Tutorial I of DO

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TUTORIAL I DETERMINISTIC OPTIMIZATION

• Model building of LP• Various Type of LP through study case• Study case

(All materials Are Taken From K. Gita Ayu’ Lecture Notes)

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Optimization Model (LP)

Identify the decision variables

Define the constraints

Quantify the decision consequencesTo be maximized or minimized through Objective function

3 Step to Formulate Any Optimization Models (Linear Programming)

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STUDY CASE (OIL REFINERY)

An oil refinery produces gasoline and heating oil from 2 crude materials A and B. Suppose there are 3 processor available and the consumption and output per production period is as follows

Suppose the profit for each barrel of gasoline and heating oil is $4 and $3 respectively. Determine the units of production using the three processors that will produce the highest profit for the company, assuming there are 8 million barrels of crude A and 5 million barrels of crude B.

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SOLUTION

Step 1 : Identify the decision variables Processor

Processor 1 = x1

Processor 2 = x2

Processor 3 = x3

Step 2 : Define the constraints crude A and crude B

Step 3 : Objective functions

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Step 3 : Objective functions profit from the processor i & each barrel of gasoline and heating oil from the processor i

SOLUTION

Processor

Gasoline

Heating oil

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TWO CRUDE PETROLEUM

Two Crude Petroleum (TCP) runs a small refinery on the Texas coast. The refinery distills crude petroleum from two sources, Saudi Arabia and Venezuela, into three main products: gasoline, jet fuel, and lubricants.

The two crudes differ in chemical composition and thus yield different product mixes. Each barrel of Saudi crude yields 0.3 barrel of gasoline, 0.4 barrel of jet fuel, and 0.2 barrel of lubricants. On the other hand, each barrel of Venezuelan crude yields 0.4 barrel of gasoline but only 0.2 barrel of jet fuel and 0.3 barrel of lubricants. The remaining 10% of each barrel is lost to refining.

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The crudes also differ in cost and availability. TCP can purchase up to 9000 barrels per day from Saudi Arabia at $20 per barrel. Up to 6000 barrels per day of Venezuelan petroleum are also available at the lower cost of $15 per barrel because of the shorter transportation distance.

TCP’s contracts with independent distributors require it to produce 2000 barrels per day of gasoline, 1500 barrels per day of jet fuel, and 500 barrels per day of lubricants. How can these requirements be fulfilled most efficiently?

TWO CRUDE PETROLEUM

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SOLUTION Step 1 : Identify the decision variables refined

oil by Saudi Arabia (x1) & refined oil by Venezuela (x2)

Define the constraints gasoline, jet fuel, lubricant, availability of Saudi Arabia and Venezuela

Objective functions TCP purchased up Saudi Arabia and Venezuela

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CARDBOARD

A paper company produces cardboard boxes. Suppose cardboard comes in a fixed area, c. We would like to produce a cardboard box of maximum volume. Write the model!

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SOLUTION

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FENCING

Suppose that we wish to enclose a rectangular equipment yard by at most 80 meters of fencing. Formulate an optimization model to find the design of maximum area.

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SOLUTION

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FARMER JANE

Farmer Jane owns 45 acres of land. She is going to plant each acre with wheat or corn. Each acre planted with wheat yields $200 profit; and each with corn yields $300 profit. The labor and fertilizer used for each acre are given below. One hundred workers and 20 tons fertilizers are available. Determine how Jane can maximize profit from her land!

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SOLUTION

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AUTO COMPANY

An auto company manufactures cars and trucks. Each vehicle must be processed in the paint shop and body assembly shop. If the paint shop were only painting trucks, 40 per day could be painted. If it were only painting cars, 60 per day could be painted. The body shop can process 50 cars alone or 50 trucks alone per day. Assume the return from truck and car are 300 and 200, respectively. Find the production schedule for the company !

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SOLUTION

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SOLUTION

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AUTO COMPANY

Suppose that auto dealers require that the auto company produce at least 30 trucks and 20 cars. Write the constraint !

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DIET PROBLEM

My diet requires that all the food I eat come from one of the four “basic food groups” (chocolate cake, ice cream, soda, and cheesecake). At present, the following four foods are available for consumption: brownies, chocolate ice cream, cola, and pineapple cheesecake.

Each brownie costs 50¢, each scoop of chocolate ice cream costs 20¢, each bottle of cola costs 30¢, and each piece of pineapple cheesecake costs 80¢. Each day, I must ingest at least 500 calories, 6 oz. of chocolate, 10 oz. of sugar, and 8 oz. of fat.

The nutritional content per unit of each food is shown below. Formulate the LP model that can be used to satisfy my daily nutritional requirements at minimum cost

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SOLUTION

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SOLUTION

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WORK SCHEDULING

# of full-time employees required

Mon 17Tue 13Wed 15Thu 19Fri 14Sat 16Sun 11

A post office requires different numbers of full-time employees on different days of the week. The number of full-time employees required on each day is given on the left.

Union rules state that each full-time employee must work five consecutive days and then receive two days off. For example, an employee who works Monday to Friday must be off on Saturday and Sunday.

The post office wants to meet its daily requirements using only full-time employees. Formulate an LP that the post office can use to minimize the number of full-time employees that must be hired.

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SOLUTION

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SOLUTION

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PROJECT SELECTION

Star Oil Company is considering five different investment opportunities. The cash outflows and net present values (in millions of dollars) are given in table below. Star Oil has $40 million available for investment at the present time (time 0); it estimates that one year from now (time 1) $20 million will be available for investment.

Star Oil may purchase any fraction of each investment. In this case, the cash outflows and NPV are adjusted accordingly. For example, if Star Oil purchases one fifth of investment 3, then a cash outflow is 1/5(5) = $1 million would be required at time 1. The one-fifth share of investment 3 would yield an NPV of 1/5(16) = $3.2 million.

Star Oil wants to maximize the NPV that can be obtained by investing in investments 1-5. Formulate an LP that will help achieve this goal. Assume that any funds left over at time 0 cannot be used at time 1

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Inv. 1 Inv. 2 Inv. 3 Inv. 4 Inv. 5Time 0 cash outflow 11 53 5 5 29

Time 1 cash outflow 3 6 5 1 34

NPV 13 16 16 14 39

PROJECT SELECTION

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CAPITAL BUDGETING

Consider two investments with varying cash flows:

Assume at time 0, $10,000 cash is available and at time 1, $7000 is available. Suppose the interest rate is 0.1. Formulate an LP so as to obtain a solution which maximizes the NPV from these investment

Cash flow (in 1000) at time0 1 2 3

Investment 1 -6 -5 7 9 Investment 2 -8 -3 9 7

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SOLUTION

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SOLUTION

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BLENDING PROBLEM

Sunco Oil (SO) manufactures three types of gasoline (gas 1, gas 2, and gas 3). Each type is produced by blending three types of crude oil (crude 1, crude 2, and crude 3). Sunco can purchase up to 5000 barrels of each type of crude oil daily.

The three types of gasoline differ in their octane rating and sulfur content. The crude oil blended to form gas 1 must have an average octane rating of at least 10 and contain at most 1% sulfur. The crude oil blended to form gas 2 and gas 3 must have an average octane rating of at least 8 and contain at most 2% sulfur and an average octane rating of at least 6 and contain at most 1% sulfur, respectively. It costs $4 to transform one barrel of oil into one barrel of gasoline and SO refinery can produce up to 14,000 barrels of gasoline daily.

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SO customers require the following amounts of each gasoline: gas 1 – 3000 barrels/day; gas 2 – 2000 barrels/day; gas 3 – 1000 barrels/day. Demand must be met. SO also has the option of advertising to stimulate demand for its products. Each dollar spent daily in advertising a particular type of gas increases the daily demand for that type of gas by 10 barrels. Formulate an LP that will enable Sunco to maximize daily profits

BLENDING PROBLEM

Sales Price/Barre

l

Purchase Price/Barre

l

Octane Rating

Sulfur Content

Gas 1 $ 70 Crude 1 $ 45 12 0.5%Gas 2 $ 60 Crude 2 $ 35 6 2.0%Gas 3 $ 50 Crude 3 $ 25 8 3.0%

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SOLUTION

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SOLUTION

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PRODUCTION PROCESS

Rylon Corp manufactures Brute and Chanelle perfumes. The raw material needed to manufacture each type of perfume can be purchased for $3/pound. Processing 1 lb of raw material requires 1 hour of laboratory time. Each pound of processed raw material yields 3 oz of Regular Brute Perfume and 4 oz of Regular Chanelle Perfume. Regular Brute can be sold for $7/oz and Regular Chanelle for $6/oz.

Rylon also has the option of further processing Regular Brute and Regular Chanelle to produce Luxury Brute, sold at $18/oz, and Luxury Chanelle, sold at $14/oz, respectively. Each ounce of Regular Brute processed further requires an additional 3 hours of laboratory time and $4 processing cost and yields 1 oz of Luxury Brute. Each ounce of Regular Chanelle processed further requires an additional 2 hours of laboratory time and $4 processing cost and yields 1 oz of Luxury Chanelle.

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Each year, Rylon has 6000 hours of laboratory time available and can purchase up to 4000 lb of raw material. Formulate an LP that can be used to determine how Rylon can maximize profits. Assume that the cost of the laboratory hours is a fixed cost

PRODUCTION PROCESS

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SOLUTION

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SOLUTION

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INVENTORY PROBLEM

Sailco Corp must determine how many sailboats should produce during each of the next four quarters. The demand during each of the next four quarters is as follows: first quarter, 40 sailboats; second quarter, 60 sailboats; third quarter, 75 sailboats; fourth quarter, 25 sailboats. Sailco must meet demands on time. The beginning of the first quarter, Sailco has an inventory of 10 sailboats.

At the beginning of each quarter, Sailco must decide how many sailboats should be produced during that quarter. For simplicity, we assume that sailboats manufactured during a quarter can be used to meet demand for that quarter. During each quarter, Sailco can produce up to 40 sailboats with regular-time labor at a total cost of $400 per sailboat.

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By having employees work overtime during a quarter, Sailco can produce additional sailboats with overtime labor at a total cost of $450 per sailboat. At the end of each quarter (after production has occurred and the current quarter’s demand has been satisfied), a carrying or holding cost of $20 per sailboat is incurred. Use linear programming to determine a production schedule to minimize the sum of production and inventory costs during the next four quarters.

INVENTORY PROBLEM

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SOLUTION

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SOLUTION

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MULTI PERIOD WORK SCHEDULINGCSL is a chain of computer service stores. The number of hours of skilled repair time that CSL requires during the next five months is as follows:

Jan: 6000 hoursFeb: 7000 hoursMar: 8000 hoursApr: 9500 hoursMay 11,000 hours

At the beginning of January, 50 skilled technician work for CSL. Each skilled technician can work up to 160 hours per month. In order to meet future demands, new technicians must be trained. It takes one month to train a new technician. During the month of training, a trainee must be supervised for 50 hours by an experienced technician. Each experienced technician is paid $2000 a month (even if he or she does not work the full 160 hours). During the month of training, a trainee is paid $1000 a month.

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At the end of each month, 5% of CSL’s experienced technicians quit to join Plum Computers. Formulate an LP whose solution will enable CSL to minimize the labor cost incurred in meeting the service requirements for the next five months

MULTI PERIOD WORK SCHEDULING

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SOLUTION

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MULTI PERIOD FINANCIAL MODEL

Finco Investment Corporation must determine investment strategy for the firm during the next three years. At present (time 0), $100,000 is available for investment. Investment A, B, C, D, and E are available. The cash flow associated with investing $1 in each investment is given below. To ensure that the company’s portfolio is diversified, Finco requires that at most $75,000 be placed in any single investment.

Cash Flow at TimeA - $1.00 $0.50 $1.00 $0.00B $0.00 - $1.00 $0.50 $1.00C - $1.00 $1.20 $0.00 $0.00D - $1.00 $0.00 $0.00 $1.90E $0.00 $0.00 - $1.00 $1.50

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In addition to investments A-E, Finco can earn interest at 8% per year by keeping uninvested cash in money market funds. Returns from investments may be immediately reinvested. For example, the positive cash flow received from investment C at time 1 may immediately be reinvested in investment B. Finco cannot borrow funds, so the cash available for investment at any time is limited to cash on hand. Formulate an LP that will maximize cash on hand at time 3.

MULTI PERIOD FINANCIAL MODEL

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SOLUTION

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PRODUCTION PROCESSAn Oil company has three different processes that can be used to manufacture various types of gasoline. Each process involves blending oils in the company’s catalytic cracker. Running process 1 for an hour costs $5 and requires 2 barrels of crude oil 1 and 3 barrels of crude oil 2. The output from running process 1 for an hour is 2 barrels of gas 1 and 1 barrel of gas 2.

Running process 2 for an hour costs $4 and requires 1 barrel of crude 1 and 3 barrels of crude 2. The output from running process 2 for an hour is 3 barrels of gas 2. Running process 3 for an hour costs $1 and requires 2 barrels of crude 1 and 3 barrels of gas 2. The output from running process 3 for an hour is 2 barrels of gas 3. Each week, 200 barrels of crude 1, at $2 per barrel, and 300 barrels of crude 2, at $3 per barrel may be purchased.

All gas produced can be sold at the following per-barrel prices: gas 1, $9; gas 2, $10; gas 3, $24. Formulate an LP whose solution will maximize revenue less costs. Assume only 100 hours of time on the catalytic cracker are available each week

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SOLUTION

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ORANGE JUICE COMPANY

O. J. Juice Company sells bags of oranges and cartons of orange juice. O. J. grades oranges on a scale of 1 (poor) to 10 (excellent). At present, O. J. has on hand 100,000 pounds of grade 9 oranges and 120,000 pounds of grade 6 oranges. The average quality sold in a bag must be at least 7, and the average quality of the oranges used to make orange juice must be at least 8.

Each pound of oranges that is used to produce juice yields a revenue of $1.50 and incurs a variable cost (consisting of labor costs, inventory costs, variable overhead costs, etc.) of $1.05. Each pound of oranges sold in bags yields a revenue of $0.50 and incurs a variable cost of $0.20. Formulate an LP to help O. J. maximize profit

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SOLUTION

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NOTIP TABLE COMPANY

The Notip Table Company sells two models of its patented five-leg tables. The basic version uses a wood top, requires .6 hour to assemble and sells for a profit of $200. The deluxe model takes 1.5 hours to assemble because of its glass top and sells for a profit of $350. Over the next week the company has 300 legs, 50 wood tops, 35 glass tops, and 63 hours of assembly available.

Notip wishes to determine a maximum profit production plan assuming that everything produced can be sold. Formulate the mathematical programming.

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SOLUTION

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REFERENCES

K. Gita Ayu. 2012. Model Formulation. Lecture Notes. Bina Nusantara University.