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Question
Holiday Mean Turkey Ranch is considering buying two different types of turkey feed. Each feed contains, in varying proportions, some or all of the three nutritional ingredients essential for fattening turkeys. Brand Y feed costs the ranch 0.2perpound.BrandZcosts .03 per pound. The rancher would like to determine the lowest-cost diet that meets the minimum monthly intake requirement for each nutritional ingredient.
The following table contains relevant information about the composition of brand Y and brand Z feeds, as well as the minimum monthly requirement for each nutritional ingredient per turkey.
Composition of Each Pound of Feed
Ingredient Brand Y Feed Brand Z Feed Minimum Monthly Requirement A 5 oz 10 oz 90 oz B 4 oz 3 oz 48 oz C .5 oz 0 1.5 oz Cost/lb 0.2$ .03$
The following table contains relevant information about the composition of brand Y and brand Z feeds, as well as the minimum monthly requirement for each nutritional ingredient per turkey.
Composition of Each Pound of Feed
| Ingredient | Brand Y Feed | Brand Z Feed | Minimum Monthly Requirement |
| A | 5 oz | 10 oz | 90 oz |
| B | 4 oz | 3 oz | 48 oz |
| C | .5 oz | 0 | 1.5 oz |
| Cost/lb | 0.2$ | .03$ |
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Solution
If we let
X1= number of pounds of brand Y feed purchased
X2= number of pounds of brand Z feed purchased
then we may proceed to formulate this linear programming problem as follows:
Minimize cost (in cents) =2X1+3X2
subject to these constraints
5X1+10X2≥90 oz (ingredient A constraint)
4X1+3X2≥48 oz (ingredient B constraint)
12X1≥112 oz (ingredient C constraint)
Figure illustrates these constraints
The iso-cost line approach may be used to solve LP minimization problems such as that of the Holiday Meal Turkey Ranch. As with iso-profit lines, we need not compute the cost lines. The lowest cost line (that is, the one closest in toward the origin) to couch the feasible region provides us with the optimal solution corner.
For example, we start in figure by drawing a 54 cents cos line, namely, 54=2X1+32. Obviously, there are many points in the feasible region that would yield a lower total cost. We proceed to move our iso-cost line toward the lower left, in plane parallel to the 54 cent solution line. The lase point we touch while still in contact with the feasible region is the same as corner point b of figure. It has the coordinates (X1=8.4,X2=4.8) and an associated cost of 31.2 cents
X1= number of pounds of brand Y feed purchased
X2= number of pounds of brand Z feed purchased
then we may proceed to formulate this linear programming problem as follows:
Minimize cost (in cents) =2X1+3X2
subject to these constraints
5X1+10X2≥90 oz (ingredient A constraint)
4X1+3X2≥48 oz (ingredient B constraint)
12X1≥112 oz (ingredient C constraint)
Figure illustrates these constraints
The iso-cost line approach may be used to solve LP minimization problems such as that of the Holiday Meal Turkey Ranch. As with iso-profit lines, we need not compute the cost lines. The lowest cost line (that is, the one closest in toward the origin) to couch the feasible region provides us with the optimal solution corner.
For example, we start in figure by drawing a 54 cents cos line, namely, 54=2X1+32. Obviously, there are many points in the feasible region that would yield a lower total cost. We proceed to move our iso-cost line toward the lower left, in plane parallel to the 54 cent solution line. The lase point we touch while still in contact with the feasible region is the same as corner point b of figure. It has the coordinates (X1=8.4,X2=4.8) and an associated cost of 31.2 cents
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