Sunday, April 20, 2014

WHAT IS TSUNAMI ? ----- BY. MWL. JAPHET MASATU.

WHAT   IS   TSUNAMI  ?

INTRODUCTION:
For other uses, see Tsunami (disambiguation).
2011 Tōhoku earthquake and tsunami, An aerial view of damage in the Sendai region with black smoke coming from the Nippon Oil Sendai oil refinery
A tsunami (plural: tsunamis or tsunami; from Japanese: 津波, lit. "harbour wave";[1] English pronunciation: /suːˈnɑːmi/ soo-NAH-mee or /tsuːˈnɑːmi/ tsoo-NAH-mee[2]) is a series of water waves caused by the displacement of a large volume of a body of water, generally an ocean or a large lake. Earthquakes, volcanic eruptions and other underwater explosions (including detonations of underwater nuclear devices), landslides, glacier calvings, meteorite impacts and other disturbances above or below water all have the potential to generate a tsunami.[3]
Tsunami waves do not resemble normal sea waves, because their wavelength is far longer. Rather than appearing as a breaking wave, a tsunami may instead initially resemble a rapidly rising tide, and for this reason they are often referred to as tidal waves. Tsunamis generally consist of a series of waves with periods ranging from minutes to hours, arriving in a so-called "wave train".[4] Wave heights of tens of metres can be generated by large events. Although the impact of tsunamis is limited to coastal areas, their destructive power can be enormous and they can affect entire ocean basins; the 2004 Indian Ocean tsunami was among the deadliest natural disasters in human history with at least 290,000 people killed or missing in 14 countries bordering the Indian Ocean.
The Greek historian Thucydides suggested in his late 5th century BC, History of the Peloponnesian War, that tsunamis were related to submarine earthquakes,[5][6] but the understanding of a tsunami's nature remained slim until the 20th century and much remains unknown. Major areas of current research include trying to determine why some large earthquakes do not generate tsunamis while other smaller ones do; trying to accurately forecast the passage of tsunamis across the oceans; and also to forecast how tsunami waves would interact with specific shorelines.
Taken at Ao Nang, Krabi Province, Thailand, during the 2004 Indian Ocean earthquake and tsunami in Thailand

Etymology

Tsunami warning bilingual sign in Ulee Lheue, Banda Aceh in Acehnese and Indonesian
The term tsunami comes from the Japanese 津波, composed of the two kanji 津 (tsu) meaning "harbour" and 波 (nami), meaning "wave". (For the plural, one can either follow ordinary English practice and add an s, or use an invariable plural as in the Japanese.[7])
Tsunami are sometimes referred to as tidal waves, which are unusually high sea waves that are triggered especially by earthquakes.[8] In recent years, this term has fallen out of favor, especially in the scientific community, because tsunami actually have nothing to do with tides. The once-popular term derives from their most common appearance, which is that of an extraordinarily high tidal bore. Tsunami and tides both produce waves of water that move inland, but in the case of tsunami the inland movement of water is much greater and lasts for a longer period, giving the impression of an incredibly high tide. Although the meanings of "tidal" include "resembling"[9] or "having the form or character of"[10] the tides, and the term tsunami is no more accurate because tsunami are not limited to harbours, use of the term tidal wave is discouraged by geologists and oceanographers.
There are only a few other languages that have an equivalent native word. In Acehnese language, the words are ië beuna[11] or alôn buluëk[12] (depending on the dialect). In Tamil language, it is aazhi peralai. On Simeulue island, off the western coast of Sumatra in Indonesia, in Devayan language the word is smong, while in Sigulai language it is emong.[13] In Singkil (in Aceh province) and surrounding, the people name tsunami with word gloro.[14]

History

See also: List of historic tsunamis
Lisbon earthquake and tsunami in 1755
The Russians of Pavel Lebedev-Lastochkin in Japan, with their ships tossed inland by a tsunami, meeting some Japanese in 1779
As early as 426 BC the Greek historian Thucydides inquired in his book History of the Peloponnesian War about the causes of tsunami, and was the first to argue that ocean earthquakes must be the cause.[5][6]
"The cause, in my opinion, of this phenomenon must be sought in the earthquake. At the point where its shock has been the most violent the sea is driven back, and suddenly recoiling with redoubled force, causes the inundation. Without an earthquake I do not see how such an accident could happen."[15]
The Roman historian Ammianus Marcellinus (Res Gestae 26.10.15-19) described the typical sequence of a tsunami, including an incipient earthquake, the sudden retreat of the sea and a following gigantic wave, after the 365 AD tsunami devastated Alexandria.[16][17]
While Japan may have the longest recorded history of tsunamis, the sheer destruction caused by the 2004 Indian Ocean earthquake and tsunami event mark it as the most devastating of its kind in modern times, killing around 230,000 people. The Sumatran region is not unused to tsunamis either, with earthquakes of varying magnitudes regularly occurring off the coast of the island.[18]

Generation mechanisms

The principal generation mechanism (or cause) of a tsunami is the displacement of a substantial volume of water or perturbation of the sea.[19] This displacement of water is usually attributed to either earthquakes, landslides, volcanic eruptions, glacier calvings or more rarely by meteorites and nuclear tests.[20][21] The waves formed in this way are then sustained by gravity. Tides do not play any part in the generation of tsunamis.

Seismicity

Tsunami can be generated when the sea floor abruptly deforms and vertically displaces the overlying water. Tectonic earthquakes are a particular kind of earthquake that are associated with the Earth's crustal deformation; when these earthquakes occur beneath the sea, the water above the deformed area is displaced from its equilibrium position.[22] More specifically, a tsunami can be generated when thrust faults associated with convergent or destructive plate boundaries move abruptly, resulting in water displacement, owing to the vertical component of movement involved. Movement on normal faults will also cause displacement of the seabed, but the size of the largest of such events is normally too small to give rise to a significant tsunami.
Tsunamis have a small amplitude (wave height) offshore, and a very long wavelength (often hundreds of kilometres long, whereas normal ocean waves have a wavelength of only 30 or 40 metres),[23] which is why they generally pass unnoticed at sea, forming only a slight swell usually about 300 millimetres (12 in) above the normal sea surface. They grow in height when they reach shallower water, in a wave shoaling process described below. A tsunami can occur in any tidal state and even at low tide can still inundate coastal areas.
On April 1, 1946, a magnitude-7.8 (Richter Scale) earthquake occurred near the Aleutian Islands, Alaska. It generated a tsunami which inundated Hilo on the island of Hawai'i with a 14-metre high (46 ft) surge. The area where the earthquake occurred is where the Pacific Ocean floor is subducting (or being pushed downwards) under Alaska.
Examples of tsunami originating at locations away from convergent boundaries include Storegga about 8,000 years ago, Grand Banks 1929, Papua New Guinea 1998 (Tappin, 2001). The Grand Banks and Papua New Guinea tsunamis came from earthquakes which destabilised sediments, causing them to flow into the ocean and generate a tsunami. They dissipated before traveling transoceanic distances.
The cause of the Storegga sediment failure is unknown. Possibilities include an overloading of the sediments, an earthquake or a release of gas hydrates (methane etc.).
The 1960 Valdivia earthquake (Mw 9.5) (19:11 hrs UTC), 1964 Alaska earthquake (Mw 9.2), 2004 Indian Ocean earthquake (Mw 9.2) (00:58:53 UTC) and 2011 Tōhoku earthquake (Mw9.0) are recent examples of powerful megathrust earthquakes that generated tsunamis (known as teletsunamis) that can cross entire oceans. Smaller (Mw 4.2) earthquakes in Japan can trigger tsunamis (called local and regional tsunamis) that can only devastate nearby coasts, but can do so in only a few minutes.

Landslides

In the 1950s, it was discovered that larger tsunamis than had previously been believed possible could be caused by giant submarine landslides. These rapidly displace large water volumes, as energy transfers to the water at a rate faster than the water can absorb. Their existence was confirmed in 1958, when a giant landslide in Lituya Bay, Alaska, caused the highest wave ever recorded, which had a height of 524 metres (over 1700 feet).[24] The wave didn't travel far, as it struck land almost immediately. Two people fishing in the bay were killed, but another boat amazingly managed to ride the wave.
Another landslide-tsunami event occurred in 1963 when a massive landslide from Monte Toc went into the Vajont Dam in Italy. The resulting wave overtopped the 262 m (860 ft) high dam by 250 metres (820 ft) and destroyed several towns. Around 2,000 people died.[25][26] Scientists named these waves megatsunami.
Devastation wrought by Hurricane Ike's meteotsunamic storm surge over the Bolivar Peninsula in 2008.
Scientists discovered that extremely large landslides from volcanic island collapses can generate megatsunamis that can cross oceans.

Meteotsunamis

Some meteorological conditions, especially deep depressions such as tropical cyclones, can generate a type of storm surge called a meteotsunami which raises water heights above normal levels, often suddenly at the shoreline.[27]
In the case of deep tropical cyclones, this is due to very low atmospheric pressure and inward swirling winds causing an uplifted dome of water to form under and travel in tandem with the storm. When these water domes reach shore, they rear up in shallows and surge laterally like earthquake-generated tsunamis, typically arriving shortly after landfall of the storm's eye.[28][29]

Characteristics

When the wave enters shallow water, it slows down and its amplitude (height) increases.
The wave further slows and amplifies as it hits land. Only the largest waves crest.
Tsunamis cause damage by two mechanisms: the smashing force of a wall of water travelling at high speed, and the destructive power of a large volume of water draining off the land and carrying a large amount of debris with it, even with waves that do not appear to be large.
While everyday wind waves have a wavelength (from crest to crest) of about 100 metres (330 ft) and a height of roughly 2 metres (6.6 ft), a tsunami in the deep ocean has a much larger wavelength of up to 200 kilometres (120 mi). Such a wave travels at well over 800 kilometres per hour (500 mph), but owing to the enormous wavelength the wave oscillation at any given point takes 20 or 30 minutes to complete a cycle and has an amplitude of only about 1 metre (3.3 ft).[30] This makes tsunamis difficult to detect over deep water, where ships are unable to feel their passage.
The reason for the Japanese name "harbour wave" is that sometimes a village's fishermen would sail out, and encounter no unusual waves while out at sea fishing, and come back to land to find their village devastated by a huge wave.
As the tsunami approaches the coast and the waters become shallow, wave shoaling compresses the wave and its speed decreases below 80 kilometres per hour (50 mph). Its wavelength diminishes to less than 20 kilometres (12 mi) and its amplitude grows enormously. Since the wave still has the same very long period, the tsunami may take minutes to reach full height. Except for the very largest tsunamis, the approaching wave does not break, but rather appears like a fast-moving tidal bore.[31] Open bays and coastlines adjacent to very deep water may shape the tsunami further into a step-like wave with a steep-breaking front.
When the tsunami's wave peak reaches the shore, the resulting temporary rise in sea level is termed run up. Run up is measured in metres above a reference sea level.[31] A large tsunami may feature multiple waves arriving over a period of hours, with significant time between the wave crests. The first wave to reach the shore may not have the highest run up.[32]
About 80% of tsunamis occur in the Pacific Ocean, but they are possible wherever there are large bodies of water, including lakes. They are caused by earthquakes, landslides, volcanic explosions, glacier calvings, and bolides.

Drawback

An illustration of the rhythmic "drawback" of surface water associated with a wave. It follows that a very large drawback may herald the arrival of a very large wave.
All waves have a positive and negative peak, i.e. a ridge and a trough. In the case of a propagating wave like a tsunami, either may be the first to arrive. If the first part to arrive at shore is the ridge, a massive breaking wave or sudden flooding will be the first effect noticed on land. However if the first part to arrive is a trough, a drawback will occur as the shoreline recedes dramatically, exposing normally submerged areas. Drawback can exceed hundreds of metres, and people unaware of the danger sometimes remain near the shore to satisfy their curiosity or to collect fish from the exposed seabed.
A typical wave period for a damaging tsunami is about 12 minutes. This means that if the drawback phase is the first part of the wave to arrive, the sea will recede, with areas well below sea level exposed after 3 minutes. During the next 6 minutes the tsunami wave trough builds into a ridge, and during this time the sea is filled in and destruction occurs on land. During the next 6 minutes, the tsunami wave changes from a ridge to a trough, causing flood waters to drain and drawback to occur again. This may sweep victims and debris some distance from land. The process repeats as the next wave arrives.

Scales of intensity and magnitude

As with earthquakes, several attempts have been made to set up scales of tsunami intensity or magnitude to allow comparison between different events.[33]

Intensity scales

The first scales used routinely to measure the intensity of tsunami were the Sieberg-Ambraseys scale, used in the Mediterranean Sea and the Imamura-Iida intensity scale, used in the Pacific Ocean. The latter scale was modified by Soloviev, who calculated the Tsunami intensity I according to the formula
\,\mathit{I} = \frac{1}{2} + \log_{2} \mathit{H}_{av}
where \mathit{H}_{av} is the average wave height along the nearest coast. This scale, known as the Soloviev-Imamura tsunami intensity scale, is used in the global tsunami catalogues compiled by the NGDC/NOAA[34] and the Novosibirsk Tsunami Laboratory as the main parameter for the size of the tsunami.

Magnitude scales

The first scale that genuinely calculated a magnitude for a tsunami, rather than an intensity at a particular location was the ML scale proposed by Murty & Loomis based on the potential energy.[33] Difficulties in calculating the potential energy of the tsunami mean that this scale is rarely used. Abe introduced the tsunami magnitude scale \mathit{M}_{t}, calculated from,
\,\mathit{M}_{t} = {a} \log h + {b} \log R = \mathit{D}
where h is the maximum tsunami-wave amplitude (in m) measured by a tide gauge at a distance R from the epicentre, a, b and D are constants used to make the Mt scale match as closely as possible with the moment magnitude scale.[35]

Warnings and predictions

See also: Tsunami warning system
Tsunami warning sign
Drawbacks can serve as a brief warning. People who observe drawback (many survivors report an accompanying sucking sound), can survive only if they immediately run for high ground or seek the upper floors of nearby buildings. In 2004, ten-year old Tilly Smith of Surrey, England, was on Maikhao beach in Phuket, Thailand with her parents and sister, and having learned about tsunamis recently in school, told her family that a tsunami might be imminent. Her parents warned others minutes before the wave arrived, saving dozens of lives. She credited her geography teacher, Andrew Kearney.
In the 2004 Indian Ocean tsunami drawback was not reported on the African coast or any other east-facing coasts that it reached. This was because the wave moved downwards on the eastern side of the fault line and upwards on the western side. The western pulse hit coastal Africa and other western areas.
A tsunami cannot be precisely predicted, even if the magnitude and location of an earthquake is known. Geologists, oceanographers, and seismologists analyse each earthquake and based on many factors may or may not issue a tsunami warning. However, there are some warning signs of an impending tsunami, and automated systems can provide warnings immediately after an earthquake in time to save lives. One of the most successful systems uses bottom pressure sensors, attached to buoys, which constantly monitor the pressure of the overlying water column.
Regions with a high tsunami risk typically use tsunami warning systems to warn the population before the wave reaches land. On the west coast of the United States, which is prone to Pacific Ocean tsunami, warning signs indicate evacuation routes. In Japan, the community is well-educated about earthquakes and tsunamis, and along the Japanese shorelines the tsunami warning signs are reminders of the natural hazards together with a network of warning sirens, typically at the top of the cliff of surroundings hills.[36]
The Pacific Tsunami Warning System is based in Honolulu, Hawaiʻi. It monitors Pacific Ocean seismic activity. A sufficiently large earthquake magnitude and other information triggers a tsunami warning. While the subduction zones around the Pacific are seismically active, not all earthquakes generate tsunami. Computers assist in analysing the tsunami risk of every earthquake that occurs in the Pacific Ocean and the adjoining land masses.
As a direct result of the Indian Ocean tsunami, a re-appraisal of the tsunami threat for all coastal areas is being undertaken by national governments and the United Nations Disaster Mitigation Committee. A tsunami warning system is being installed in the Indian Ocean.
One of the deep water buoys used in the DART tsunami warning system
Computer models can predict tsunami arrival, usually within minutes of the arrival time. Bottom pressure sensors relay information in real time. Based on these pressure readings and other seismic information and the seafloor's shape (bathymetry) and coastal topography, the models estimate the amplitude and surge height of the approaching tsunami. All Pacific Rim countries collaborate in the Tsunami Warning System and most regularly practice evacuation and other procedures. In Japan, such preparation is mandatory for government, local authorities, emergency services and the population.
Some zoologists hypothesise that some animal species have an ability to sense subsonic Rayleigh waves from an earthquake or a tsunami. If correct, monitoring their behavior could provide advance warning of earthquakes, tsunami etc. However, the evidence is controversial and is not widely accepted. There are unsubstantiated claims about the Lisbon quake that some animals escaped to higher ground, while many other animals in the same areas drowned. The phenomenon was also noted by media sources in Sri Lanka in the 2004 Indian Ocean earthquake.[37][38] It is possible that certain animals (e.g., elephants) may have heard the sounds of the tsunami as it approached the coast. The elephants' reaction was to move away from the approaching noise. By contrast, some humans went to the shore to investigate and many drowned as a result.
Along the United States west coast, in addition to sirens, warnings are sent on television and radio via the National Weather Service, using the Emergency Alert System.

Forecast of tsunami attack probability

Kunihiko Shimazaki (University of Tokyo), a member of Earthquake Research committee of The Headquarters for Earthquake Research Promotion of Japanese government, mentioned the plan to public announcement of tsunami attack probability forecast at Japan National Press Club on 12 May 2011. The forecast includes tsunami height, attack area and occurrence probability within 100 years ahead. The forecast would integrate the scientific knowledge of recent interdisciplinarity and aftermath of the 2011 Tōhoku earthquake and tsunami. As the plan, announcement will be available from 2014.[39][40][41]

Mitigation

See also: Tsunami barrier
Photo of seawall with building in background
A seawall at Tsu, Japan
In some tsunami-prone countries earthquake engineering measures have been taken to reduce the damage caused onshore.
Japan, where tsunami science and response measures first began following a disaster in 1896, has produced ever-more elaborate countermeasures and response plans.[42] That country has built many tsunami walls of up to 12 metres (39 ft) high to protect populated coastal areas. Other localities have built floodgates of up to 15.5 metres (51 ft) high and channels to redirect the water from incoming tsunami.
However, their effectiveness has been questioned, as tsunami often overtop the barriers. The 2011 Fukushima nuclear disaster was directly triggered by a tsunami that exceeded the height of the plant's sea wall.[43] The Okushiri, Hokkaidō tsunami which struck Okushiri Island of Hokkaidō within two to five minutes of the earthquake on July 12, 1993 created waves as much as 30 metres (100 ft) tall—as high as a 10-story building. The port town of Aonae was completely surrounded by a tsunami wall, but the waves washed right over the wall and destroyed all the wood-framed structures in the area. The wall may have succeeded in slowing down and moderating the height of the tsunami, but it did not prevent major destruction and loss of life.[44] Iwate Prefecture, which is an area at high risk from tsunami, had tsunami barriers walls totalling 25 kilometres (16 mi) long at coastal towns. The 2011 tsunami toppled more than 50% of the walls and caused many damages.[45]

As a weapon

There have been studies and at least one attempt to create tsunami waves as a weapon. In World War II, the New Zealand Military Forces initiated Project Seal, which attempted to create small tsunamis with explosives in the area of today's Shakespear Regional Park; the attempt failed.[46]

See also

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Footnotes

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LINEAR PROGRAMMING PROBLEMS AND SOLUTIONS ----- BY. MWL. JAPHET MASATU.

LINEAR PROGRAMMING PROBLEMS AND SOLUTIONS 













PROBLEM NUMBER 1

A farmer can plant up to 8 acres of land with
wheat and barley. He can earn $5,000 for every
acre he plants with wheat and $3,000 for every
acre he plants with barley. His use of a
necessary pesticide is limited by federal
regulations to 10 gallons for his entire 8 acres.
Wheat requires 2 gallons of pesticide for every
acre planted and barley requires just 1 gallon
per acre.

What is the maximum profit he can make?

SOLUTION TO PROBLEM NUMBER 1

let x = the number of acres of wheat
let y = the number of acres of barley.

since the farmer earns $5,000 for each acre of wheat and $3,000 for each acre of barley, then the total profit the farmer can earn is 5000*x + 3000*y.

let p = total profit that can be earned. your equation for profit becomes:

p = 5000x + 3000y

that's your objective function. it's what you want to maximize

the constraints are:
number of acres has to be greater than or equal to 0.
number of acres has to be less than or equal to 8.
amount of pesticide has to be less than or equal to 10.

your constraint equations are:
x >= 0
y >= 0
x + y <= 8
2x + y <= 10

to graph these equations, solve for y in those equations that have y in them and then graph the equality portion of those equations.

x >= 0
y >= 0
y <= 8-x
y <= 10 - 2x

x = 0 is a vertical line that is the same line as the y-axis.
y = 0 is a horizontal line that is the same line as the x-axis.

the area of the graph that satisfies all the constraints is the region of feasibility.

the maximum or minimum solutions to the problem will be at the intersection points of the lines that bound the region of feasibility.

the graph of your equations looks like this:
$$$$

the region of feasibility is the shaded area of the graph.

you can see from this graph that the region of feasibility is bounded by the following (x,y) coordinate points:
(0,0)
(0,8)
(2,6)
(5,0)

the point (0,0) is the intersection of the line x-axis with the y-axis.
the point (0,8) is the intersection of the line y = 8 - x with the y-axis.
the point (5,0) is the intersection of the line y = 10 - 2x with the x-axis.
the point (2,6) is the intersection of the line y = 8 - x with the line y = 10 - 2x.

the point (2,6) was solved for in the following manner:
equations of the intersecting lines are:
y = 8 - x
y = 10 - 2x
subtract the first equation from the second equation and you get:
0 = 2 - x
add x to both sides of this equation and you get:
x = 2
substitute 2 for x in either equation to get y = 6.
that makes the intersection point (x,y) = (2,6).

the objective equation is:
p = 5000x + 3000y

profit will be maximum at the intersection points of the region of feasibility on the graph.
the profit equation is evaluated at each of these points as shown in the following table.
intersection point of (x,y)         p
             (0,0)                  $0
             (0,8)                  $24,000
             (2,6)                  $28,000 *****
             (5,0)                  $25,000

the maximum profit occurs when the farmer plants 2 acres of wheat and 6 acres of barley.
number of acres of wheat is 2 and number of acres of barley is 6 for a total of 8 acres which is the maximum number of acres available for planting.
number of gallons of pesticide used for wheat is 4 and number of gallons of pesticide used for barley is 6 for a total of 10 gallons of pesticide which is the maximum amount of pesticide that can be used.

PROBLEM NUMBER 2

A painter has exactly 32 units of yellow dye and 54 units of green dye.
He plans to mix as many gallons as possible of color A and color B.
Each gallon of color A requires 4 units of yellow dye and 1 unit of green dye.
Each gallon of color B requires 1 unit of yellow dye and 6 units of green dye.

Find the maximum number of gallons he can mix.

SOLUTION TO PROBLEM NUMBER 2

the objective function is to determine the maximum number of gallons he can mix.

the colors involved are color A and color B.

let x = the number of gallons of color A.
let y = the number of gallons of color B.

if we let g = the maximum gallons the painter can make, then the objective function becomes:

g = x + y

make a table for color A and color B to determine the amount of each dye required.
your table will look like this:

each gallon of color A or B will require:
              units of yellow dye        units of green dye

color A                4                         1
color B                1                         6


total units of yellow dye available are 32
total units of green dye available are 54

your constraint equations are:
x >= 0
y >= 0
4x + y <= 32
x + 6y <= 54

x and y have to each be greater than or equal to 0 because the number of gallons can't be negative.

in order to graph these equations, you solve for y in those equations that have y in them.

the equtions for graphing are:

x >= 0
y >= 0
y <= 32 - 4x
y <= (54 - x)/6

x = 0 is a vertical line that is the same line as the y-axis.
y = 0 is a horizontal line that is the same line as the x-axis.

the graph will look like this:
$$$$

the region of feasibility is the shaded area of the graph.
all points within the feasibility region meet the constraint of the problem.

the intersection points of the region of feasibility are:
(0,0)
(0,9)
(6,8)
(8,0)

the maximum or minimum value of the objective function will be at these points of intersection.

solve the objective function at each of these intersection points to determine which point contains the maximum number of gallons.

the objective function is:
g = x + y
the table with the value of g at each of these intersection points is shown below:
intersection point (x,y)    gallons of paint
(0,0)                               0
(0,9)                               9
(6,8)                               14 *****
(8,0)                               8


the maximum gallons of paint for color A and B, given the constraints, is equal to 14.
this is comprised of 6 gallons of color A and 8 gallons of color B.
6 gallons of color A uses 24 gallons of yellow dye and 8 gallons of color B uses 8 gallons of yellow dye for a total of 32 gallons of yellow dye which is the maximum amount of yellow dye that can be used.
6 gallons of color A user 6 gallons of green dye and 8 gallons of color B uses 48 gallons of green dye for a total of 54 gallons of green dye which is the maximum amount of green dye that can be used.


PROBLEM NUMBER 3

The Bead Store sells material for customers to make their own jewelry. Customer can select beads from various bins. Grace wants to design her own Halloween necklace from orange and black beads. She wants to make a necklace that is at least 12 inches long, but no more than 24 inches long. Grace also wants her necklace to contain black beads that are at least twice the length of orange beads. Finally, she wants her necklace to have at least 5 inches of black beads.

Find the constraints, sketch the problem and find the vertices (intersection points)

SOLUTION TO PROBLEM NUMBER 3

let x = the number of inches of black beads.
let y = the number of inches of orange beads.

the objective function is the length of the necklace
there is a maximum length and a minimum length.

if you let n equal the length of the necklace, then the objective function becomes:

n = x + y

since the problem is looking for the number of inches of black beads and the number of inches of orange beads, we will let:

the constraint equations for this problem are:
x >= 0
y >= 0
x + y >= 12
x + y <= 24
x >= 2y
x >= 5

x >= 0 is there because the number of inches of black beads can't be negative.
y >= 0 is there because the number of inches of orange beads can't be negative.
x + y >= 12 is there because the total length of the necklace has to be greater than or equal to 12 inches.
x + y <= 24 is there because the total length of the necklace has to be less than or equal to 24 inches.
x >= 2y is there because the length of the black beads has to be greater than or equal to twice the length of the orange beads.
x >= 5 is there because the number of inches of black beads has to be greater than or equal to 5.

to graph these equations, we have to solve for y in each equation that has y in it and then graph the equality portion of each of them.

your equations for graphing are:
x >= 0
y >= 0
y >= 12 - x
y <= 24 - x
y <= x/2
x >= 5

x = 0 is a vertical line that is the same line as the y-axis.
y = 0 is a horizontal line that is the same line as the x-axis.
x = 5 is a vertical line at x = 5.

a graph of you equations is shown below:
$$$$

the region of feasibility is the shaded area of the graph.
all points within the region of feasibility meet the constraint requirements of the problem.

the intersection points bounding the region of feasibility are:
(8,4)
(12,0)
(16,8)
(24,0)

(8,4) is the intersection of the lines y = x/2 and y = 12 - x
to find the point of intersection, set x/2 and 12-x equal to each other and solve for xx.
you get:
x/2 = 12-x
multiply both sides of the equation by 2 to get:
x = 24-2x
add 2x to both sides of the equation to get:
3x = 24
divide both sides of the equation by 3 to get:
x = 8.
substitute 8 for x in either equation to get y = 4.

(12,0) is the intersection of the line y = 12 - x with the x-axis.
(24,0) is the intersection of the line y = 24 - x with the x-axis.
to find the point of intersection, set y equal to 0 in each equation and solve for x.

(16,8) is the intersection of the lines y = x/2 and y = 24 - x.
to find the intersection point, set x/2 equal to 24-x and solve for x.
you get:
x/2 = 24-x
multiply both sides of this equation by 2 to get:
x = 48 - 2x
add 2x to both sides of this equation to get:
3x = 48
divide both sides of this equation by 3 to get:
x = 16
substitute 16 for x in either equation to get:
y = 8.

the maximum / minimum necklace length will be at the intersection points of the boundaries of the region of feasibility.

evaluation of the objective function at these intersections yields the following:
objective function is:
x + y = n where n is the length of the necklace in inches. 

intersection points       number of inches
(8,4)                           12
(12,0)                          12
(16,8)                          24
(24,0)                          24

the number of inches of black beads is at least twice the number of inches of orange beads.
the number of inches of black beads is at least 5.
the total length of the necklace is greater than or equal to 12 inches or less than or equal to 24 inches.

all the constraints have been met.
the maximum length the necklace can be and still meet the constraints is 24 inches.
the minimum length the necklace can be and still meet the constraints is 12 inches.

PROBLEM NUMBER 4

A garden shop wishes to prepare a supply of special fertilizer at a minimal cost by mixing two fertilizers, A and B.
The mixture is to contain:
at least 45 units of phosphate
at least 36 units of nitrate
at least 40 units of ammonium
Fertilizer A costs the shop $.97 per pound.
Fertilizer B costs the shop $1.89 per pound.
fertilizer A contains 5 units of phosphate and 2 units of nitrate and 2 units of ammonium.
fertilizer B contains 3 units of phosphate and 3 units of nitrate and 5 units of ammonium.

how many pounds of each fertilizer should the shop use in order to minimize their cost.

SOLUTION TO PROBLEM NUMBER 4

let x = the number of pounds of fertilizer A.
let y = the number of pounds of fertilizer B.

the objective function is to minimize the cost.

the objective function becomes:

c = .97x + 1.89y

the constraint equations are:

since the number of pounds of each fertilizer can't be negative, 2 of the constraint equations become:

x >= 0
y >= 0

since the number of units of phosphate has to be at least 45, the constraint equation for phosphate becomes:

5x + 3y >= 45

since the number of units of nitrate must be at least 36, the constraint equation for nitrates becomes:

2x + 3y >= 36

since the number of units of ammonium must be at least 40, the constraint equation for ammonium becomes:

2x + 5y >= 40

the constraint equations for this problem become:

x >= 0
y >= 0
5x + 3y >= 45
2x + 3y >= 36
2x + 5y >= 40

in order to graph these equations, you have to solve for y in each equation that has y in it and then graph the equality portion of those equations.

the equations to be graphed are:

x >= 0
y >= 0
y >= (45-5x)/3
y >= (36 - 2x)/3
y >= (40-2x)/5

x = 0 is a vertical line that is the same line as the y-axis.
y = 0 is a horizontal line that i the same line as the x-axis.

a graph of your equation is shown below:
$$$$

the feasibility region is the area of the graph that is shaded.
all points within the region of feasibility meet the constraint requirements of the problem.

the intersection points at the boundaries of the feasibility region are:
(0,15)
(3,10)
(15,2)
(20,0)


the intersection points of the boundaries of the region of feasibility contain the minimum cost solution for the objective function in this problem.

now that you have the intersection points, you can solve for the minimum cost equation which is the objective function of:

c = .97x + 1.89y

the following table shows the value of the cost equation at each of the intersection points. 

          intersection points (x,y)          c = .97x + 1.89y        minimum solution

                   (0,15)                          28.35
                   (3,10)                          21.81
                   (15,2)                          18.33                 *****
                   (20,0)                          19.40


the table suggests that we have a minimum cost solution when the value of x is equal to 15 and the value of y is equal 2.

when x = 15 and y = 2, the number of pounds of potassium, nitrates, and ammonium are:

phosphate = 5x + 3y = 5*15 + 3*2 = 75 + 6 = 81
nitrate = 2x + 3y = 2*15 + 3*2 = 30 + 6 = 36
ammonium = 2x + 5y = 2*15 + 5*2 = 30 + 10 = 40

all the constraints associated with the minimum cost objective have been met.
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Saturday, April 19, 2014

INTRODUCTION TO LINEAR PROGRAMMING ---- BY. MWL. JAPHET MASATU.


INTRODUCTION    TO  LINEAR   PROGRAMMING.

This section covers Review of Inequalities, Bounded and Unbounded Regions, Inequality Word Problem, Linear Programming Terms, and Linear Programming Word Problems.
Linear Programming sounds really difficult, but it’s just a neat way to use math to find out the best way to do things – for example, how many things to make or buy.  It usually involves a system of linear inequalities, called constraints, but in the end, we want to either maximize something (like profit) or minimize something (like cost).   Whatever we’re maximizing or minimizing is called the objective function.
Linear programming was developed during the second World War for solving military logistic problems.  It is used extensively today in business to minimize costs and maximize profits.
Before we start Linear Programming, let’s review Graphing Linear Inequalities with Two Variables.

Review of Inequalities

Let’s go back and revisit graphing linear inequalities on the coordinate system.
To graph inequalities on the coordinate system, we need to get “y” by itself on the left hand side, so it’s best to use the slope-intercept or “y = mx + b” formula.  This is because we’ll easily know which way to shade the graph, and we’ll make fewer errors doing this.  The shaded areas will be where the equation “works”; in other words, where the solutions are.
When we have “y  < ”, we always shade in under the line that we draw (or to the left, if we have a vertical line).  So think of “less than” as “raining down” from the graph.
When we have “y  > ”, we always shade above the line that we draw (or to the right, if we have a vertical line).  So think of “greater than” as “raining up” from the graph.  (I know – it doesn’t really “rain up”, but I still like to explain the graphs that way.)
Note that you can always plug in an (x, y) ordered pair to see if it shows up in the shaded areas (which means it’s a solution), or the unshaded areas (which means it’s not a solution.)  For an example of this, see the first inequality below.
With “<” and “>” inequalities, we draw a dashed (or dotted) line to indicate that we’re not  including that line (but everything up to it), whereas with “Less Than or Equal To” and “Greater Than or Equal”, we draw a regular line, to indicate that we are including it in the solution.  To remember this, I think about the fact that “<” and “>” have less pencil marks than “Less Than or Equal To” and “Greater Than or Equal”, so there is less pencil used when you draw the lines on the graph.  You can also remember this by thinking the line under the “Less Than or Equal To” and “Greater Than or Equal” means you draw a solid line on the graph.
Note that the last example is a “Compound Inequality” since it involves more than one inequality.  The solution set is the ordered pairs that satisfy both inequalities; it is indicated by the darker shading.
Here are some examples:

Again, note that the last example is a “Compound Inequality” since it involves more than one inequality.  The solution set is the ordered pairs that satisfy both inequalities; it is indicated by the darker shading.

Bounded and Unbounded Regions

With our Linear Programming examples, we’ll have a set of compound inequalities, and they will be bounded inequalities, meaning the inequalities will have both maximum and minimum values.  (We’ll show examples below, but think of bounded meaning that you could draw a “circle” around the feasible region, which is the solution set to the inequalities).
Here are what some typical Systems of Linear Inequalities might look like in Linear Programming:
Bounded and Unbounded Inequalities

Inequality Word Problem:

Linear Programming problems are typically word problems – not cool.   But most will fit in the same mold: for these beginning problems, they will have two types of unknowns or variables, like earrings and necklaces, and they will involve inequalities.
You’ll just put the first variable on the x axis and the second on the y axis.  We will be solving for the number of each type that either gives the maximum profit, or minimum cost.
Jewelry
Let’s first see an example of just graphing an inequality in a real-life situation:
Lisa has an online jewelry shop where she sells earrings and necklaces.  She sells earrings for $30 and necklaces for $40.  To make a profit, she must sell at least $1200 of jewelry per month.
Define the variables, write an inequality for this situation, and graph the solutions to the inequality.
It’s best to make the first item (pairs of earrings) the x, and the second item (necklaces) y.  (Usually we can look at what the problem is asking to get these variables).   So let’s plug in some real numbers to try to get our equations.  If she sells 1 pair of earrings and 1 necklace, she’ll only make $70, which is way below her goal for a profit.  How about 20 pairs of earrings and 15 necklaces?  Do you see what we’re doing here?   The number of pairs of earrings she sells times $30 plus the number of necklaces she sells times $40 has to be greater than $1200.
Also, she can’t sell less than 0 pairs of earrings or necklaces, so we need to include those inequalities, too.
So here are the inequalities, and we’ll also draw a graph:
Jewelry Inequality Problem

Linear Programming Terms

Again, the linear programming problems we’ll be working with have the first variable on the x axis and the second on the y axis.   In our example, x is the number of pairs of earrings and y is the number of necklaces. Typically you can look at what the problem is asking to determine what the variables are.
Typically, we will be solving for the number of each type that either gives the maximum profit, or minimum cost.  The maximum profit or minimum cost expression is called the objective function.  We’ll do an example where we want to maximize profit, so our objective function will be “Maximize 30x + 40y”.
The inequalities of the problem are called the constraints, since we are limiting what we have, such as time or resources.  We’ll do an example where we are limiting our time to make jewelry.  We also will always have our non-negative constraints, where the x and y values have to be greater than or equal to 0.  Some constraints will involve greater than inequalities, for example, if a certain number of things need to be sold.  Usually there will be a sentence or phrase in the word problem for each constraint.  
We have to make sure our inequality is bounded, in order to find a maximum profit (for minimizing cost, it doesn’t have to be bounded, but it usually is).  Again, the bounded region (solutions to the system of inequalities) is called the feasible region, which will be the double-shaded region..
After we graph the inequalities, we’ll want to find the corner points.  The corner points are the vertices of the feasible region, which are the intersections of the lines of the feasible region.   The solution to the linear programming will occur at one of the corner points.  (There is something called a Corner Point Theorem that proves this, but we won’t have to worry about it).
Then we’ll substitute our corner points into the objective function to see which point yields the largest (for maximizing profit) or smallest (for minimizing cost) value.  We can do this with a table, as we’ll see later.

Linear Programming Word Problems

Now let’s put it all together and solve “real” linear programming problems!

Problem:

Lisa has an online jewelry shop where she sells earrings and necklaces.  She sells earrings for $30 and necklaces for $40.  It takes 30 minutes to make a pair of earrings and 1 hour to make a necklace, and, since Lisa is a math tutor, she only has 10 hours a week to make jewelry.  In addition, she only has enough materials to make 15 total jewelry items per week.  She makes a profit of $15 on each pair of earrings and $20 on each necklace.  How many pairs of earrings and necklaces should Lisa make each week in order to maximize her profit, assuming she sells all her jewelry?
Define the variables, write an inequality for this situation, and graph the solutions to the inequality.

Solution:

Variables:  Let’s look at what the problem is asking: how many pairs of earrings and how many necklaces should Lisa make to make a profit?  So let’s let x = the number of pairs of earrings, and y = the number of necklaces.
Objective Function:  Since we are maximizing profit, this will be a maximum, and it will be total dollars.  So the objective function is Maximize 15x + 20y.
Constraints:   Lisa’s constraints are based on her time, and also her materials.  Always make sure all the units match; we had to change 30 minutes into .5 hours.  Also note that we didn’t need to know what the jewelry sells for; sometimes they will give you extra information!
Let’s set up a separate constraint for each sentence, and first put it in a table:
Constraint Table
Let’s turn these into inequalities, and also add the non-negative constraints:
Jewelry Graph
Corner Points:  Think of the corner points as on the outside of the shaded area, but where there is a turn in the graph (the vertices).  In this case, we can easily see what they are from the graph; if we can’t, we’ll have to solve a system of equations (see next example).
Then take these points and plug in the x and y values into the objective function.  Here is what we get:
Corner Point Table
So, in order to maximize her profits, Lisa should make 10 pair of earrings and 5 necklaces per week, and her weekly profit is $250.

Problem:

Anchor
Bountiful Boats has to produce at least 5000 cabin cruisers and 12,000 pontoons each year; they can produce at most 30,000 jet skis in a year.  The company has two factories:  one in Michigan, and one in Wisconsin; each factory is open for a maximum of 240 days per year.  The Michigan factory makes 20 cabin cruisers, 40 pontoons, and 60 jet skis per day.  The Wisconsin factory makes 10 cruisers, 30 pontoons, and 50 jet skis per day.  The cost to run the Michigan factory per day is $960,000; the cost to run the Wisconsin factory per day is $750,000.  How many days of the year should each factory run in order to meet the boat production, yet do so at a minimum cost?

Solution:

Variables:  Let’s look at what the problem is asking: how many days of the year should each factory run?  So let’s let x = the number days per year that the Michigan factory should run, and y = the number days per year that the Wisconsin factory should run.
Objective Function:  Since we are minimizing cost per day, this will be a minimum, and it will be total dollars.  So the objective function is Minimize 960000x + 750000y.
Constraints:   Bountiful Boats’ constraints are based on how much they need to produce (both less than and greater than inequalities), how many days the factories are open (less than inequality), along with the non-negative constraints.
If you can’t find the constraints from sentences in the problem, we can look for another set of items (like cabin cruisers, pontoons, and jet skis), and these will usually each represent an inequality.  Let’s set up a separate constraint for each of the boats, and put them in a table.  Note that the first two inequalities are less than (“at least”), and the last one is greater than (“at most”).
Boat Problem Table
Let’s turn these into inequalities, and also add the factory days open and non-negative constraints.   Notice how we use compound inequalities for the number of days open.
Boat Linear Programming Graph
Corner Points:  Think of the corner points as on the outside of the shaded area, but where there is a turn in the graph (the vertices).  In this case, we can easily see what they are from the graph; if we can’t, we’ll have to solve a system of equations.
Let’s say we couldn’t see the exact intersection of Equations to get Intersection from the graph.  We could turn the equations into equalities, and use Linear Combination of Systems to solve:
Linear Combination
Then take these points and plug in the x and y values into the objective function.  Here is what we get:
Corner Point Table Boats
Note that you can see that the point (240, 240) won’t provide a minimum since the point (240, 80) will provide a smaller amount.  So, technically, you wouldn’t even have to plug this point into the objective function.
So, in order to minimize their costs, Bountiful Boats should make boats 240 days a year at their Michigan plant, and 80 days a year at their Wisconsin plant.  This will yield a cost of $290,400,000.  So you can see that your answer may be surprising!
Learn these rules, and practice, practice, practice!
On to Rational Functions and Equations  – you are ready!
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LINEAR PROGRAMMING : A WORD PROBLEM WITH FOUR VARIABLES ----- BY. MWL. JAPHET MASATU

Linear Programming:
  A Word Problem with Four Variables .
Sections: Optimizing linear systems, Setting up word problems
INTRODUCTION:

  • A building supply has two locations in town. The office receives orders from two customers, each requiring 3/4-inch plywood. Customer A needs fifty sheets and Customer B needs seventy sheets.
    • The warehouse on the east side of town has eighty sheets in stock; the west-side warehouse has forty-five sheets in stock. Delivery costs per sheet are as follows: $0.50 from the eastern warehouse to Customer A, $0.60 from the eastern warehouse to Customer B, $0.40 from the western warehouse to Customer A, and $0.55 from the western warehouse to Customer B.
    Find the shipping arrangement which minimizes costs.
    Hmm... I've got four things to consider:
      east warehouse to Customer A
      east warehouse to Customer B
      west warehouse to Customer A
      west warehouse to Customer B
    But I only have two variables. How can I handle this?
    The variables obviously need to stand for the number of sheets being shipped, but I have four different sets of sheets. This calls for subscripts and explicit labelling:
      shipped from east warehouse to Customer A: Ae
      shipped from west warehouse to Customer A:       Aw
      shipped from east warehouse to Customer B:       Be
      shipped from west warehouse to Customer B:       Bw
    Since Customer A wants 50 sheets and Customer B wants 70 sheets, then:
      Ae + Aw = 50, so Aw = 50 – Ae   (I'll call this Equation I)
      Be       + Bw = 70,       so Bw = 70 – Be   (I'll call this Equation II)
    Since the eastern warehouse can ship no more than eighty sheets and the western warehouse can ship no more than forty-five sheets, then:
      0 < Ae + Be < 80
      0 < Aw + Bw < 45
    And the optimization equation will be the shipping cost:
      C = 0.5Ae + 0.6Be + 0.4Aw + 0.55Bw
    Where does this leave me? The equations (labelled as Equations I and II above) allow me to substitute and get rid of two of the variables in the second inequality above, so:
      0 < Ae + Be < 80
      0 < (50 – Ae) + (70 – Be) < 45



    Simplifying the second inequality above gives me the new system:
      0 < Ae + Be < 80
      0 < 120 – Ae – Be < 45
    Multiplying through by –1 (thereby flipping the inequality signs) and adding 120 to all three "sides" of the second inequality, I get the new system:
      0 < Ae + Be < 80
      120 > Ae + Be > 75
    Since Ae + Be is no less than 75 and is no more than 80, then these two inequalities reduce to one:

      75 < Ae + Be < 80
    I can also simplify the optimization equation:
      C = 0.5Ae + 0.6Be + 0.4Aw + 0.55Bw
                = 0.5Ae + 0.6Be + 0.4(50 – Ae) + 0.55(70 – Be)
          = 0.1Ae + 0.05Be + 58.50
    Due to the size of the orders, I have:
      0 < Ae < 50
      0 < Be < 70
    Since I have only two variables now, and since I'll be graphing with x and y, I'll rename the variables:   Copyright © Elizabeth Stapel 2006-2011 All Rights Reserved
      x = Ae
      y = Be
    Then entire system is as follows:
      Minimize C = 0.1x + 0.05y + 58.50, subject to the following:
        x > 0
        y > 0
        y > –x + 75
        x < 50
        y < 70
        y < –x + 80
    The feasibility region graphs as:
      feasibility region
When you test the corner points, (5, 70), (10, 70), (50, 30), and (50, 25), you should get the minimum cost when you ship as follows:
  5 sheets from the eastern warehouse to Customer A70 sheets from the eastern warehouse to Customer B45 sheets from the western warehouse to Customer A
  0 sheets from the western warehouse to Customer B
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LINEAR PROGRAMMING: WORD PROBLEMS ----FORM FOUR BY. MWL. JAPHET MASATU.

LINEAR   PROGRAMMING: WORD   PROBLEMS.





INTRODUCTION:

  • 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 200 scientific 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?
    The question asks for the optimal number of calculators, so my variables will stand for that:
      x: number of scientific calculators producedy: number of graphing calculators produced



    Since they can't produce negative numbers of calculators, I have the two constraints, x > 0 and y > 0. But in this case, I can ignore these constraints, because I already have that x > 100 and y > 80. The exercise also gives maximums: x < 200 and y < 170. The minimum shipping requirement gives me x + y > 200; in other words, y > –x + 200. The revenue relation will be my optimization equation: R = –2x + 5y. So the entire system is:
      R = –2x + 5y, subject to:
      100 < x < 200 
      80 <  y < 170       
      y > –x + 200       
    The feasibility region graphs as:   Copyright © Elizabeth Stapel 2006-2011 All Rights Reserved
      feasibility region
When you test the corner points at (100, 170), (200, 170), (200, 80), (120, 80), and (100, 100), you should obtain the maximum value of R = 650 at (x, y) = (100, 170). That is, the solution is "100 scientific calculators and 170 graphing calculators".


  • 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?
    • The question ask for the number of cabinets I need to buy, so my variables will stand for that:
      x: number of model X cabinets purchasedy: number of model Y cabinets purchased
    Naturally, x > 0 and y > 0. I have to consider costs and floor space (the "footprint" of each unit), while maximizing the storage volume, so costs and floor space will be my constraints, while volume will be my optimization equation.
      cost: 10x + 20y < 140, or y < –( 1/2 )x + 7
      space:       6x + 8y < 72, or y < –( 3/4 )x + 9
      volume:       V = 8x + 12y
    This system (along with the first two constraints) graphs as:
      feasibility region
When you test the corner points at (8, 3), (0, 7), and (12, 0), you should obtain a maximal volume of 100 cubic feet by buying eight of model X and three of model Y .               Linear Programming: More Word Problems Sections: Optimizing linear systems, Setting up word problems


  • In order to ensure optimal health (and thus accurate test results), a lab technician needs to feed the rabbits a daily diet containing a minimum of 24 grams (g) of fat, 36 g of carbohydrates, and 4 g of protien. But the rabbits should be fed no more than five ounces of food a day.
    • Rather than order rabbit food that is custom-blended, it is cheaper to order Food X and Food Y, and blend them for an optimal mix. Food X contains 8 g of fat, 12 g of carbohydrates, and 2 g of protein per ounce, and costs $0.20 per ounce. Food Y contains 12 g of fat, 12 g of carbohydrates, and 1 g of protein per ounce, at a cost of $0.30 per ounce.
    What is the optimal blend?
    Since the exercise is asking for the number of ounces of each food required for the optimal daily blend, my variables will stand for the number of ounces of each:



      x: number of ounces of Food Xy: number of ounces of Food Y
    Since I can't use negative amounts of either food, the first two constrains are the usual ones: x > 0 and y > 0. The other constraints come from the grams of fat, carbohydrates, and protein per ounce:
      fat:        8x + 12y > 24
      carbs:        12x + 12y > 36
      protein:        2x +   1y >   4
    Also, the maximum weight of the food is five ounces, so:

      x + y < 5
    The optimization equation will be the cost relation C = 0.2x + 0.3y, but this time I'll be finding the minimum value, not the maximum.
    After rearranging the inequalities, the system graphs as:
      feasibility region
    (Note: One of the lines above is irrelevant to the system. Can you tell which one?)
When you test the corners at (0, 4), (0, 5), (3, 0), (5, 0), and (1, 2), you should get a minimum cost of sixty cents per daily serving, using three ounces of Food X only.

Sometimes you'll have more than just two things to deal with. The next example has three things to juggle; the next page provides an example of juggling four things.

  • You have $12,000 to invest, and three different funds from which to choose. The municipal bond fund has a 7% return, the local bank's CDs have an 8% return, and the high-risk account has an expected (hoped-for) 12% return. To minimize risk, you decide not to invest any more than $2,000 in the high-risk account. For tax reasons, you need to invest at least three times as much in the municipal bonds as in the bank CDs. Assuming the year-end yields are as expected, what are the optimal investment amounts?
    • Since the question is asking me to find the amount of money for each account, my variables will need to stand for those amounts. Since I'd like to deal with smaller numbers, I'll count by thousands, so:
      x: amount (in thousands) invested in bondsy: amount (in thousands) invested in CDs
    Um... now what? I only have two variables, but I have three accounts. To handle this, I need the "how much is left" construction:
      12 – x – y: amount (in thousands) invested in the high-risk account
    I can't invest negative amounts of money, so the first two constraints are the usual ones: x > 0 and y > 0. The amount in the high-risk account can't be negative either, so 12 – x – y > 0, which simplifies as:
      y < –x + 12
    Also, the upper limit on the high-risk account gives me the inequality (12 – x – y) < 2. This simplifies as:   Copyright © Elizabeth Stapel 2006-2011 All Rights Reserved
      y > –x + 10
    And the tax requirements give me y < ( 1/3 )x. The optimization equation will be the total investment yield, Y = 0.07x + 0.08y + 0.12(12 – x – y) = 1.44 – 0.05x – 0.04y. The entire system is then as follows:
      Maximize Y = 1.44 – 0.05x – 0.04y, subject to:
              x > 0
        y > 0
        y > –x + 10
        y < –x + 12
        y < ( 1/3 )x
    The feasibility region graphs as:

feasibility region
When you test the corner points at (9, 3), (12, 0), (10, 0), and (7.5, 2.5), you should get an optimal return of $965 when you invest $7,500 in municipal bonds, $2,500 in CDs, and the remaining $2,000 in the high-risk account.
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