web solver integration

8/12/2019 Web Solver Integration

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Solver

Finding maximum, minimum, or value by

changing other cells

Can add constraints

Dont need to guess and check


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Solver


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Using Solver, Excels Solver

1. EXCELSSOLVER

The utility Solveris one ofExcels

most useful tools for businessanalysis. This allows us to maximize, minimize, or find a predetermined value

for the contents of a given cell by changing the values in other cells.

Moreover, this can be done in such a way that it satisfies extra constraints that

we might wish to impose.

Example 1. The size limitations on boxes shipped by your plant areas follows. (i) Their circumference is at most 100 inches. (ii) The sum of their

dimensions is at most 120 inches. You would like to know the dimensions of

such a box that has the largest possible volume. LetH, W, andLbe the

height, width, and length of a box; respectively; measured in inches. We wish

to maximizethe volume of the box, V= HW

L , subject to the limitations thatthe circumference C= 2H+ 2W 100and the sum S= H+ W+ L 120.

This problem is set up in theExcelfile Shipping.xls. We will outline

its solution with screen captures and directions. First, enter any reasonable

values for the dimensions of the box in Cells B7:D7.

Shipping.xls

Using Solver. Excels Solver

(material continues) IT C
http://localhost/var/www/apps/conversion/tmp/Shipping.xlshttp://localhost/var/www/apps/conversion/tmp/Shipping.xlshttp://localhost/var/www/apps/conversion/tmp/MBD%20Part%202.ppthttp://localhost/var/www/apps/conversion/tmp/MBD%20Part%202.ppthttp://localhost/var/www/apps/conversion/tmp/Shipping.xls


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To use Solver, click on Data, then

Solver in the Analysisbox. In older

versions ofExcelselect Toolsin the main

Excelmenu, then click on Solver.

Using Solver, Solver

H

L

FRAGILE

Crush slowly

Shipping.xls

Using Solver. Excels Solver: page 2

(material continues)

Computer Problem?

ICT

To use Solver, click on Data, then

Solver in the Analysisbox. In older

versions ofExcelselect Toolsin the main

Excelmenu, then click on Solver.

Enter cell that computes volume.

Select Max.

Enter cells that contain

dimensions

Click on Add.
http://localhost/var/www/apps/conversion/tmp/Shipping.xlshttp://localhost/var/www/apps/conversion/tmp/Shipping.xlshttp://localhost/var/www/apps/conversion/tmp/MBD%20Part%202.ppthttp://localhost/var/www/apps/conversion/tmp/MBD%20Part%202.ppthttp://localhost/var/www/apps/conversion/tmp/MBD%20Part%202.ppthttp://localhost/var/www/apps/conversion/tmp/MBD%20Part%202.ppthttp://localhost/var/www/apps/conversion/tmp/MBD%20Part%202.ppthttp://localhost/var/www/apps/conversion/tmp/Shipping.xls


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Shipping.xls


(material continues) ICT

Enter cell that computes circumference.

Select


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Click on Keep Solver Solution.

Click on OK.

Click on Solve.



Shipping.xls (material continues)ICT


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Using Solver, SolverThe dimensions that maximize

volume are now shown in Cells B8:D8. The maximum volume, the value of

the circumference and the sum of the dimensions are now displayed. For a

maximum volume of 43,750 cubic inches, the box should be 25 inches high,

25 inches wide, and 70 inches long.

Shipping.xls


(material continues) I

In rare cases; such as very large or small initial values ofH, W, orL;

you may need to add the constraints B7 >= 0, C7 >= 0and D7 >= 0.

CT


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Using Solver, SolverUsing Solver. Excels Solver: page 6


Rush!shipping company limits the size of the

boxes that it accepts by limiting their volume to at most 16 cubic feet(27,648 cubic inches). For it to ship a box, each dimension must be between

3 and 54 inches. (i) Modify Shipping.xlsand use Solverto find thedimensionsof a Rush!box which will accept the longest possible item.

H int: Use dif ferent in iti al values for each dimension. (ii) What is the

maximum lengthof such an item? Note that the longest item which can beshipped in a box has a length of

.222 LWH

Shipping.xls ICT

Show ex3-sep14-shipping.xls
http://localhost/var/www/apps/conversion/tmp/Shipping.xlshttp://localhost/var/www/apps/conversion/tmp/Shipping.xlshttp://localhost/var/www/apps/conversion/tmp/Shipping.xlshttp://localhost/var/www/apps/conversion/tmp/Shipping.xlshttp://localhost/var/www/apps/conversion/tmp/Shipping.xlshttp://localhost/var/www/apps/conversion/tmp/Shipping.xlshttp://localhost/var/www/apps/conversion/tmp/Shipping.xlshttp://localhost/var/www/apps/conversion/tmp/Shipping.xlshttp://localhost/var/www/apps/conversion/tmp/Shipping.xlshttp://localhost/var/www/apps/conversion/tmp/Shipping.xlshttp://localhost/var/www/apps/conversion/tmp/MBD%20Part%202.ppthttp://localhost/var/www/apps/conversion/tmp/MBD%20Part%202.ppthttp://localhost/var/www/apps/conversion/tmp/Shipping.xls


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Solver

Sensitive to initial value

Use graphical approximation to help solveproject

Use to verify/solve Questions 1 - 3

Use to solve Questions 6 - 8


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Integration

Revenue as an area underDemandfunction

qDqqR

-1.2

-10 q

D(q)

Demand Function

Revenue

qD(q)


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Integration

Total possible revenue-The total poss ib le revenue is the m oney thatthe producer wo uld receive if everyone who wanted the good, bought i t at the maximum

price that he or she was w il l ing to pay. This is the greatest possible revenue that a seller or

producer could obtain when operating with a given demand function

-1.2

-8

Demand Function

Total PossibleRevenue


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Integration

Approximating area under graph

- Counting rectangles (by hand)- Using Midpoint Sum (by hand)

- UsingMidpoint Sums.xls(usingExcel)

- UsingIntegrating.xls(usingExcel)


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Integration

Approximating area (Counting Rectangles)

Ex.Approx. 9 rectangles

Each rectangle is 0.25

square units

Total area is approx.

2.25 square units

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2


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Integration

Approximating area (Midpoint Sums)

- Notation

- Meaning

bafSn ,,

giveninterval:,givenfunction:

rectangleswithsum:

ba

fnSn


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Integration


- ProcessFind endpoints of each subinterval

Find midpoint of each subinterval

nxxxxx ...,,,,, 3210

nmmmm ...,,,, 321


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Integration


- Process (continued)Find function value at each midpoint

Multiply each by and add them all

This sum is equal to

nmfmfmfmf ...,,,, 321

imf x xmfxmfxmfxmf n ...321 bafSn ,,


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Integration


Ex1.Determine where.

5.1,0,3 fS 246 xxxf

5.10.1

5.0

0

3

2

1

0

xx

x

x

25.1

75.0

25.0

3

2

1

m

m

m


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Integration


Ex1.(Continued)

375.25.025.15.025.25.025.1

5.025.15.075.05.025.0

5.1,0, 3213

fff

xmfxmfxmffS


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IntegrationApproximating area (Midpoint Sums.xls)

Ex1.(Continued)

=6*x-4*x^2


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Show

ex2-n-100Area

Example.xlsm

23

EXAMPLE 2 - Modify sheet n = 20in Area Example.xls,

so that it computes the sum S100(f, [0, 4]), with 100 subintervals, for f(x) = 2x

x2/2.


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Integration-9/28

Approximating area (Integrating.xls)

- File is similar toMidpoint Sums.xls

- Notation: or or

b

adxxf

b

adttf

Integration


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Integrationshow ex3-Integrating.xlsm


Ex3.UseIntegrating.xlsto compute

4

1

6/

61 dxe x


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IntegrationApproximating area (Integrating.xls)


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Integration


Ex3.(Continued)

So . Note that is the

p.d.f.of an exponential random variable withparameter . This area could be calculate

using the c.d.f.function .

3331.04

1

6/

61 dxe x 6/

61 xe

6

aFbF XX


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Integration


Ex3.(Continued)

3331.01535182751.0486582881.0

11

14

6/16/4

ee

FFaFbF XxXX


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Integration, Integrals

2. INTEGRALS

What would happen if we computed midpoint sums for a functionwhich might assume negative values in the interval [a, b]?

Wheref(mi) < 0, the product

f(mi)xis also negative. Thus, the

midpoint sums

will approximate the signed area

of the region between the x-axi s andthe graph of f , over [a, b]. This is the

algebraic sum of the area above the

axis, minus the area below the axis.

xmfbafS

n

i

in 1

)(]),[,(


Integration. Integrals

a

b

+

ICT
http://localhost/var/www/apps/conversion/tmp/MBD%20Part%202.ppthttp://localhost/var/www/apps/conversion/tmp/MBD%20Part%202.ppt


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Integration, Integrals

the integral of f over [a, b] is

and i t represents the algebraic sum of the signed areas of the regions

between the hor izontal axis and the graph of f , over [a, b] .

b

a

dxxf )(

xmfdxxfn

i

in

b

a

1

)(lim)(


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Integration

Approximating area

- Values fromMidpoint Sums.xlscan be positive,negative, or zero

- Values fromIntegrating.xlscan be positive,negative, or zero

I t ti A li ti

Integration A l i t i 6


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dqqDqDqdqqD

maxq

soldq

soldsold

soldq

)(SoldNot)()(

SurplusConsumer

0

.)(

Revenue

PossibleTotal

0

dqqD

maxq

Integration, Applications


Integration. App l icat ions: page 6

Revenue computations for an arbitrary demand function work in the

same way as those for the buffalo steak dinners.

Let D(q) give the price per unit for a good,that would result in the sale

of qunits, and let qmaxbe the maximum number of units that could be sold atany price. That is, D(qmax) = 0. The total possib le revenueis given by

If qsoldunits are sold, then the revenuewill be qsoldD(qsold). Thefollowing formulas give consumer surp lusand lost revenue from units not

sold.

It is clear that

revenue+ consumer surp lus + not so ld =total po ssib le revenu e.

ICT
http://localhost/var/www/apps/conversion/tmp/scratch_1/MBD%20Part%202.ppthttp://localhost/var/www/apps/conversion/tmp/scratch_1/MBD%20Part%202.ppthttp://localhost/var/www/apps/conversion/tmp/scratch_1/MBD%20Part%202.ppthttp://localhost/var/www/apps/conversion/tmp/scratch_1/MBD%20Part%202.ppthttp://localhost/var/www/apps/conversion/tmp/scratch_1/MBD%20Part%202.ppt


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Integration

Ex4.Suppose a demand function was found to

be

. Determine

the consumer surplus at a quantity of 400 units

produced and sold.

196.321225.00001392. 2 qqqD


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revenue+ consumer surp lus at 400 units


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Integration

Ex.(Continued)

Calculate Revenue at 400 units

60.83569$

924.208400400400

)(

D

qDqqR

qDqqR soldsoldsold


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Integration

Ex.(Continued)

$107,508.80$83,569.60 = $23,939.20

So, the consumer surplus is $23,939.20

)(})({

Surplus

Consumer

0

soldsold

q

qDqdqqDso ld


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Integration

Formula for consumer surplus

Income stream

- revenue enters as a stream

- take integral of income stream to get totalrevenue

0

0 0

q

qRdqqD


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Integration Applications-oct1st

Fundamental Theorem of Calculus

-

Example : applies top.d.f.s and c.d.f.s

Recall from Math 115a

aFbFdxxf XXb

a X

Fundamental Theorem of Calculus. For many of the functions,f,

which occur in business applications, the derivative of with

respect tox, isf(x). This holds for any number aand anyx, such that the

closed interval between aandxis in the domain off.

,)( duuf

x

a

Integration Applications


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Example 4.

The Plastic-Is-UsToy Company

incoming revenue -as an income stream(rather than a collection of discretepayments)

At a time tyears from the start of its fiscal year on July 1 the company expects

to receive revenue at the rate ofA(t) million dollars per year

Records from past years indicate that Plastic-Is-Uscan model its revenuerate

A(t) = 110t5+ 330t4330t3+ 110t2+3.174 million dollars per year.


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0

2468

0 0.25 0.5 0.75 1

t

A(t)

Oct. 1 Jan. 1 April 1 July 1July 1

The chief financial officer wants

to compute the total amount of revenue that Plastic-Is-Uswill receive in one

year.

The income stream,A(t), is a rate of changein money, given in million dollars

per year.

the units along the t-axis are years

the area of a region under the graph ofA(t) is given in

(mil li ons of dollars/year)(years) = mil l ions of dollars.


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Since gives the area between

the t-axis and the graph of

A(t), over the interval [0, T], it can beshown that the integral gives the total

amoun t of money, in millions of dollars,

that will be received from the income

stream in the first Tyears.

T

dttA

0

)(

I t ti A li ti


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dollarsmillion007.5174.3110330330110

1

0

2345 dxxxxx

Use Integrating.xlsto compute the total income received by Plastic-

Is-Us during the period from 0 to 1 year. (Remember that we must use x, nott, as the var iable of integration in I ntegrating.xls.)

I t ti A li ti

Integration. Applications: page12


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The total revenue, in dol lars, received fr om an income stream of

A(t) dollars per year, starting now and continuing for the next T years is

given by .)(0T

dttA

Integration, ApplicationsIntegration. Applications: page 12

In addition to the total revenue, a company would often like to know

the present valueof its income stream during the next Tyears (0 tT),

assuming that money earns interest at some annual rate r, compoundedcontinuously.

Suppose that money earns at an annual r ate, r, compounded

conti nuously. The present dollar value of an income stream of A(t)

dol lars per year, starting now and continuing for the next T years is

given by .)(

0

T tr dtetA

Integration Applications


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Example 5. We return to the Plastic-Is-UsToy Company

that we considered inExample 4. Recall that they have an

income stream ofA(t) = 110t5+ 330t4330t3+ 110t2+3.174

million dollars per year. The management of Plastic-Is-Uswouldlike to know the present valueof its income stream during the

next year (0 t1), assuming that money earns interest at an

annual rate of 5.5%, compounded continuously.

Applying the integral formula for present value to Plastic-Is-Us, we

use Integrating.xlsto find that the present value of their income stream for one

year, starting on July 1, is

million dollars.

879.4174.31103303301101

0

055.02345

dtetttt t

Integration, Calculus


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the inverse connection between integration and differentiation is

called the Fundamental Theorem of Calculus.

Fundamental Theorem of Calculus. For many of the functions,f,

which occur in business applications, the derivative of with

respect tox, isf(x). This holds for any number aand anyx, such that the

closed interval between aandxis in the domain off.

,)( duuf

x

a

Example 7. Letf(u) = 2 for all values of u. Ifx1, then

integral offfrom 1 toxis the area of the region over the interval [1,x],between the u-axis and the graph off.

Integration, Calculus3


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g ,

The region whose area

is represented by the integral is

rectangular, with height 2 and

widthx 1. Hence, its area is2(x1) = 2x2, and

0

1

2

3

0 1 2 3 4 5

u

f(u )

(1, 2) (x, 2)

x

2

x

duuf

1

)(

x1

.22)(

1

xduufx

In the sectionProperties and ApplicationsofDifferentiation, we saw

that the derivative off(x) = mx+ bis equal to m, for all values ofx. Thus, the

derivative of with respect tox, is equal to 2. As predicted by the

Fundamental Theorem of Calculus, this is also the value off(x).

The next example uses the definition of a derivative as the limit of

difference quotients.

,)(

1

x

duuf



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g ,

Example 8. Recall the income stream ofA(t) = 110t5+ 330t4

330t3+ 110t2+3.174 million dollars per year that was expected by the

Plastic-Is-Ustoy company inExample 4ofApplications. Let G(T) be the total

income that is expected during the first Tyears, for 0 T1. Picking a time T= 0.5 years, we will check that the instantaneous rate of change of G(T), with

respect to T, is the same asA(T).

Note that We now wish to compute G(0.5). Recall

that G(T) is approximated by the difference quotient

for small values of h. We will let h= 0.0001, and use Integrating.xlsto

evaluate G(0.5 + 0.0001) and G(0.5 0.0001). Integrating.xlsrounds the

numerical values of integrals to four decimal places. For the present

calculation, we gain extra precision by copying the values from Cell N20andkeeping all of their decimal places.

G(0.5 + 0.0001) = G(0.5001) = 2.79078611562868

G(0.5 0.0001) = G(0.4999) = 2.78946381564699

.)()(

0

T

dttATG

,2

)()(

h

hTGhTG



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g

These give a value of 6.6115 for the difference quotient

rounded to four decimal places. This is the

instantaneous rate of change in total income after 0.5 years. I ntegrating.xls

shows the same value forA(0.5).

Noting that we have

verified the Fundamental Theorem of Calculus. At T= 0.5, the derivative of

with respect to T, is equal toA(T).

,)(

0

T

dttA

,)()(

0

T

dttATG

,

0002.0

)4999.0()5001.0( GG

web solver integration

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