail  to  see  the  special  quality  of  this  dish,  and  they   loathingly   refer   to   it   as   the   Killer   Casserole.   The   students   reluctantly   eat   the   casserole,   however,   because   the   cafeteria   provides   only   a   limited   selection   of   dishes  for  Thursday's  lunch  (namely,  the  casserole).     Maria   Gonzalez,   the   cafeteria   manager,   is   looking   to   cut   costs   for   the   coming   year,  and  she  believes  that  one  sure  way  to  cut  costs  is  to  buy  less  expensive  and   perhaps  lower  quality  ingredients.  Because  the  casserole  is  a  weekly  staple  of  the   cafeteria   menu,   she   concludes   that   if   she   can   cut   costs   on   the   ingredients   purchased   for   the   casserole,   she   can   significantly   reduce   overall   cafeteria   operating   costs.   She   therefore   decides   to   invest   time   in   determining   how   to   minimise   the   costs   of   the   casserole   while   maintaining   nutritional   and   taste   requirements.     Maria  focuses  on  reducing  the  costs  of  the  two  main  ingredients  in  the  casserole,   the   potatoes   and   green   beans.   These   two   ingredients   are   responsible   for   the   greatest  costs,  nutritional  content,  and  taste  of  the  dish.     Maria  buys  the  potatoes  and  green  beans  from  a  wholesaler  each  week.  Potatoes   cost  $0.40  per  pound  (lb),  and  green  beans  cost  $1.00  per  lb.     All-­‐State  University  has  established  nutritional  requirements  that  each  main  dish   of   the  cafeteria  must  meet.  Specifically,   the  dish  must  contain  180  grams  (g)  of   protein,   80  milligrams   (mg)   of   iron,   and   1,050  mg   of   vitamin   C.   For   simplicity   when   planning,   Maria   assumes   that   only   the   potatoes   and   green   beans   contribute  to  the  nutritional  content  of  the  casserole.     Because   Maria   works   at   a   cutting-­‐edge   technological   university,   she   has   been   exposed  to  the  numerous  resources  on  the  World  Wide  Web.  She  decides  to  surf   the   Web   to   find   the   nutritional   content   of   potatoes   and   green   beans.   Her   research  yields  the  following  nutritional  information  about  the  two  ingredients.     Nutrient   Potatoes   Green  Beans   Protein   1.5  g  /  100g   5.67  g  /  10  oz   Iron   0.3  mg  /  100  g   3.402  mg  /  10  oz   Vitamin  C   12  mg  /  100  g   28.35  mg  /  10  oz     (There  are  28.35  g  in  one  ounce,  and  16  ounces  in  one  pound)     Edson   Branner,   the   cafeteria   cook   who   is   surprisingly   concerned   about   taste,   informs  Maria  that  an  edible  casserole  must  contain  at  least  a  six-­‐to-­‐five  ratio  in   the  weight  of  potatoes  to  green  beans.  

  Given   the   number   of   students   who   eat   in   the   cafeteria,   Maria   knows   that   she   must   purchase   enough  potatoes   and   green  beans   to  prepare   a  minimum  of   10   kilograms  (kg)  of  casserole  each  week.  (There  are  1,000  g  in  one  kg.)  Again,  for   simplicity   in   planning,   she   assumes   that   only   the   potatoes   and   green   beans   determine   the   amount   of   casserole   that   can   be   prepared.   Maria   does   not   establish  an  upper  limit  on  the  amount  of  casserole  to  prepare  since  she  knows   all  leftovers  can  be  served  for  many  days  thereafter  or  can  be  used  creatively  in   preparing  other  dishes.  

a. Determine   the   amount   of   potatoes   and   green   beans   Maria   should   purchase   each   week   for   the   casserole   to   minimise   the   ingredient   costs   while  meeting  nutritional,  taste,  and  demand  requirements.  

Before   she   makes   her   final   decision,   Maria   plans   to   explore   the   following   questions  independently,  except  where  otherwise  indicated.  

b. Maria   is  not  very  concerned  about   the  taste  of   the  casserole;  she   is  only   concerned  about  meeting  nutritional  requirements  and  cutting  costs.  She   therefore   forces   Edson   to   change   the   recipe   to   allow   only   for   at   least   a   one-­‐to-­‐two  ratio  in  the  weight  of  potatoes  to  green  beans.  Given  the  new   recipe,  determine  the  amount  of  potatoes  and  green  beans  Maria  should   purchase  each  week.  

c. Maria   decides   to   lower   the   iron   requirement   to   65   mg   since   she   determines   that   the   other   ingredients,   such   as   the   onions   and   cream  of   mushroom  soup,  also  provide  iron.  Determine  the  amount  of  potatoes  and   green   beans   Maria   should   purchase   each   week   given   this   new   iron   requirement.   To   answer   this   part,   first   conduct   sensitivity   analysis   to   check  whether   the  part-­‐a  solution  still   remains  optimal.   If   it   is  not,   then   you  must  find  a  new  one.  

d. Maria   learns   that   the   wholesaler   has   a   surplus   of   green   beans   and   is   therefore  selling  the  green  beans  for  a  lower  price  of  $0.50  per  lb.  Using   the  same  iron  requirement  from  part  c  and  the  new  price  of  green  beans,   determine   the   amount   of   potatoes   and   green   beans   Maria   should   purchase  each  week.  To  answer  this  part,  first  conduct  sensitivity  analysis   to  check  whether  the  part-­‐c  solution  still  remains  optimal.  If  it  is  not,  then   you  must  find  a  new  one.  

e. Maria   decides   that   she   wants   to   purchase   lima   beans   instead   of   green   beans  since  lima  beans  are  less  expensive  and  provide  a  greater  amount   of   protein   and   iron   than   green   beans.   Maria   again   wields   her   absolute   power   and   forces   Edson   to   change   the   recipe   to   include   lima   beans   instead   of   green   beans.   Maria   knows   she   can   purchase   lima   beans   for   $0.60  per  lb  from  the  wholesaler.  She  also  knows  that  lima  beans  contain   22.68  g  of  protein  and  6.804  mg  of  iron  per  10  ounces  of  lima  beans  and   no   vitamin  C.  Using   the  new   cost   and  nutritional   content   of   lima  beans,   determine  the  amount  of  potatoes  and  lima  beans  Maria  should  purchase   each   week   to   minimise   the   ingredient   costs   while   meeting   nutritional,   taste,   and   demand   requirements.   The   nutritional   requirements   include   the  reduced  iron  requirement  from  part  c.  

f. Will  Edson  be  happy  with  the  solution  in  part  e?  Why  or  why  not?   g. An  All-­‐State  student   task   force  meets  during  Body  Awareness  Week  and  

determines   that   All-­‐State   University's   nutritional   requirements   for   iron  

are  too  lax  and  that  those  for  vitamin  C  are  too  stringent.  The  task  force   urges   the   university   to   adopt   a   policy   that   requires   each   serving   of   an   entrée  to  contain  at  least  120  mg  of  iron  and  at  least  500  mg  of  vitamin  C.   Using  potatoes  and   lima  beans  as   the   ingredients   for   the  dish  and  using   the  new  nutritional  requirements,  determine  the  amount  of  potatoes  and   lima  beans  Maria  should  purchase  each  week.  

  In  each  part  you  must:  

• Define  decision  variables  and  describe  the  mathematical  model;  and   • Except   part   f,  write   a   recommendation   to   the   problem  by   including   the  

total   cost,   the   amount  of   each   casserole’s   ingredient,   and   the   amount  of   each  nutrient.