Richard BRONSON, 1982, "Theory and Problems of Operations Research", Schaum, McGraw-Hill Book Comp. (including 310 solved problems), viii+328 pp
· Part I, "Mathematical Programming"; Chapter 1, "Math. Progr.", Solved Problems, p 2

Problem 1.1
    The Village Butcher Shop traditionally makes its meat loaf from a combination of lean ground beef* and ground pork.  The ground beef contains 80 % meat and 20 % fat, and costs the shop 80 ¢ / lb; the ground pork contains 68 % meat and 32 % fat, and costs 60 ¢ / lb.  How much of each kind of meat should the shop use in each pound of meat loaf if it wants to minimize its cost and to keep the fat content of the meat loaf to no more than 25 % ?

Resolution
    x1 = poundage of ground beef used in each pound of meat loaf
    x2 = poundage of ground pork used in each pound of meat loaf

Objective: [min]   z = 80 x1 + 60 x2
subject to: 0.20 x1 + 0.32 x2 £ 0.25
    x1 + x2 = 1
"Hidden" constraints: x1, x2 ³ 0
   Solution:
    x1 = 7/12; x2 = 5/12;     z = 80 ´ 7/12 + 60 ´ 5/12 = 860/12 = 71 ¢ / lb (ca. 130 $ / lb)

Comentário
    Como só há 2 variáveis, este problema pode ser resolvido pelos métodos clássicos.  Fazendo a substituição x2 = 1 – x1, temos
z = 80 x1 + 60 x2 = 80 x1 + 60 (1 – x1) = 60 + 20 x1
    Para minimizar z, terá de ser x1 (positivo ou nulo) tão pequeno quanto possível.  Ora, do outro constrangimento, temos, sucessivamente:

0.20 x1 + 0.32 x2 £ 0.25
0.20 x1 + 0.32 (1 – x1) £ 0.25
0.12 x1 ³ 0.07
x1 ³ 7/12

Logo,
x1 = 7/12 e, portanto, x2 = 5/12

    Como seria de esperar, não é este o método geral para resolver problemas de Programação Linear.  Apresentar-se-á, para tal, o algoritmo ou método do simplex, de George Dantzig (a quem se faz referência nestas páginas Internet).

*lean ground beef: carne de vaca, magra picada; ground pork, carne de porco picada.   n

Actualizado em: 17-Mar-2001