Petunjuk : Pergunakanlah garis selidik atau titik pojok (sudut) untuk

menyelesaikan permasalahan berikut.

 

01.  Rokok jenis “arma” yang harganya 200 dukat per bungkus dijual dengan laba 40

dukat per bungkus, sedangkan rokok jenis “armin” yang harganya 100 dukat per bungkus

dijual dengan laba 30 dukat per bungkus. Seorang pedagang yang mempunyai modal

80.000 dukat, dan kiosnya mampu menampung 500 bungkus rokok. Agar pedagang

tersebut memperolah laba maksimum, tentukan :

1. Berapa bungkus rokok yang harus dibeli.

2. Berapa keuntungan maksimumnya

02.  Seorang penjaja kue membeli kue A dengan harga 100 cent, kemudian menjualnya

dengan harga 130 cent per potongnya. Sedangkan kue B dibeli dengan harga 200 cent, dan

dijual 220 cent. Jika modal yang dimiliki hanya 4000 cent, dan setiap hari hanya mampu

menjual 30 kue, maka laba maksimal setiap harinya adalah ?

03.  Untuk dapat diterima di suatu PT ternama, seorang calon harus lulus tes matematika

tidak kurang dari 8, tes logika tidak kurang dari 5. Si Steven memiliki hasil ujian dimana

jumlah dua kali nilai matematika dan tiga kali nilai logika sama denga 30. Menurut anda

diterima atau tidak calon tersebut ? (=ditolak, karena ? )

04.  Seorang saudagar menjual dua jenis barang A dan B. Harga pembelian barang A Rp

1000 per satuan, barang B Rp 400 per satuan. Modal yang dmiliki Rp 250.000, sementara

jumlah barang yang dibeli tidak lebih 400 satuan. Jika keuntungan barang A adalah dua

kali keuntungan barang B, maka agar diperoleh keuntungan maksimal, pedagang harus

membeli barang sebanyak ?

05.  Seorang yang ingin sehat bermaksud untuk minum sedikitnya 36 sat vitamin A per

hari, 28 sat vit C dan 32 sat vitamin D. Multivitamin jenis pertama berharga 3 sat uang

menyediakan 2 sat vitamin A per hari, 2 sat vitamin C dan 8 sat vitamin D. Multivitamin

jenis kedua berharga 4 sat uang menyediakan 3 sat vitamin A per hari, 2 sat vitamin C dan

2 sat vitamin D. Tentukan jumlah vitamin yang harus diminum agar kebutuhan akan

vitamin terpenuhi.

06.  Sebuah perusahaan truk mempunyai 2 jenis truk. Jenis I mempunyai 20 m3 kotak

berpendingin dan 40 m3 tanpa pendingin. Kendaraan jenis II mempunyai 30 m3 kotak

pendingin, dan 30 m3 tanpa pendingin. Petani ingin mengirim 900 m3 sayuran yang harus

dikirim dengan cara mendinginkan, dan 1200 m3 tanpa harus didinginkan. Tentukan 

jumlah truk yang harus disewa agar ongkos sewa minimum, jika :

1. Ongkos truk jenis I adalah 30 sat.uang, dan jenis II adalah 40 sat.uang

2. Ongkos truk jenis I adalah 20 sat.uang, dan jenis II adalah 30 sat.uang

07.  Seorang petani memerlukan 3 jenis zat kimia A,B, dan C masing-masing sebesar 10,12,

dan 12 satuan per satuan luas. Produk dalam bentuk kering memuat zat A,B, dan C masing-

masing 5,2 dan 1 satuan per karton. Produk dalam bentuk basah memuat zat A,B, dan C

masing-masing 1,2 dan 4 per botol. Jika produk cair dijual dengan harga 3 satuan uang per

botol dan produk kering dijual 2 satuan uang per karton, tentukan jumlah produk yang

harus dibeli dengan ongkos sekecil mungkin.

08.  Suatu perusahaan penambangan mempunyai 2 tempat. Tiap hari, penambangan A

menghasilkan masing-masing 1 ton, 3 ton dan 5 ton biji besi kualitas kurang,sedang dan

baik. Penambangan B menghasilkan masing-masing 2 ton biji besi. Perusahaan

memerlukan masing-masing 80 ton, 160 ton dan 200 ton biji besi berkualitas kurang,

sedang dan baik. Berapa hari penambangan tersebut harus bekerja agar ongkos yang

dikeluarkan sekecil mungkin jika ongkos penambangan 200 satuan uang per hari.

09.  Pedagang roti memiliki tiga bahan A,B, dan C masing-masing sebanyak 150,90 dan 150

satuan. Roti polos memerlukan bahan A,B, dan C masing-masing sebanyak 1,1, dan 2

satuan, sedangkan kue memerlukan bahan A,B, dan C masing-masing sebanyak 5,2, dan 1

satuan. Tentukan jumlah roti dan kue yang harus dibuat agar pendapatan maksimum.

10.  A plant makes aluminum and copper wire. Each pound of aluminum wire requires 5

kwh of electricity and 0.25 hour of labor. Each pound of copper wire requires 2 kwh of

electricity and 0.5 hour of labor. Production of copper wire is restricted by the fact that raw

materials are available to produce at most 60 lbs/ day. Electricity is limited to 500 kwh/ day

and labor to 40 person – hour/ day. If the profit from aluminum wire is $ 0.25/lb, and the

profit from copper is $ 0.40/ lb., how much of each should be produced to maximize profit

and what is the maximum profit?

11.  A farmer has a 320 acre farm on which she plants two crops: corn and soybeans. For

each acre of corn planted, her expenses are $ 50 and for each acre of soybeans planted,

her expenses are $ 100. Each acre of corn requires 100 bushels of storage and yields a

profit of $ 60; each acre of soybeans requires 40 bushels of storage and yields a profit of $

90. If the total amount of storage space available is 19,200 bushels and the farmer has only

$ 20,000 on hand, how many acres of each crop should she plant in order to maximize her

profit? What will her profit be if she follows this strategy?

12.  A potter is making cups and plates. It takes her 6 minutes to make a cup and 3 minutes

to make a plate. Each cup uses 0.75 lb of clay and each plate uses one lb of clay. She has

20 hours available for making the cups and plates and has 250 lb of clay on hand. She

makes a profit of $ 2 on each cup and $ 1.50 on each plate. How many cups and how many

plates should she make in order to maximize her profit?

Linear Programming Problems

 

1.  A farmer has 10 acres to plant in wheat and rye. He has to plant at least 7 acres.

However, he has only $1200 to spend and each acre of wheat costs $200 to plant

and each acre of rye costs $100 to plant. Moreover, the farmer has to get the

planting done in 12 hours and it takes an hour to plant an acre of wheat and 2 hours

to plant an acre of rye. If the profit is $500 per acre of wheat and $300 per acre of

rye how many acres of each should be planted to maximize profits?

2. A gold processor has two sources of gold processor, source A and source B. In order

to keep his plant running, at least three tons of ore must be processed each day. Ore

from source A costs $20 per ton to process, and ore from source B costs $10 per ton

to process. Costs must be kept to less than $80 per day. Moreover, Federal

Regulations require that the amount of ore from source B cannot exceed twice the

amount of ore from source A. If ore from source A yields 2 oz. of gold per ton, and

ore from source B yields 3 oz. of gold per ton, how many tons of ore from both

sources must be processed each day to maximize the amount of gold extracted

subject to the above constraints?

3. A publisher has orders for 600 copies of a certain text from San Francisco and 400

copies from Sacramento. The company has 700 copies in a warehouse in Novato and

800 copies in a warehouse in Lodi. It costs $5 to ship a text from Novato to San

Francisco, but it costs $10 to ship it to Sacramento. It costs $15 to ship a text from

Lodi to San Francisco, but it costs $4 to ship it from Lodi to Sacramento. How many

copies should the company ship from each warehouse to San Francisco and

Sacramento to fill the order at the least cost?

4. A calculator company produces a scientific calculator and a graphing calculator.

Long-term projections indicate an expected demand of at least 100 scientific

and 80 graphing calculators each day. Because of limitations on production

capacity, no more than 200scientific and 170 graphing calculators can be made

daily. To satisfy a shipping contract, a total of at least 200 calculators much be

shipped each day. If each scientific calculator sold results in a $2 loss, but each

graphing calculator produces a $5 profit, how many of each type should be made

daily to maximize net profits?

5. You need to buy some filing cabinets. You know that Cabinet X costs $10 per unit,

requires six square feet of floor space, and holds eight cubic feet of files. Cabinet Y

costs $20 per unit, requires eight square feet of floor space, and holds twelve cubic

feet of files. You have been given $140 for this purchase, though you don’t have to

spend that much. The office has room for no more than 72 square feet of cabinets.

How many of which model should you buy, in order to maximize storage volume?

        

Hints:

1. Every group just solves three problems; but you must answer question number four.

2. Due date: 31 August 2009 by email.

 

Steps:

1. Define your unknowns

2. Determine the mathematic model of each problem

3. Express the objective and the constraints.

4. Write the constraints and the objective function.

5. Graph the solution area of the problems, and find the feasible corner points

6. Evaluate the objective function in all of the feasible corner points