Group 1 Midterm Report Economical Math (E).Pdf

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UNIVERSITY OF ECONOMICS AND LAW FACULTY OF ACCOUNTING AND AUDITING GROUP 1 MIDTERM REPORT ECONOMICAL MATH (E) Supervisor: Ph.D Nguyen Phuc Son

No Name Student ID Class Question

1 |Nguyen Hung K234050653 K23405E 1,7

2 | Huynh Nhat Nam K234050655 K23405E 2

3_ | Nguyen Thuy Khanh Bmh | K234050643 K23405E 3

4_ | Chu Vo Dan Thanh K234050663 K23405E 4

5_ | Dang Hoang Thanh Thuy K234050668 K23405E 5

6_ | Nguyen Dao Minh Thao K234050664 K23405E 6

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TABLE OF CONTENTS TABLE OF FIGURES QUESTION

TABLE OF CONTENTS

ANSWER

Question 1: Question 2: Question 3: Question 4: Question 5: Question 6: Question 7:

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TABLE OF FIGURES

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QUESTION

Question 1 Shale Oil, located on the island of Aruba, has a capacity of 1,500,000 bbl of crude oil per day The final products from the refinery include three types of unleaded gasoline with different octane numbers (ON): regular with ON =87, premium with ON =89, and super with ON =92

The refining process encompasses three stages:

(1) a distillation tower that produces feedstock (ON =82) at the rate of 2 bbl per bbl

of crude oil

(2) a cracker unit that produces gasoline stock (ON =98) by using a portion of the feedstock produced from the distillation tower at the rate of 5 bbl per bbl of feedstock

(3) a blender unit that blends the gasoline stock from the cracker unit and the feedstock from the distillation tower

The company estimates the net profit per barrel of the three types of gasoline to be $6.70, $7.20, and $8.10, respectively The inputcapacity of the cracker unit is 200,000 bbl of feedstock a day The demand limits for regular, premium, and super gasoline are 50,000, 30,000, and 40,000 bbl, respectively, per day Develop a model for determining the optimum production schedule for the refinery

Question 2 Alumco manufactures aluminum sheets and aluminum bars The maximum production capacity is estimated at either 800 sheets or 600 bars per day The maximum daily demand is 550 sheets and 560 bars The profit per ton is $40 per sheet and $35 per bar Determine the optimal daily production mix

Question 3 The Continuing Education Division at the Ozark Community College offers a total of 30 courses each semester The courses offered are usually of two types: practical and humanistic To satisfy the demands of the community, at least 10 courses of each type must be offered each semester The division estimates that the revenues of offering practical and humanistic courses are approximately $1500 and $1000 per course, respectively

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b) Show that the worth per additional course is $1500, which is the same as the revenue per practical course What does this result mean in terms of offering additional courses ?

Question 4 Wyoming Electric Coop owns a steam-turbine power-generating plant Because Wyoming is rich in coal deposits, the plant generates its steam from coal This, however, may result in emission that does not meet the Environmental Protection Agency (EPA) standards EPA regulations limit sulfur dioxide discharge to 2000 parts per million per ton of coal burned and smoke discharge from the plant stacks to 20 Ib per hour The Coop receives two grades of pulverized coal, Cl and C2, for use in the steam plant The two grades are usually mixed together before buming For simplicity, it can be assumed that the amount of sulfur pollutant discharged (an parts per million) is a weighted average of the proportion of each grade used in the mixture The following data is based on the consumption of | ton per hr of each of the two coal grades

Sulfur discharge Smoke discharge Steam generated Coal grade in parts per million in Ib per hour in lb per hour

Cc 1800 2.1 12,000 2100 9 9.000

(a) Determine the optimal ratio for mixing the two coal grades

(b) Determine the effect of relaxing the smoke discharge limit by 1 Ib on the amount of generated steam per hour

Question 5, An assembly line consisting of three consecutive stations produces two radio models: HiFi-1 and HiFi-2 The following table provides the assembly times for the three workstations

Minutes per unit

Workstation HiFi-1 HiFi-2

1 6 4

2 5 5

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The daily maintenance for stations 1, 2, and 3 consumes 10%, 14%, and 12%, respectively,

of the maximum 480 minutes available for each station each day Determine the optimal product mix that will minimize the idle (or unused) times in the three workstations Question 6 An industrial recycling center uses two scrap aluminum metals, A and B, to produce a special alloy Scrap A contains 6% aluminum, 3% silicon, and 4% carbon Scrap B has 3% aluminum, 6% silicon, and 3% carbon The costs per ton for scraps A and B are $100 and $80, respectively The specifications of the special alloy require that (1) the aluminum content must be at least 3% and at most 6%, (2) the silicon content must be between 3% and 5%, and (3) the carbon content must be between 3% and 7% Determine the optimum mix of the scraps that should be used in producing 1000 tons of the alloy Question 7 Fox Enterprises is considering six projects for possible construction over the next four years Fox can undertake any of the projects partially or completely A partial undertaking of a project will prorate both the return and cash outlays proportionately The expected (present value) returns and cash outlays for the projects are given in the

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ANSWER Question 1:

In a nutshell, a factory has output is 1,500,000 bbl of crude oil per day They have 3 type:

(1) Regular with ON =87 (2) Premium with ON =89 (3) Super with ON =92

They also have 3 stage of refining process:

(1) a distillation tower that produces feedstock (ON =82) at the rate of 2 bbl per bbl of crude oil

(2)a cracker unit that produces gasoline stock (ON=98) by using a portion of the feedstock produced from the distillation tower at the rate of 5 bbl per bbl of feedstock

(3)a blender unit that blends the gasoline stock from the cracker unit and the feedstock from the distillation tower

The price of three type of oil: (1) Regular is $6.70 (2) Premium is $7.20 (3) Super is $8.10

Max output that the cracker unit can produce 200,000 bbl of feedstock per day Max output of regular oil is 50,000 per day

Max output of premium oil is 30,000 per day Max output of super oil is 40,000 per day Make a Matrix to maxims to bring the most profit P/S-

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Octane rating does not relate directly to the power output or the energy content of the fuel per unit mass or volume, but simply indicates gasoline's capability against compression Whether or not a higher octane fuel improves or impairs an engine's performance depends on the design of the engine In broad terms, fuels with a higher octane rating are used in higher-compression gasoline engines, which may yield higher power for these engines Such higher power comes from the fuel's higher compression by the engine design, and not directly from the gasoline

With the data that is given, we have:

XytX») +X 1:] Xi †+X;¡,ON=& 3: 2 X,, +Xy +X Bender ==> X,, +X,, ,ON=8 Distillatio ON =82 F————> ON =98 —_) ¬ ON! Feedstock X; : bbl/ day used in stage to produce oil type 1=1,2 j=1,2,3

The number of regular (ON =87): X,, +X, The number of premium (ON =89); X, +X, The number of super (ON =92): X,, +X, Equation:

5(X1+ X pt X43)42( XQ, +X py +X; = 1,500,00

2(X;¡+X„+X„;)<200,00( Xu +X¿, <50,00/ X 1) +X <30,00 X 3 +X>, <40,00

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82 Xt 98 Xz=89(Xi¿TX¿ 82 X13 +98 X,,=92( X13 +X

$6.70(X + X21 )+$7.20 (Xizt Xq2)+ $8 10(X 13+ Xy3)=Ma

= Be 7° op Wrap Tet Fweer Gene , BIY- Be Be B- SSS CE Beergeacerter- S- 9 + Qộ g Com

D & & fl Ena Sy it

inset Delete Format Sort & Find & ial norAg

Font 2 Alignment a Number

B Cc D E F 6 H | J K M

A 1 Oil crude

INPUT

4 Distillation (ON=82) Cracker (ON=98) Profit

12 Total of crude oil

OUTPUT

4 Total profit

Figure | So the most profit the factory can get is $875.000 Question 2:

The optimal daily production is 550 sheets and 387.5 bars Step-by-step explanation:

Let: x be the no of sheets

y be the no of bars

The profit per ton is $40 per sheet and $35 per bar So, Profit function 440 x+35 The maximum daily demand is 550 sheets and 560 bars, so: x < 550A y < 5(

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+ c <« B 600/800x + y < 800 @ 600 < 600/800: +» (53.333, 560) (320, 560) @ «<550 @ y<560 (550, 387.5) (550, 187.5) Figure 2 So, The points of feasible region are: (53.333 ,560, (320 ,560, (550 ,387.5 and (550 , 187.5 Profit function ¿ 40 x+35 At (53.333 ,560° => Profit £40 <53.333+35 x 560=21733.32 At (320 ,560 = Proft ¿40 x320+35 x560=32400 At (550 ,387.5 => Profit ,40 x 550+35 x 387.5=35562.5

At(550 ,187.5 = Proñt ¿40 x550+35x 187.5=28562.5

Since profit is maximum at (550 ,387.5`

So, the optimal daily production is 550 sheets and 387.5 bars Question 3:

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> Th Cortina Eduestion Oininion ol Ozark Communi tu,

l darn total of 30) courts Ngự, TÌ, caurnes ques, ose mal {a podel ond hums To ail

\ he cry “ot bioxt 0 cours du pe runt be {ued wath Aunevlor a dhm dlwdle kd Ít nad ic practi on huni QW ít and $1500

$1000 pe ee F

_Íq) Dose ono 0QMA, krv to lbog,

(b) Show la eb pu “cl ph Ñ ie wbich i

fhe same ad fe rosrue per proctoal qoune What doar fi rol

_ a IH “a du additional ưu — PN ts

Thedleg na kHÍ d 8) pướn eo mày | cm Practical —> Rownue : appcainly GINO per count

The or ÂN ( es

- Humane —> Ruau:appnonll 1000 pu tow Nu (a) Doix on opti bụng s

(b) Por additional, coune cots $1500

The rst sham of dụ additional qoute, ?

Figure 3.1

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_Andutr; ; www.vhu.edu,

s)k: a iced QUA hes a hohe gle (2 %10)

8 'womlv suøe th xà dị (>0) Ra mM «3i 2neuuilc: bạn I ot scvb Siấgtixgidlssbs, 44044

keo ti R= | S001 + N0dy deste k abet | cS gr: Sa) the keni ite Y that potilly oll his Am ạv (l0, vì and te SS 1 es hà inl So : mộ | -Tkmaiem đ.É à dan Ủ- (000) ond ah R= 90x HOO +10 1000 = W000 _ — khung, rR wan aD bet, |

ÔOW[Nh

Figure 3.2

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Question 4:

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1 “Shed “tis | Ki 2imaxikeakion _ prob lemy theo opt: mization

direction for igo ploFit ig away from tPe origin he 4Pe ido profit line Veoues | Moving in thet direction,

Hopi r0 Phi +] ae em) ak the uel

“apt mại eeluction is as folbw

er : ud rasa TP a6 ke ~—=+teimtdtrrtrrrn Pha

L——® 2 > May.= 18.000 0 BAK te) 9000.» ` RC Lele

Ld Let gh #46¬ tr lis tour al Saas?

Figure 4.2

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L, = wamber of HiFi-d units

%= number of HiFi- a units

Stat 4a, € 480,90% Garr hag < 432

Constraints » Ne Bras Sag < 480,86% => 4 Su, + Sa, < 440.8 (4276) — LAG + 6x; < be x 88% hay + 64, < 420.4 641, + 4454 Sy = 432 We have: {¬ + 5a, +S = AALS 4a, + 6 ty 43, = 4aa.4 => ryt ba 4 Ss, 4 Sey + Saat $+ baat Ga, +S; = ABQ + ANAS + 422.4 > ASH, 4 ADU + Sha Sy + Sy = ARGTL = Sy + Spt S,2 A863, 3 —(A5 a4 + ‘5%, ) V+, ~~ ue TN P E z MinP = MaxZ > Ort hays 430 (1) #10 72 %;| A0E 0 9,5%, + 5a, = #A9.% (2)

%| 0 32.56

| 8256 0 Dd Aaya Guy = 490.4 (3)

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Question 6:

6 An wduinal recycling Center Uren THO Acrap aurntnum metals , Pr & B, 4

rey ALecon, ond 5⁄ carbon, Te cost eae?

for AvopA AB ong $100 and $80, napect iy The Apecificad ows of the Apeaiol

pe ee đe tr gu Dh cache Comdtrd mudt be ot Rost À7, & at mom bY,

(2) +he AiQtcon codent munt be hetusem 37% SY ond CRY ah mache: ended pate | Re between y 4 TY | Determune the optimism mux of the Acrrapd that should be ured

| tw Predacing \CƠO toms of RL allo Tn Aun

— Re induarnral eo Cates

~ They howe 3 be, of Acre |

VN inckades CY Rann

os cai, SY ahi caine

ew hor eukput Ww 1000 tons of tRe oo

SY AUUece~ :W/cenkew-, er, Akicon ; 7 corbor ~ Cot of each Ara sper tow!

* Cort of fr pew ton $100 se số Ui pie ASS 880

a The cesxponedx te Re Moy ane requlasedl ors foros :

`." + oA sosA BY and ok most Gy ` CAN : +h Arco betveen 2% ond OY

= €3)+ the carror = betuew 37 amd 4Y

ci Variables : ReaD,

%y + proportion of 4620p xe, v Row allow

Xe + ptOpewAdx O soap RM TỦ vr olay

Function obyechwve

Maw Z = 1000 - (1009-4) 4 1000+ (90) + x, Rostructions : * ©,0Gx, + O,O%x, % O03 * O,06xy 4 0,03¥, £0, { coded oan - ©,OSx; * O,Oby, 9 6,03 € OS, 45 ĐyOEo + 005 en “O94, 7S, 58x > O08 + 2 Qy

"` Loos t Bowdien che

"UX Sor CaP Aur of the propotivon muht be equal \)

\ 1X7 (now ~ raqoacwry ) eo

Figure 6.1

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Pow UHLAL be oS, su Êxakä C§£„„

À King, Wt Beck a : \ Atk el 2 \ ly

C iG ` %y c

wee ae

; ` # =~ O,O} = i *Ÿ x C\ FZ

KS ©, À

Nà o nod = ISSO ( OD -(100)- x, 4 1600 (ge) , S L

ee ene 2 The optimal reno in the alle, ah

rae

- Ahould be o 183 ADop K 10,3 10,6 lsc coed Ond O06 Acrop ® crop B The hy Ninn cot per I900 tons

Of We okey woud be gic cce, & ¢

Figure 6.3 Form the data that we have:

X,; -X: the pertencage of work that Fox wanna undertake i: the number order of the project

Explain:

The number between year and project is the number of fund that Fox have to pay to do the undertake depend on the pertentage of Fox wanna undertake completely (¢1) or partically (0< .<1

I Minimize the fund that Fox have to pay