Operations Research IIntroduction

September 4, 2012

Operations Research

What is Operations Research?Discipline by means of which it is possible to allocate scarce resources tothe operations of a firm.

I How to conduct and coordinate the operations (activities) of anorganization

I Research: Use of the scientific method to investigate the problemof interest

OR ModelsCharacteristics

All OR models share the same characteristics:

I Decision Alternatives

I Restrictions, due to the scarcity of resources available

I Objective Criterion (Function)

OR ModelsGeneral Format

Optimize an Objective Function

subject to

Constraints

Optimize: either Minimize or Maximize (according to the case at hand).

OR ModelsAn Example

We are to produce soda cans with a volume of 355 ml and with smallestsurface. Determine

I Decision Alternatives

I Restrictions (Constraints)

I The objective function

OR ModelsAn Example − Solution

Model: The can can be thought of as a cylinder, with radius r andheight h. Thus,

Objective Min z = 2πr2 + 2πrh

Constraint 1 πr2h = 355

Constraint 2 r ≥ 0, h ≥ 0

(In the initial modeling phase, physical units are not considered.)Which are all the feasible decision alternatives?This is an example of a nonlinear programming model.

Standard OR Tools

I Linear Programming (Dantzig, late 40’s)

I Dynamic Programming (Bellmann, 50’s)

I Queueing Theory

I Inventory Theory

I Nonlinear Programming (Economics)

I etc. . .

This course of Operations Research 1 will be devoted to the study ofLinear Programming (LP) and how LP can be used to solve relativelysimple problems in Engineering.

Introduction to Linear ProgrammingToy Model

Flinks Furniture produces inexpensive tables and chairs.The production process is similar in that both products require a certainnumber of carpentry work and a certain number of labor hours in thepainting and varnishing department.

Each chair requires 3 hours in carpentry and 1 hour in painting andvarnishing.

Each table takes 4 hours of carpentry and 2 hours in the painting andvarnishing shop.

Introduction to Linear ProgrammingToy Model

During the current production time, 240 hours of carpentry time areavailable, and 100 hours in painting and varnishing time are available.

Each table sold yields a profit of $7; each chair produced is sold for a $5profit.

Determine the best possible combination of tables and chairs in order toreach the maximum profit.

Toy Model

The mathematical formulation of this problem is as follows1: We want to

Max z = 5x1 + 7x2

subject to3x1 +4x2 ≤ 240x1 +2x2 ≤ 100

with xi ≥ 0, i = 1, 2.

1We shall use x1 to denote chairs and x2 to denote tables

Linear Programming Properties

I There must be one objective function

I Presence of constraints

I There must be alternatives available

I Mathematical relationships are linear

Linear Programming Assumptions

I Certainty

I Proportionality

I Additivity

I Divisibility

I Nonnegative variables

Problem 1

The Apex Television Company has to decide on the number of 27- and20-inch sets to be produced at one of its factories. Market researchindicates that at most 40 of the 27-inch sets and 10 of the 20-inch setscan be sold per month. The maximum number of work-hours available is500 per month. A 27-inch set requires 20 work-hours and a 20-inch setrequires 10 work-hours. Each 27-inch set sold produces a profit of $120and each 20-inch set produces a profit of $80. A wholesaler has agreed topurchase all the television sets produced if the numbers do not exceedthe maxima indicated by the market research.

Formulate a linear programming model for this problem.

Problem 1Solution

Let x1 (resp. x2) be the number of 27-inch (resp. 20-inch) TV setsproduced. Let Z be the total profit per month.The mathematical formulation of the model is

Max Z = 120x1 + 80x2

subject tox1 ≤ 40

x2 ≤ 1020x1 +10x2 ≤ 500

with x1 ≥ 0, x2 ≥ 0.

Problem 2Dwight is an elementary school teacher who also raises pigs forsupplemental income. He is trying to decide what to feed his pigs. He isconsidering using a combination of pig feeds available from localsuppliers. He would like to feed the pigs at minimum cost while alsomaking sure each pig receives an adequate supply of calories andvitamins. The cost, calorie content, and vitamin content of each feed aregiven in the table below.

Contents Feed Type A Feed Type BCalories (per pound) 800 1,000Vitamins (per pound) 140 units 70 unitsCost (per pound) $0.40 $0.80

Each pig requires at least 8,000 calories per day and at least 700 units ofvitamins. A further requirement (constraint) is that no more thanone-third of the diet (by weight) can consist of Feed Type A, since itcontains an ingredient which is toxic if consumed in too large a quantity.

Formulate a linear programming model for this problem.

Problem 2Solution

Let A and B be the quantity (pounds) of Feed Type A and Feed Type B,respectively, used per day.The mathematical formulation is

A Diet ProblemWinston, Chapter 3

My diet requires that all the food I eat come from one of the four ”basicfood groups” (chocolate cake, ice cream, soda and cheesecake). Atpresent, the following four foods are available for consumption: brownies,chocolate ice cream, cola and pineapple cheesecake. Each brownie costs$0.50, each scoop of chocolate ice cream costs $0.20, each bottle of colacosts $0.30 and each piece of pineapple cheesecake costs $0.80. Eachday, 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 inthe table below.Formulate a linear programming model that can be used to satisfy mydaily nutritional requirements at minimum cost.

A Diet ProblemWinston, Chapter 3

Table: Nutritional Values

Calories Chocolate (oz) Sugar (oz) Fat (oz)Brownie 400 3 2 2Chocolate Ice Cream(1 scoop) 200 2 2 4Cola (1 bottle) 150 0 4 1PineappleCheesecake 500 0 4 5

Problem 1, Problem Set 2.3A

A realtor is developing a rental housing and retail area. The housing areaconsists of efficiency departments, duplexes and single-family homes.Maximum demand by potential renters is estimated to be 500 efficiencydepartments, 300 duplexes and 250 single-family homes, but the numberof duplexes must equal at least 50% of the number of efficiencydepartments and single-family homes. Retail space is proportional to thenumber of home units at the rates of at least 10 ft2, 15 ft2 and 18 ft2 perefficiency departments, duplexes and single-family homes, respectively.However, land availability limits retail space to no more than 10,000 ft2.The monthly rental income is estimated at $600, $750 and $1200 forefficiency-, duplex- and single-family units, respectively. The retail spacerents for $100/ft2.

Formulate a linear programming model that can be used to determine theoptimal retail space area and the number of family residences.

Problem 4, Problem Set 2.3D

The demand for ice cream during the three summer months (June, Julyand August) at All-Flavors Parlor is estimated at 500, 600 and 40020-gallon cartons, respectively. Two wholesalers, 1 and 2, can supplyAll-Flavors with its ice cream. Although the flavors from the twosuppliers are different, they are interchangeable. The maximum numberof cartons either supplier can provide is 400 per month. Also, the priceseach supplier charges from one month to the next varies, according tothe schedule

June July AugustSupplier 1 $100 $110 $120Supplier 2 $115 $108 $125

(price per carton)

Problem 4, Problem Set 2.3D

To take advantage of price fluctuations, All-Flavors can purchase morethan is needed for a month and store the surplus to satisfy the demand ina later month. The cost of refrigerating an ice cream carton is $5 permonth. It is realistic in the present situation to suppose that therefrigeration cost is a function of the average number of cartons on handduring the month.

Develop an optimal schedule for buying ice cream from the two suppliers.

Problem 2, Problem Set 2.3F

A hospital employs volunteers to staff the reception desk between 08:00and 22:00. Each volunteer works three consecutive hours except thosestarting at 20:00 who works for two hours only.The minimum need forvolunteers is approximated by a step function over 2-hour intervalsstarting at 08:00 as 4, 6, 8, 6, 4, 6, 8.Because most volunteers are retired individuals, they are willing to offertheir services at any hour of the day (08:00-22:00).However, because of the large number of charities competing for theirservices, the number must be kept as low as possible.

Formulate a linear programming model that can be used to find anoptimal schedule for the start time of the volunteers.

Problem 3.4-8, HL

Web Mercantile sells many household products through an onlinecatalog. The company needs substantial warehouse space for storing itsgoods. Plans are now being made for leasing warehouse storage spaceover the next 5 months. Just how much space will be required in each ofthese months is known. However, since these space requirements arequite different, it may be most economical to lease only the amountneeded each month, on a month-by-month basis. On the other hand, theadditional cost for leasing space for additional months is much less thanfor the first month, so it may be less expensive to lease the maximumamount needed for the entire 5 months. Another option is theintermediate approach of changing the total amount of space leased (byadding a new lease and/or having an old lease expire) at least once, butnot every month. The space requirements and the leasing costs for thevarious leasing periods are as follows.

Problem 3.4-8, HL

Month Required space Leasing period Cost per sq ft(sq ft) (months) Leased

1 30,000 1 $652 20,000 2 $1003 40,000 3 $1354 10,000 4 $1605 50,000 5 $190

Formulate a linear programming model that will minimize the totalleasing cost for meeting the space requirements.

Transportation Problem

Three orchards supply crates of oranges to four retailers. The dailydemand amounts at the four retailers are 150, 150, 400, and 100 crates,respectively. Supplies at the three orchards are dictated by the availableregular labor and are estimated at 150, 400, and 250 crates daily. Thetransportation costs per crates from the orchards to the retailers aregiven in the next slide.Formulate the problem as a transportation model.

Problem 11, Problem Set 5.1A

Retailer 1 Retailer 2 Retailer 3 Retailer 4Orchard 1 $1 $2 $3 $2Orchard 2 $2 $4 $1 $2Orchard 3 $1 $3 $5 $3

Problem 11, Problem Set 5.1A

Three orchards supply crates of oranges to four retailers. The dailydemand amounts at the four retailers are 150, 150, 400, and 100 crates,respectively. Supplies at the three orchards are dictated by the availableregular labor and are estimated at 150, 200, and 250 crates daily.However, Orchards 1 and 2 have indicated that they can supply morecrates, if necessary, by using overtime labor. Orchard 3 does not offerthis option. The transportation costs per crates from the orchards to theretailers are given in the next slide.Formulate the problem as a transportation model.

Problem 11, Problem Set 5.1A

Retailer 1 Retailer 2 Retailer 3 Retailer 4Orchard 1 $1 $2 $3 $2Orchard 2 $2 $4 $1 $2Orchard 3 $1 $3 $5 $3

Problem 3.4-9, HL

Larry Edison is the director of the Computer Center for Buckly College.He now needs to schedule the staffing of the center. It is open from 8AMuntil midnight. Larry has monitored the usage of the center at varioustime of the day and determined that the following number of consultantsare required:

Time of day Minimum number of consultantsrequired to be on duty

8AM – Noon 4Noon – 4PM 84PM – 8PM 10

8PM – Midnight 6

Problem 3.4-9, HL

Two types of computer consultants can be hired: full-time and part-time.The full-time consultants work for 8 consecutive hours in any of thefollowing shifts: morning (8AM–4PM), afternoon (noon–8PM), andnight (4MP–midnight). Full-time consultants are paid $14 per hour.Part-time consultants can be hired to work any of the you shifts listed inthe table above and are paid $12 per hour.An additional requirement is that during any time period, there must beat least 2 full-time consultants on duty for every part-time consultant onduty.Formulate a linear programming model that will allow Larry to determinehow many full-time and part-time consultants should work each shift tomeet the above requirement at the minimum possible cost.

Investment ProblemTaha, Problem 1, Problem Se 2.3C

Fox Enterprises is considering six projects for possible construction overthe next four years. The expected (present value) returns and cashoutlays for the projects are given below. Fox can undertake any of theprojects partially or completely. A partial undertaking of a project willprorate both the return and the cash outlay proportionately.

Investment ProblemTaha, Problem 1, Problem Se 2.3C

Table: Cash Outlays

Project Year 1 Year 2 Year 3 Year 4 Return($1000)

1 10.5 14.4 2.2 2.4 32.42 8.3 12.6 9.5 3.1 35.83 10.2 14.2 5.6 4.2 17.754 7.2 10.5 7.5 5 14.95 12.3 10.1 8.3 6.3 18.26 9.2 7.8 6.9 5.1 12.35

Available 60 70 35 20Funds ($1000)

Investment ProblemTaha, Problem 1, Problem Se 2.3C

1. Formulate the problem as a linear program. Ignore the time value ofmoney.

2. Suppose that if a portion of project 2 is undertaken, then at least anequal portion of project 6 must be undertaken as well. Modify theformulation of the model.

3. In the original model, suppose that any funds left at the end of ayear are used in the next year. Modify the formulation of the linearprogram.

Blending Model

The Metalco Compan desires to blend a new alloy of 40 percent zinc, 35percent tin, and 24 percent lead from several available alloys having thefollowing properties:

Table: Alloy

Property 1 2 3 4 5% zinc 60 25 45 20 50% tin 10 15 45 50 40% lead 30 60 10 30 10Cost ($/lb) 22 20 25 24 27

The objective is to determine the proportions of these alloys that shouldbe blended to produce the desired new alloy at minimum cost. Formulatea linear programming model that ca be used to that purpose.

Transportation ModelTaha, Problem Set 5.1A, Problem 6

Three electric power plants with capacities of 25, 40, and 30 million kWhsupply electricity to three cities. The maximum demands at the threecities are estimated at 30, 35, and 25 million kWh. The price per millionkWh at the three cities is given by the table below:

Table: Price/Million kWh

City 1 $600 $700 $400City 2 $320 $300 $350City 3 $500 $480 $450

During the month of August, there is a 20% increase in demand at eachof the three cities, which can be met purchasing electricity from anothernetwork at a premium rate of $1000 per million kWh. The network is notlinked to City 3, however. The company wishes to determine the mosteconomical plan for the distribution and purchase of the electricity.Formulate the problem as a transportation model.

Transportation ModelTaha, Problem Set 5.2A, Problem 6

The demand for a special small engine over the next five quarters is 200,150, 300, 250, and 400 units, respectively. The manufacturer supplyingthe engine has different production capacities estimated at 180, 230, 430,300, and 300 units for the five quarters. Backordering is not allowed, butthe manufacturer can use overtime to fill the immediate demand, ifnecessary. The overtime capacity for each period is half the regularcapacity.The production costs per unit for the five periods are $100, $96, $116,$102, and $106, respectively. The overtime production costs are 50%higher than the regular production costs.If an engine is produced now for use in later periods, an additionalstorage cost of $4per engine per period is incurred.Formulate the problem as a transportation model.

A Multiperiod Model

An earth-moving company is planning a training program so that its newemployees are able to operate the large earth-moving machines properly.Employees who are already trained will be the instructors of newly hiredemployees; the new employees must complete the training successfully inorder to stay on the payroll of the company. The ratio instructor:newemployee will be 1:10 and, from existing data, seven out of ten newemployees will complete successfully the training program.

A Multiperiod Model

The number of trained employees that the company needs in the nextfour months (January through April) are:

I January: 100

I February: 150

I March: 200

I April: 250

At the beginning of January the company has 130 trained operators.

A Multiperiod Model

Payroll costs are as follows:

I New employee in training: $400.

I Trained employee: $700

I Idle trained employee: $500

(Idle trained employees are not fired, as per company’s policy.)

The company wishes to find the way to have 250 trained employees atthe end of April while minimizing total costs of hiring, training andoperating the machines.

A Multiperiod ModelFormulation − Decision Variables

The decision variables are the number of trained operators who act asinstructors and the number of idle trained operators. In any given period(month) the number of trained operators working the machines is givenby the corresponding monthly requirements.So, we define xT ,j as the number of trained operators employed asinstructors, and xId,j as the number of trainde operators who are idle inmonth j = January (J), February (F), March (M) and April (A).

A Multiperiod ModelFormulation − Constraints

The total number of trained operators at the beginning of each monthhas to equal the number of instructors plus the number of idle trainedoperators plus the number of trained employees operating the machines.Thus:

100 + xT ,J + xId,J = 130 (January)

150 + xT ,F + xId,F = 130 + 7xT ,J (February)

200 + xT ,M + xId,M = 130 + 7xT ,J + 7xT ,F (March)

250 = 130 + 7xT ,J + 7xT ,F + 7xT ,M (April)

A Multiperiod ModelFormulation − Objective Function

The objective function need not include the payroll cost of the trainedemployees working the machines because it is a constant cost. Pertinentcosts are

I Training costs (instructors and trainees)

I Idle trained operators

Thus, the linear programming model is

A Multiperiod ModelFormulation − The LP Model

Min z =400 (10xT ,J + 10xT ,F + 10xT ,M) + 700 (xT ,J + xT ,F + xT ,M)

+ 500 (10xId,J + 10xId,F + 10xId,M)

subject to

xT ,J +xId,J = 307xT ,J −xT ,F −xId,F = 207xT ,J 7xT ,F −xT ,M −xId,M = 707xT ,J +7xT ,F +7xT ,M = 120

All variables are positive or zero.

Multi-period Production Model

A company has contracted to produce two products, A and B, over themonths of June, July and August. The total production capacity(expressed in hours) varies monthly and is equal to 3000, 3500 and 3000hours for the month of June, July and August, respectively.The demand for product A (units) is 500, 500 and 750, while the demandof product B (units) is 1000, 1200, 1200 (June, July and August). Theproduction rates (units per hour) are 1.25 and 1.00 per products A andB, respectively.All demand must be met. However, demand for a later month may befilled from the production in an earlier period. For any carryover from onemonth to the next, holding costs of $0.90 and $0.75 per unit per monthare charged for product A and B, respectively.The unit production costs for the two products are $30 and $28 forproducts A and B, respectively.Determine the optimum production schedule for the two products.

Optimal Allocation of Aircrafts to Routes

Aircraft type Capacity (passengers) Number of aircrafts 1 2 3 4

1 50 5 3 2 2 1

2 30 8 4 3 3 2

3 20 10 5 5 4 2

Daily number of customers 1000 2000 900 1200

Associated costs, including penalties for losing customers because of space unavailability, are:

Aircraft type 1 2 3 4

1 1000 1100 1200 1500

2 800 900 1000 1000

3 600 800 800 900

Penalty ($) per lost customer 40 50 45 70

Develop a linear model for determining the optimum allocation of aircrafts to routes.

Number of daily trips on route

Operating costs ($) per trip on route

Optimal Allocation of Aircrafts to Routes

Let xij = number of aircraft of type i allocated to route j , i = 1, 2, 3,j = 1, 2, 3, 4. Let Sj = the number of passengers not served on route j ,j = 1, 2, 3, 4.The objective function is to minimize total operating costs plus thepenalty for each lost customer

Table: Objective function

Minimize z = 1000(3x11 +1100(2x12) +1200(2x13) +1500(x14)800(4x21) +900(3x12) +1000(3x13) +1000(2x24)600(5x31) +800(5x32) +800(4x13) +900(2x14)40S1 +50S2 +45S3 +70S4

The constraints are given in the following slides.

Optimal Allocation of Aircrafts to Routes

Number of aircrafts available:

4∑j=1

x1j ≤ 5,4∑

j=1

x2j ≤ 8,4∑

j=1

x3j ≤ 10

Daily number of customers:

50(3x11) + 30(4x21) + 20(5x31)− S1 = 1000

50(2x12) + 30(3x22) + 20(5x32)− S2 = 2000

50(2x13) + 30(3x23) + 20(4x33)− S3 = 900

50(x14) + 30(2x24) + 20(2x34)− S4 = 1200

Nonnegativity: xij ≥ 0, Sj ≥ 0, i = 1, 2, 3, j = 1, 2, 3, 4.

Blending Models

Hawaii Sugar Company produces brown sugar, processed (white) sugar,powdered sugar and molasses from sugar cane syrup. The companypurchases 4000 tons of syrup weekly and is contracted to deliver at least25 tons weekly of each type of sugar. The production process starts bymanufacturing brown sugar and molasses from the syrup. A ton of syrupproduces 0.3 tons of brown sugar and 0.1 ton of molasses. White sugar isproduced by processing brown sugar; it takes 1 ton of brown sugar toproduce 0.8 tons of white sugar. Powdered sugar is produced from whitesugar through a special grinding process that has a 95% conversionefficiency (1 ton of white sugar produces 0.95 ton of powdered sugar).The profit per ton for brown sugar, white sugar, powdered sugar andmolasses are $150, $200, $230 and $35, respectively. Formulate theproblem of determining the optimal weekly production schedule as alinear program.

Blending ModelsSolution

Let x1 = tons of brown sugar produced per week; let x2 = tons of whitesugar produced per week; let x3 = tons of powdered sugar produced perweek; and let x4 = tons of molasses produced per week.A schematic of the production process is given below:

1:0.8

0.10.3

One tonof syrup

0.1

1:0.95

x4

x3

x2

x3/0.95

x2+(x3/0.95)

x1

1/(0.8)(x2+(x3/0.95))

x1+

(1/(

0.8)

(x2+

(x3

/0.9

5)))

Blending ModelsSolution

The complete LP model is Max z = 150x1 + 200x2 + 230x3 + 35x4

subject to

x1 +1

0.8

[x2 +

( x3

0.95

)]≤ 0.3(4000)

x4 ≤ 0.1(4000), x1 ≥ 25, x2 ≥ 25, x3 ≥ 25

x4 ≥ 0.

Mixing Model

Two alloys are made from four metals M1, M2, M3 and M4 according tothe following specifications

Alloy Specifications Selling price

At most 80% of M1

At most 30% of M2

At least 50% of M4

Between 40% and 60% 0f M2

At least 30% of M3

At most 70% of M4

A

B

$200

$300

Mixing Model

The four metals, in turn, are extracted from three ores according to thefollowing data:

Max Qty Price/ton

Ore (tons) M1 M2 M3 M4 Others ($)

1 1000 20 10 30 30 10 30

2 2000 10 20 30 30 10 40

3 3000 5 5 70 20 0 50

Develop a linear programming model for determining the optimum alloy production.

Constituents (%)

Mixing ModelSolution

Define xij = tons of ore i allotted to alloy k ; wk = tons of alloy kproduced.The objective function is

Max z = 200W1 + 300W2−30(x1A +x1B)−40(x2A +x2B)−50(x3A +x3B)

subject toSpecs constraints . . .&Ore constraints . . .

Linear Programming Models

September 4, 2012

Production Models

Toolco has contracted with AutoMate to supply their automotivediscount stores with wrenches and chisels. AutoMate’s weekly demandconsists of at least 1500 wrenches has 1200 chisels. Toolco’s presentone-shift capacity is not large enough to produce the requested units andit must use overtime and possibly subcontracting with other tool shops.The result is an increase in the production cost per unit, as shown in thefollowing table. Market demand restricts chisels to wrenches ratio to aratio of at least 2:1.

Production Models

Table: Relevant Data

Tool Production type Weekly production Unit costs ($)range (units)

Wrenches Regular 0− 550 2.00Overtime 551− 800 2.80Subcontracting 801−∞ 3.00

Chisels Regular 0− 620 2.10Overtime 621− 900 3.20Subcontracting 901−∞ 4.20

Formulate the problem as a linear program.

Production ModelsSolution

Let Wr , Cr be the number of wrenches and chisels manufactured usingregular time; let Wo , Co be the number of wrenches and chiselsmanufactured using overtime; finally, let Ws , Cs be the number ofwrenches and chisels manufactured using subcontracting.The objective function is Maxz = 2Wr + 2.80Wo + 3Ws + 2.10Cr + 3.20Co + 4.20Cs .The proportion of chisels to wrenches gives

(Cr + Co + Cs) ≥ 2(Wr + Wo + Ws)

Other constraints are

Wr ≤ 550, Cr ≤ 620

Wo ≤ 250, Co ≤ 280

Wr + Wo + Ws ≥ 1500, Cr + Co + Cs ≥ 1200

All variables are ≥ 0.

Investment Problem

A business executive has the option of investing money in three plans:Plan A guarantees that each dollar invested will earn $0.70 one year later;plan B guarantees that each dollar invested will earn $3.00 after twoyears; and plan C guarantees $4.50 after four years.Investments can be made annually in the three plans.How should the executive invest $100,000 to maximize earnings at theend of 5 years? Propose a linear programming model for this problem.

Blending ModelTaha, Section 2.3

HI-V produces three types of canned juice drinks, A, B, and C, usingfresh strawberries, grapes and apples. The daily supply is limited to 200tons of strawberries, 100 tons of grapes and 150 tons of apples. The costper ton of strawberries, grapes and apples is $200, $100, and $90,respectively. Each ton makes 1500 lb of strawberry juice, 1200 lb of grapejuice and 1000 lb of apple juice. Drink A is a 1:1 mix of strawberry andgrape juice. Drink B is a 1:1:2 mix of strawberry, grape and apple juice.Drink C is a 2:3 mix of grape and apple juice. All drinks are canned in16-oz (1 lb) cans. The price per can is $1.15, $1.25 and $1.20 for drinksA, B and C. Determine the optimal production mix of the three drinks.

Blending ModelSolution–Objective Function

Define the following decision variables:

I xs , xg and xa tons of strawberry, grapes and apples purchased daily,respectively.

I xA, xB , xC cans of drink A, B, C produced daily. (Each can holds 16oz.)

I xsA, xsB pounds of strawberries used in drink A, B; xgA, xgB , xgCpounds of grapes used in drink A, B, C ; xaB , xaC pounds of applesused in drink B, C . (1 lb = 16 oz.)The objective function is Maximize

z = 1.15xA + 1.25xB + 1.2xC − 200xs − 100xg − 90xa

Blending ModelSolution–Constraints

We have three types of constraints:

I Raw material.

I Transformation from raw material to fruit juice.

I Mixing the different fruit juices to prepare the three types of drinks(proportion rates).

Blending ModelSolution–Constraints

Raw Material

xs ≤ 200, xg ≤ 100, xa ≤ 150

Transformation

xsA+xsB = 1500xs , xgA+xgB+xgC = 1200xg , xaB+xaC = 1000xa

xA = xsA + xgA, xB = xsB + xgB + xaB , xC = xgC + xaC ,

Proportion rates

xsA = xgA, xsB = xgB , xgB = 0.5xaB , 3xgC = 2xaC

All variables are positive.

Blending ModelTaha, Section 2.3–Extra Homework

Same as in the previous problem, except that we are using metric units.Consider that one liter of fruit juice is approximately one kilogram. Thecans hold 355 ml. Modify the LP model for this new situation.

Multiperiod Financial ModelsWinston, Section 3.11

Finco Investment Corporation must determine the investment strategy forthe firm during the next three years. At present (time 0) $100,000 isavailable for investment. Investments A, B, C, D, E are all available. Thecash flow associated with investing $1 in each investment is given in thetable below:

0 1 2 3A –$1 +$0.50 +$1 $0B $0 –$1 +$0.50 +$1C –$1 +$1.20 $0 $0D –$1 $0 $0 +$1.9E $0 $0 –$1 +$1.50

For example, $1 invested in investment B requires $1 cash outflow attime 1 and returns $0.50 at time 2 and $1 at time 3.

Multiperiod Financial ModelsWinston, Section 3.11

To ensure that the company’s portfolio is diversified, Finco requires thatat most $75,000 be placed in any single investment. In addition toinvestments A–E, Finco can earn interest at 8% per year by keepinguninvested 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 beimmediately reinvested in investment B.Finco cannot borrow funds, so the cash available for investment at anytime is limited to the cash on hand.Formulate a linear program that will maximize cash on hand at time 3.

Programming Human ResourcesProblem 2

Oxbridge University maintains a powerful mainframe computer forresearch use by its faculty, PhD students and research associates. Duringall work hours, an operator must be available to operate and maintainthe computer, as well as to perform some programming services. BerylIngram, the director of the computer facility, oversees the operation.It is now the beginning of the semester and Beryl is confronted with theproblem of assigning different working hours to her operators. Because allthe operators are currently enrolled at the university, they are able to workonly a limited number of hours each day, as shown in the table below.

Programming Human Resources

Table: Maximum Hours of Availability

Operator Wage Rate Mon Tue Wed Thurs FriK.C. $10.00/hour 6 0 6 0 6D.H. $10.10/hour 0 6 0 6 0H.B. $9.90/hour 4 8 4 0 4S.C. $9.80/hour 5 5 5 0 5K.S. $10.80/hour 3 0 3 8 0N.K. $11.30/hour 0 0 0 6 2

There are six operators (four undergraduate students and two graduatestudents). They all have different wages because of their experience inwith computers and in their programming ability. The above table showstheir wage rates, together with the maximum number of hours that eachcan work each day.

Programming Human Resources

Each operator is guaranteed a minimum number of hours each week thatwill maintain an adequate knowledge of the operation. This level is setarbitrarily at 8 hours per week for the undergraduate students and 7hours per week for the graduate students.The computer facility is to be open from 8AM to 10PM Monday throughFriday with exactly one operator on duty between these hours. OnSaturday and Sunday, the center is operated by other staff.Because of a tight budget, Beryl has to minimize cost. She wishes todetermine the number of hours she should assign to each operator oneach day. Formulate a linear programming model for this problem.

Programming Human Resources

A large department store operates 7 days a week. The manager estimatesthat the minimum number of salespersons required to provide promptservice is 12 for Monday, 18 for Tuesday, 20 for Wednesday, 28 forThursday, 32 for Friday and 40 for each Saturday and Sunday. Eachsalesperson works 5 days a week, with the two consecutive off-daysstaggered throughout the week. For example, if 10 workers starts onMonday, then a possible allocation of off-days is that 2 salespersons cantake their off-days on Tuesday and Wednesday, 5 on Wednesday andThursday and 3 on Saturday and Sunday. How many salespersons shouldbe contracted and how should their off-days be allocated?

Programming Human Resources

Let yij the number starting their working day on day i and having theirtwo days off on day j , j 6= i . The total number starting on day i(regardless of when they have their days off) is

xi =7∑

j=1

yij , j 6= i

The objective function is to minimize the total number of workers:

Min z = x1 + x2 + x3 + x4 + x5 + x6 + x7

subject to . . .(Complete the LP model.)

Blending Model

Shale Oil, located on the island of Aruba, has a capacity of 600,000barrels of crude oil per day. The final products from the refinery includetwo types of unleaded gasoline: regular and premium. The refiningprocess encompasses three stages: (1) a distillation tower that producesfeedstock; (2) a cracker unit that produces gasoline stock by using aportion of the feedstock produced from the distillation tower; and (3) ablender unit that blends the gasoline stock from the cracker unit and thefeedstock from the distillation tower.Both regular and premium gasoline can be blended from either thefeedstock or the gasoline stock at different production costs. Thecompany estimates that the net profit per barrel of regular gasoline is$7.70 and $5.20, depending on whether it is blended from feedstock orfrom gasoline stock. The corresponding profit values for the premiumgrade are $10.40 and $12.30.

Blending Model

Design specifications require 5 barrels of crude oil to produce 1 barrel offeedstock. The capacity of the cracker unit is 40,000 barrels of feedstocka day. All remaining feedstock is used directly in the blender unit toproduce end-product gasoline. The demand limits for regular andpremium gasoline is 80,000 and 50,000 barrels a day, respectively.Develop a linear program for determining the optimum productionschedule for the refinery.

Blending ModelSolution

Consider the following picture, which sums up the production process:

Distillation

R

R

P

P

Cracker

Blender

Crude

5:1

x22

x21

x12

x11

Blending ModelSolution–LP Model

The objective is to maximize

z = 7.70x11 + 5.20x21 + 10.40x12 + 12.30x22

subject to

5 (x11 + x12 + x21 + x22) ≤ 600, 000

x21 + x22 ≤ 40, 000

x11 + x21 ≤ 80, 000

x12 + x22 ≤ 50, 000

xij ≥ 0, i , j = 1, 2.

Production Process ModelsWinston, Section 3.9

AmeriCo Oil has three different processes that can be used tomanufacture various type of gasoline (G1, G2, G3). Each process involvesblending oils in the company’s catalytic cracker.Running Process 1 for one hour costs $5 and require 2 barrels of C1 and3 barrels of C2. The output from running process one for one hour is 2barrels of G1and 1 barrel of G2.Running Process 2 for one hour costs $4 and require 1 barrel of C1 and 3barrels of C2. The output from running process one for one hour is 3barrels of G2.Running Process 3 for one hour costs $1 and require 2 barrels of C2 and3 barrels of G2. The output from running process three for one hour is 2barrels of G3.

Production Process ModelsWinston, Section 3.9

Each week, 200 barrels of C1, at $2/bbl, and 300 barrels of C2, at$3/bbl, may be purchased. All gasoline produced can be sold at thefollowing per-barrel prices: G1, $9; G2, $10; G3, $24.Formulate an LP whose solution will maximize total profit (revenueminus costs). Assume that only 100 hours are available at the catalyticcracker each week.

Production Process ModelsSolution–Objective Function

Define gi = barrels of gasoline i produced weekly, i = 1, 2, 3; ok = barrelsof oil type k, k = 1, 2; xj = hours used in process j , j = 1, 2, 3.The objective function is maximize

z = 9g1 + 10g2 + 24g3 − 5x1 − 4x2 − x3 − 2o1 − 3o2

Production Process ModelsSolution–Constraints

To better understand the constraints, let us consider the following tablewith the relevant information:

Input Output

Hours of process

Barrels of Oil

Barrels of gasoline

Barrels of gasoline

x1 2o1, 3o2 - 2g1, g2

x2 1o1, 3o2 - 3g2

x3 2o2 3g2 2g3

Thus:

o1 = 2x1 + x2o2 = 3x1 + 3x2 + 2x3g2 + 3x3 = x1 + 3x2o1 < 200o2 < 300x1 + x2 + x3 < 100g1 = 2x1g3 = 2x3All variables are positive, or zero.

Blending ModelWinston, Section 3.8

SunCo Oil manufactures three types of gasoline (G1, G2, G3). Each typeis produced by blending three tys of crude oil (C1, C2, C3). The salesprice per barrel (bbl) of gasoline and the purchase price per bbl of crudeoil are given in the table below. SunCo can purchase 5000 bbl of eachtype of crude daily.The three types of crude differ in their octane rating and sulfur content.The octane ratings and sulfur content of each type of crude is given in asecond table below. The crude oil blended to form G1 must have anaverage (per volume) octane rating of at least 10 and contain at most1% sulfur. The crude oil blended to form G2 must have an averageoctane rating of at least 8 and contain at most 2% sulfur. The crude oilblended to form G3 must have an average octane rating of at least 6 andcontain at most 1% sulfur. It costs $4 to transform one barrel of oil intoone barrel of gasoline, and SunCo’s refinery can produce up to 14,000barrels of gasoline daily.

Blending ModelWinston, Section 3.8

SunCo’s customers requires the following quantities of gasoline each day:G1–3000 barrels; G2–2000 barrels; G3–1000 barrels. The companyconsiders an obligation to meet its customers’ demands.Formulate an LP that will enable SunCo to maximize daily profits. Tosimplify matters, consider that the gasoline cannot be stored, so it mustbe sold the day it is produced.

Sales Price/bbl Purchase Price/bblG1 $70 C1 $45G2 $60 C2 $35G3 $50 C3 $25

Table: Per Barrel of Crude

Octane Rating Sulfur ContentC1 12 0.5%C2 6 2.0%C3 8 3.0%

Winston, Section 3.8Solution–Decision Variables

Let xij = the number of barrels of crude i used to produce gasoline j(i , j = 1, 2, 3).Thus,

xi1 + xi2 + xi3, i = 1, 2, 3

is the number of barrels of crude i used daily, and

x1j + x2j + x3j , j = 1, 2, 3

is the number of barrels of gasoline j produced daily.

Winston, Section 3.8Solution–Objective Function

The objective function is the daily total profit, which is calculated astotal revenues from selling gasoline minus total purchasing costs ofcrude oil minus total transformation costs.

Max z =70 (x11 + x21 + x31) + 60 (x12 + x22 + x32)

+ 50 (x13 + x23 + x33)

− 45 (x11 + x12 + x13)− 35 (x21 + x22 + x23)

− 25 (x31 + x32 + x33)

− 4 (x11 + x12 + x13 + x21 + x22 + x23 + x31 + x32 + x33)

Thus, the objective is to maximize

z = 21x11 + 11x12 + x13 + 31x21 + 21x22 + 11x23 + 41x31 + 31x32 + 21x33

Winston, Section 3.8Solution–Demand Constraints

The gasoline j , j = 1, 2, 3, produced daily should equal its demand:

x11 + x21 + x31 = 3000 Demand G1

x12 + x22 + x32 = 2000 Demand G2

x13 + x23 + x33 = 3000 Demand G3

Winston, Section 3.8Solution–Crude Supply Constraints

At most a certain quantity of crude i , i = 1, 2, 3, can be purchased daily:

x11 + x12 + x13 ≤ 5000 Supply C1

x21 + x22 + x23 ≤ 5000 Supply C2

x31 + x32 + x33 ≤ 5000 Supply C3

Winston, Section 3.8Solution–Limited Refinery Capacity

At most 14,000 barrels of gasoline can be produced daily:

x11 + x21 + x31 + x12 + x22 + x32 + x13 + x23 + x33 ≤ 14, 000

Winston, Section 3.8Solution–Octane Specifications

G1 must have an average octane rating of at least 10:

(12x11 + 6x21 + 8x31) ≥ 10 (x11 + x21 + x31)

G2 must have an average octane rating of at least 8:

(12x12 + 6x22 + 8x32) ≥ 8 (x12 + x22 + x32)

G3 must have an average octane rating of at least 62:

(12x13 + 6x23 + 8x33) ≥ 6 (x13 + x23 + x33)

2This constraint is redundant

Winston, Section 3.8Solution–Sulfur Specifications

G1 must contain at most 1% of sulfur:

(0.005x11 + 0.02x21 + 0.03x31) ≤ 0.01 (x11 + x21 + x31)

G2 must contain at most 2% of sulfur:

(0.005x12 + 0.02x22 + 0.03x32) ≤ 0.02 (x12 + x22 + x32)

G3 must contain at most 1% of sulfur:

(0.005x13 + 0.02x23 + 0.03x33) ≤ 0.01 (x13 + x23 + x33)

Blending ModelExtra Homework

Consider the same problem as above. Assume, moreover, that SunCo canincrease the sale of the gasoline it sells by advertising. More precisely, foreach dollar spent in advertising a particular type of gasoline, the dailydemand will increase by the following amounts: 10 barrels, for G1; 15barrels for G2; and 12 barrels, for G3. For example, if SunCo spends $10in advertising G2, then its demand will increase by 10(15) = 150 barrels.Modify the above LP model to take into account this extra assumption.