ch6 - linear programming

8/12/2019 Ch6 - Linear Programming

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Lesson 6.6


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Industrial managers must consider physicallimitations, standards of quality, customerdemand, availability of materials, and

manufacturing expenses as restrictions, orconstraints, that determine how much of anitem they can produce.

Then they determine the optimum, or best,amount of goods to produceusually tominimize production costs or maximizeprofit.

The process of finding a feasible region anddetermining the point that gives themaximum or minimum value to a specificexpression is called linear programming


3/22

The Elite Pottery Shoppe makes two kinds of birdbaths: a

fancy glazed and a simple unglazed. An unglazed birdbath requires 0.5 h to make using a

pottery wheel and 3 h in the kiln.

A glazed birdbath takes 1 h on the wheel and 18 h in thekiln.

The companys one pottery wheel is available for at most 8hours per day (h/d).

The three kilns can be used a total of at most 60 h/d, andeach kiln can hold only one birdbath.

The company has a standing order for 6 unglazed

birdbaths per day, so it must produce at least that many. The pottery shops profit on each unglazed birdbath is $10,

and the profit on each glazed birdbath is $40.

How many of each kind of birdbath should the companyproduce each day in order to maximize profit?


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Organize the information into a table like thisone:


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Use your table to help you write inequalitiesthat reflect the constraints given, and be sureto include any commonsense constraints. Let x represent the number of unglazed birdbaths,

and let y represent the number of glazed birdbaths. Graph the feasible region to show the combinations

of unglazed and glazed birdbaths the shop couldproduce, and label the coordinates of the vertices.

(Note: Profit is not a constraint; it is what you aretrying to maximize.)


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0.5 8x y

3 18 60x y

6x

0x

0y

Pottery-wheel hours constraint

Kiln hours constraint

At least 6 unglazed

Common sense

Common sense


7/22

It will make sense to produce only wholenumbers of birdbaths. List the coordinates ofall integer points within the feasible region.(There should be 23.) Remember that the

feasible region may include points on theboundary lines.


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(5,0)(6,0)(7,0)

(8,0)(9,0)(10,0)(11,0)(12,0)(13,0)

(14,0)(15,0)(16,0)

(6,1)(7,1)(8,1)

(9,1)(10,1)(11,1)(12,1)(13,1)(14,1)

(6,2)(7,1)(8,2)


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Write the equation that will determine profitbased on the number of unglazed and glazedbirdbaths produced. Calculate the profit that

the company would earn at each of thefeasible points you found in last step. Youmay want to divide this task among themembers of your group.

Profit 10 40x y (6,1)

Profit 10(6) 40(1) 100


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What number of each kind of birdbath shouldthe Elite Pottery Shoppe produce to maximizeprofit? What is the maximum profit possible?

Plot this point on your feasible region graph.What do you notice about this point?

Maximum profit is $180 when 14 unglazed

birdbaths and 1 glazed birdbath are produced.This point is a vertex of the feasible region.


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Suppose that you want profit to be exactly$100. What equation would express this?Carefully graph this line on your feasibleregion graph.


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Suppose that you want profit to be exactly$140. What equation would express this?Carefully add this line to your graph.


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Suppose that you want profit to be exactly$170. What equation would express this?Carefully add this line to your graph.


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How do your results from the last three steps showyou that (14, 1) must be the point that maximizesprofit?

Generalize your observations to describe a methodthat you can use with other problems to find theoptimum value.

What would you do if this vertex point did not have

integer coordinates? What if you wanted tominimize profit?

You can graph one profit line and imagine shifting it up untilyou get the highest profit possible within the feasible

region. This should occur at a vertex. If the vertex is not aninteger point, you might test the integer points near theoptimum vertex. To minimize profit, move the profit linedown until it leaves the feasible region. This would occur at(6, 0).


15/22

Marco is planning to provide a snack of grahamcrackers and blueberry yogurt at his schools trackpractice.

He wants to make sure that the snack contains no morethan 700 calories and no more than 20 g of fat.

He also wants at least 17 g of protein and at least 30%of the daily recommended value of iron.

The nutritional content of each food is given above.

Each serving of yogurt costs $0.30 and each graham

cracker costs $0.06. What combination of servings of graham crackers and

blueberry yogurt should Marco provide to minimizecost?


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First organize the constraint information into a table, thenwrite inequalities that reflect the constraints. Be sure toinclude any commonsense constraints. Let x represent thenumber of servings of graham crackers, and let y representthe number of servings of yogurt.


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60 130 700x y

2 2 20x y

2 5 17x y

6 1 30x y

0x

Calories

Fat

Protein

Iron

Common sense

0y Common sense


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Now graph the feasible region and find thevertices.


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Next, write an equation that will determinethe cost of a snack based on the number ofservings of graham crackers and yogurt.

You could try any possible combination of

graham crackers and yogurt that is in thefeasible region, but recall that in theinvestigation it appeared that optimum valueswill occur at vertices. Calculate the cost at

each of the vertices to see which vertexprovides a minimum value.

cost 0.06 0.30x y


20/22

What if Marco wants to serve only whole numbers ofservings? The points (8, 1), (9, 1), and (9, 0) are the integerpoints within the feasible region closest to (8.5, 0), so testwhich point has a lower cost.


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(8, 1) costs $0.78, (9, 1) costs $0.84, and(9, 0) costs $0.54.Therefore, if Marco wants to serve only wholenumbers of servings, he should serve 9 grahamcrackers and no yogurt.


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ch6 - linear programming

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