Quantitative Decision Models I MSCI$-2200$

EXAMPLE $4$: A BLENDING PROBLEM. For a similar practice example, try Problem $4($Coffee Production$)$ on page $176$ in the textbook $16$e$.$

Iron ore from four different mines will be blended to make treads for a new product at PROTRAC, a medium$-$size crawler tractor, the E$-6,$ designed especially to compete in the European market. Analysis has shown that in order to produce a blend with suitable tensile qualities, minimum requirements must be met on three basic elements, denoted for simplicity as $A,B,$ and $C.$ In particular, each ton of ore must contain at least $5$ pounds of basic element $A,$ at least $100$ pounds of basic element $B,$ and at least $30$ pounds of basic element $C.$ These data are summarized in Figure $4\mathrm{.}1.$ The ore from each of the four different mines possesses each of the three basic elements, but in different amounts. These compositions, in pounds per ton, are given in Figure $4\mathrm{.}2.$

FIGURE $4\mathrm{.}1$

Requirements of Basic Elements

$\backslash$table$[[,$MINIMUM REQUIREMENT$],[$BASIC ELEMENT,PER TON OF BLEND $($LB$)],[$A$,5],[$B$,100],[$C$,30]]$

FIGURE $4\mathrm{.}2$

Compositions from Each Mine $($pounds per ton$)$

$\backslash$table$[[$BASIC ELEMENT,MINE$],[1,2,3,4],[$A$,10,3,8,2],[$B$,90,150,75,175],[$C$,45,25,20,37]]$

Notice that a ton of ore from the first mine contains $10$ pounds of basic element A and hence satisfies the minimum requirement on this element of $5$ pounds per ton. Similarly, this same ton of ore contains $90$ pounds of basic element $B$ and $45$ pounds of basic element $C,$ hence satisfying the requirement on basic element $C$ but not on basic element $B.$ Similarly, you can verify that a single ton of ore from the second mine will not satisfy the requirement on $A$ or $C.$ A single ton of ore from mine $3$ will not satisfy requirements on $B$ and $C,$ and a single ton from mine $4$ will

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