Operations Research I Introduction
September 4, 2012
Operations Research
What is Operations Research? Discipline by means of which it is possible to allocate scarce resources to the operations of a firm. How to conduct and coordinate the operations (activities) of an organization Research: Use of the scientific method to investigate the problem of interest
Operations Research
What is Operations Research? Discipline by means of which it is possible to allocate scarce resources to the operations of a firm. How to conduct and coordinate the operations (activities) of an organization Research: Use of the scientific method to investigate the problem of interest
OR Models Characteristics
All OR models share the same characteristics: Decision Alternatives Restrictions, due to the scarcity of resources available Objective Criterion (Function)
OR Models General Format
Optimize an Objective Function
subject to
Constraints
Optimize: either Minimize or Maximize (according to the case at hand).
OR Models An Example
We are to produce soda cans with a volume of 355 ml and with smallest surface. Determine Decision Alternatives Restrictions (Constraints) The objective function
OR Models An Example − Solution
Model: The can can be thought of as a cylinder, with radius r and height h. Thus, Objective Min z = 2π r 2 + 2π rh Constraint 1 π r 2 h = 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
Linear Programming (Dantzig, late 40’s) Dynamic Programming (Bellmann, 50’s) Queueing Theory Inventory Theory Nonlinear Programming (Economics) etc. . .
This course of Operations Research 1 will be devoted to the study of Linear Programming (LP) and how LP can be used to solve relatively simple problems in Engineering.
Introduction to Linear Programming Toy Model
Flinks Furniture produces inexpensive tables and chairs. The production process is similar in that both products require a certain number of carpentry work and a certain number of labor hours in the painting and varnishing department. Each chair requires 3 hours in carpentry and 1 hour in painting and varnishing. Each table takes 4 hours of carpentry and 2 hours in the painting and varnishing shop.
Introduction to Linear Programming Toy Model
During the current production time, 240 hours of carpentry time are available, 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 $5 profit. Determine the best possible combination of tables and chairs in order to reach the maximum profit.
Toy Model
The mathematical formulation of this problem is as follows1 : We want to Max z = 5x 1 + 7x 2 subject to 3x 1 x 1
with x i ≥ 0, i = 1, 2.
+4x 2 +2x 2
≤ 240 ≤ 100
Linear Programming Properties
There must be one objective function Presence of constraints There must be alternatives available Mathematical relationships are linear
Linear Programming Assumptions
Certainty Proportionality Additivity Divisibility Nonnegative variables
Problem 1
The Apex Television Company has to decide on the number of 27- and 20-inch sets to be produced at one of its factories. Market research indicates that at most 40 of the 27-inch sets and 10 of the 20-inch sets can be sold per month. The maximum number of work-hours available is 500 per month. A 27-inch set requires 20 work-hours and a 20-inch set requires 10 work-hours. Each 27-inch set sold produces a profit of $120 and each 20-inch set produces a profit of $80. A wholesaler has agreed to purchase all the television sets produced if the numbers do not exceed the maxima indicated by the market research. Formulate a linear programming model for this problem.
Problem 1 Solution
Let x 1 (resp. x 2 ) be the number of 27-inch (resp. 20-inch) TV sets produced. Let Z be the total profit per month. The mathematical formulation of the model is Max Z = 120x 1 + 80x 2 subject to x 1
20x 1 with x 1 ≥ 0, x 2 ≥ 0.
x 2 +10x 2
≤ 40 ≤ 10 ≤ 500
Problem 2 Dwight is an elementary school teacher who also raises pigs for supplemental income. He is trying to decide what to feed his pigs. He is considering using a combination of pig feeds available from local suppliers. He would like to feed the pigs at minimum cost while also making sure each pig receives an adequate supply of calories and vitamins. The cost, calorie content, and vitamin content of each feed are given in the table below.
Contents Calories (per pound) Vitamins (per pound) Cost (per pound)
Feed Type A 800 140 units $0.40
Feed Type B 1,000 70 units $0.80
Each pig requires at least 8,000 calories per day and at least 700 units of vitamins. A further requirement (constraint) is that no more than one-third of the diet (by weight) can consist of Feed Type A, since it contains an ingredient which is toxic if consumed in too large a quantity.
Problem 2 Solution
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 Problem Winston, Chapter 3
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 $0.50, each scoop of chocolate ice cream costs $0.20, each bottle of cola costs $0.30 and each piece of pineapple cheesecake costs $0.80. 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 in the table below. Formulate a linear programming model that can be used to satisfy my daily nutritional requirements at minimum cost.
A Diet Problem Winston, Chapter 3
Table: Nutritional Values
Brownie Chocolate Ice Cream (1 scoop) Cola (1 bottle) Pineapple Cheesecake
Calories 400
Chocolate (oz) 3
Sugar (oz) 2
Fat (oz) 2
200 150
2 0
2 4
4 1
500
0
4
5
Problem 1, Problem Set 2.3A A realtor is developing a rental housing and retail area. The housing area consists of efficiency departments, duplexes and single-family homes. Maximum demand by potential renters is estimated to be 500 efficiency departments, 300 duplexes and 250 single-family homes, but the number of duplexes must equal at least 50% of the number of efficiency departments and single-family homes. Retail space is proportional to the number of home units at the rates of at least 10 ft 2 , 15 ft2 and 18 ft2 per efficiency departments, duplexes and single-family homes, respectively. However, land availability limits retail space to no more than 10,000 ft 2 . The monthly rental income is estimated at $600, $750 and $1200 for efficiency-, duplex- and single-family units, respectively. The retail space rents for $100/ft2 . Formulate a linear programming model that can be used to determine the optimal 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, July and August) at All-Flavors Parlor is estimated at 500, 600 and 400 20-gallon cartons, respectively. Two wholesalers, 1 and 2, can supply All-Flavors with its ice cream. Although the flavors from the two suppliers are different, they are interchangeable. The maximum number of cartons either supplier can provide is 400 per month. Also, the prices each supplier charges from one month to the next varies, according to the schedule June July August Supplier 1 $100 $110 $120 Supplier 2 $115 $108 $125 (price per carton)
Problem 4, Problem Set 2.3D
To take advantage of price fluctuations, All-Flavors can purchase more than is needed for a month and store the surplus to satisfy the demand in a later month. The cost of refrigerating an ice cream carton is $5 per month. It is realistic in the present situation to suppose that the refrigeration cost is a function of the average number of cartons on hand during 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 08:00 and 22:00. Each volunteer works three consecutive hours except those starting at 20:00 who works for two hours only.The minimum need for volunteers is approximated by a step function over 2-hour intervals starting at 08:00 as 4, 6, 8, 6, 4, 6, 8. Because most volunteers are retired individuals, they are willing to offer their services at any hour of the day (08:00-22:00). However, because of the large number of charities competing for their services, the number must be kept as low as possible. Formulate a linear programming model that can be used to find an optimal schedule for the start time of the volunteers.
Problem 3.4-8, HL Web Mercantile sells many household products through an online catalog. The company needs substantial warehouse space for storing its goods. Plans are now being made for leasing warehouse storage space over the next 5 months. Just how much space will be required in each of these months is known. However, since these space requirements are quite different, it may be most economical to lease only the amount needed each month, on a month-by-month basis. On the other hand, the additional cost for leasing space for additional months is much less than for the first month, so it may be less expensive to lease the maximum amount needed for the entire 5 months. Another option is the intermediate approach of changing the total amount of space leased (by adding a new lease and/or having an old lease expire) at least once, but not every month. The space requirements and the leasing costs for the various leasing periods are as follows.
Problem 3.4-8, HL
Month 1 2 3 4 5
Required space (sq ft) 30,000 20,000 40,000 10,000 50,000
Leasing period (months) 1 2 3 4 5
Cost per sq ft Leased $65 $100 $135 $160 $190
Formulate a linear programming model that will minimize the total leasing cost for meeting the space requirements.
Transportation Problem
Three orchards supply crates of oranges to four retailers. The daily demand amounts at the four retailers are 150, 150, 400, and 100 crates, respectively. Supplies at the three orchards are dictated by the available regular labor and are estimated at 150, 400, and 250 crates daily. The transportation costs per crates from the orchards to the retailers are given in the next slide. Formulate the problem as a transportation model.
Problem 11, Problem Set 5.1A
Orchard 1 Orchard 2 Orchard 3
Retailer 1 $1 $2 $1
Retailer 2 $2 $4 $3
Retailer 3 $3 $1 $5
Retailer 4 $2 $2 $3
Problem 11, Problem Set 5.1A
Three orchards supply crates of oranges to four retailers. The daily demand amounts at the four retailers are 150, 150, 400, and 100 crates, respectively. Supplies at the three orchards are dictated by the available regular labor and are estimated at 150, 200, and 250 crates daily. However, Orchards 1 and 2 have indicated that they can supply more crates, if necessary, by using overtime labor. Orchard 3 does not offer this option. The transportation costs per crates from the orchards to the retailers are given in the next slide. Formulate the problem as a transportation model.
Problem 11, Problem Set 5.1A
Orchard 1 Orchard 2 Orchard 3
Retailer 1 $1 $2 $1
Retailer 2 $2 $4 $3
Retailer 3 $3 $1 $5
Retailer 4 $2 $2 $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 8AM until midnight. Larry has monitored the usage of the center at various time of the day and determined that the following number of consultants are required:
Time of day 8AM – Noon Noon – 4PM 4PM – 8PM 8PM – Midnight
Minimum number of consultants required to be on duty 4 8 10 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 the following shifts: morning (8AM–4PM), afternoon (noon–8PM), and night (4MP–midnight). Full-time consultants are paid $14 per hour. Part-time consultants can be hired to work any of the you shifts listed in the table above and are paid $12 per hour. An additional requirement is that during any time period, there must be at least 2 full-time consultants on duty for every part-time consultant on duty. Formulate a linear programming model that will allow Larry to determine how many full-time and part-time consultants should work each shift to meet the above requirement at the minimum possible cost.
Investment Problem Taha, Problem 1, Problem Se 2.3C
Fox Enterprises is considering six projects for possible construction over the next four years. The expected (present value) returns and cash outlays for the projects are given below. Fox can undertake any of the projects partially or completely. A partial undertaking of a project will prorate both the return and the cash outlay proportionately.
Investment Problem Taha, Problem 1, Problem Se 2.3C
Table: Cash Outlays
Project
Year 1
Year 2
Year 3
Year 4
1 2 3 4 5 6 Available Funds ($1000)
10.5 8.3 10.2 7.2 12.3 9.2 60
14.4 12.6 14.2 10.5 10.1 7.8 70
2.2 9.5 5.6 7.5 8.3 6.9 35
2.4 3.1 4.2 5 6.3 5.1 20
Return ($1000) 32.4 35.8 17.75 14.9 18.2 12.35
Investment Problem Taha, Problem 1, Problem Se 2.3C
1. Formulate the problem as a linear program. Ignore the time value of money. 2. Suppose that if a portion of project 2 is undertaken, then at least an equal portion of project 6 must be undertaken as well. Modify the formulation of the model. 3. In the original model, suppose that any funds left at the end of a year are used in the next year. Modify the formulation of the linear program.
Blending Model The Metalco Compan desires to blend a new alloy of 40 percent zinc, 35 percent tin, and 24 percent lead from several available alloys having the following properties: Table: Alloy
Property % zinc % tin % lead Cost ($/lb)
1 60 10 30 22
2 25 15 60 20
3 45 45 10 25
4 20 50 30 24
5 50 40 10 27
The objective is to determine the proportions of these alloys that should be blended to produce the desired new alloy at minimum cost. Formulate a linear programming model that ca be used to that purpose.
Transportation Model Taha, Problem Set 5.1A, Problem 6
Three electric power plants with capacities of 25, 40, and 30 million kWh supply electricity to three cities. The maximum demands at the three cities are estimated at 30, 35, and 25 million kWh. The price per million kWh at the three cities is given by the table below: Table: Price/Million kWh
City 1 City 2 City 3
$600 $320 $500
$700 $300 $480
$400 $350 $450
During the month of August, there is a 20% increase in demand at each of the three cities, which can be met purchasing electricity from another network at a premium rate of $1000 per million kWh. The network is not linked to City 3, however. The company wishes to determine the most economical plan for the distribution and purchase of the electricity.
Transportation Model Taha, 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 supplying the engine has different production capacities estimated at 180, 230, 430, 300, and 300 units for the five quarters. Backordering is not allowed, but the manufacturer can use overtime to fill the immediate demand, if necessary. The overtime capacity for each period is half the regular capacity. 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 additional storage 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 new employees are able to operate the large earth-moving machines properly. Employees who are already trained will be the instructors of newly hired employees; the new employees must complete the training successfully in order to stay on the payroll of the company. The ratio instructor:new employee will be 1:10 and, from existing data, seven out of ten new employees will complete successfully the training program.
A Multiperiod Model
The number of trained employees that the company needs in the next four months (January through April) are: January: 100 February: 150 March: 200 April: 250 At the beginning of January the company has 130 trained operators.
A Multiperiod Model
Payroll costs are as follows: New employee in training: $400. Trained employee: $700 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 at the end of April while minimizing total costs of hiring, training and operating the machines.
A Multiperiod Model Formulation − Decision Variables
The decision variables are the number of trained operators who act as instructors and the number of idle trained operators. In any given period (month) the number of trained operators working the machines is given by the corresponding monthly requirements. So, we define x T T j as the number of trained operators employed as instructo instructors, rs, and x Id Id j as the number of trainde operators who are idle in month j = = January (J), February (F), March (M) and April (A). ,
,
A Multiperiod Model Formulation − Constraints
The total number of trained operators at the beginning of each month has to equal the number of instructors plus the number of idle trained operators plus the number of trained employees operating the machines. Thus: 100 + x T J + x Id J = 130 150 + x T F + x Id F = 130 + 7x T J 200 + x T M + x Id M = 130 + 7x T J + 7x T F 250 = 130 + 7x T J + 7x T F + 7x T M ,
,
,
,
,
,
,
,
,
,
,
,
(January) (February) (March) (April)
A Multiperiod Model Formulation − Objective Function
The objective function need not include the payroll cost of the trained employees working the machines because it is a constant cost. Pertinent costs are Training costs (instructors and trainees) Idle trained operators Thus, the linear programming model is
A Multiperiod Model Formulation − The LP Model
Min z =400 (10x T J + 10x T F + 10x T M ) + 700 (x T J + x T F + x T M ) + 500 (10x Id J + 10x Id F + 10x Id M ) ,
,
,
,
,
,
,
,
subject to x T J 7x T J 7x T J 7x T J ,
,
,
,
+x Id J ,
−x T F 7x T F +7x T F ,
,
,
All variables are positive or zero.
−x Id F ,
−x T M +7x T M ,
,
−x Id M ,
= 30 = 20 = 70 = 120
,
Multi-period Production Model A company has contracted to produce two products, A and B, over the months of June, July and August. The total production capacity (expressed in hours) varies monthly and is equal to 3000, 3500 and 3000 hours for the month of June, July and August, respectively. The demand for product A (units) is 500, 500 and 750, while the demand of product B (units) is 1000, 1200, 1200 (June, July and August). The production rates (units per hour) are 1.25 and 1.00 per products A and B, respectively. All demand must be met. However, demand for a later month may be filled from the production in an earlier period. For any carryover from one month to the next, holding costs of $0.90 and $0.75 per unit per month are charged for product A and B, respectively. The unit production costs for the two products are $30 and $28 for products A and B, respectively. Determine the optimum production schedule for the two products.
Optimal Allocation of Aircrafts to Routes
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Optimal Allocation of Aircrafts to Routes Let x ij = number of aircraft of type i allocated to route j , i = 1, 2, 3, j = 1, 2, 3, 4. Let S j = the number of passengers not served on route j , j = 1, 2, 3, 4. The objective function is to minimize total operating costs plus the penalty for each lost customer Table: Objective function
Minimize z =
1000(3x 11 +1100(2x 12 ) 800(4x 21 ) +900(3x 12 ) 600(5x 31 ) +800(5x 32 ) 40S 1 +50S 2
+1200(2x 13 ) +1000(3x 13 ) +800(4x 13 ) +45S 3
The constraints are given in the following slides.
+1500(x 14 ) +1000(2x 24 ) +900(2x 14 ) +70S 4
Optimal Allocation of Aircrafts to Routes Number of aircrafts available: 4
4
4
x 1 j ≤ 5,
j =1
j =1
x 2 j ≤ 8,
x 3 j ≤ 10
j =1
Daily number of customers: 50(3x 11 ) + 30(4x 21 ) + 20(5x 31 ) − S 1 = 1000 50(2x 12 ) + 30(3x 22 ) + 20(5x 32 ) − S 2 = 2000 50(2x 13 ) + 30(3x 23 ) + 20(4x 33 ) − S 3 = 900 50(x 14 ) + 30(2x 24 ) + 20(2x 34 ) − S 4 = 1200
Nonnegativity: x ij ≥ 0, S j ≥ 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 company purchases 4000 tons of syrup weekly and is contracted to deliver at least 25 tons weekly of each type of sugar. The production process starts by manufacturing brown sugar and molasses from the syrup. A ton of syrup produces 0.3 tons of brown sugar and 0.1 ton of molasses. White sugar is produced by processing brown sugar; it takes 1 ton of brown sugar to produce 0.8 tons of white sugar. Powdered sugar is produced from white sugar through a special grinding process that has a 95% conversion efficiency (1 ton of white sugar produces 0.95 ton of powdered sugar). The profit per ton for brown sugar, white sugar, powdered sugar and molasses are $150, $200, $230 and $35, respectively. Formulate the problem of determining the optimal weekly production schedule as a linear program.
Blending Models Solution
Let x 1 = tons of brown sugar produced per week; let x 2 = tons of white sugar produced per week; let x 3 = tons of powdered sugar produced per week; and let x 4 = tons of molasses produced per week. A schematic of the production process is given below:
Blending Models Solution
The complete LP model is Max z = 150x 1 + 200x 2 + 230x 3 + 35x 4 subject to 1 x 3 ≤ 0.3(4000) x 1 + x 2 + 0.8 0.95 x 4 ≤ 0.1(4000), x 1 ≥ 25, x 2 ≥ 25, x 3 ≥ 25
x 4 ≥ 0.
Mixing Model Two alloys are made from four metals M1, M2, M3 and M4 according to the following specifications
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Mixing Model The four metals, in turn, are extracted from three ores according to the following data:
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Mixing Model Solution
Define x ij = tons of ore i allotted to alloy k ; w k = tons of alloy k produced. The objective function is Max z = 200W 1 +300W 2 − 30(x 1A + x 1B ) − 40(x 2A + x 2B ) − 50(x 3A + x 3B ) subject to Specs constraints . . . & Ore constraints . . .
Linear Programming Models
September 4, 2012
Production Models
Toolco has contracted with AutoMate to supply their automotive discount stores with wrenches and chisels. AutoMate’s weekly demand consists of at least 1500 wrenches has 1200 chisels. Toolco’s present one-shift capacity is not large enough to produce the requested units and it must use overtime and possibly subcontracting with other tool shops. The result is an increase in the production cost per unit, as shown in the following table. Market demand restricts chisels to wrenches ratio to a ratio of at least 2:1.
Production Models
Table: Relevant Data
Tool
Production type
Wrenches
Regular Overtime Subcontracting Regular Overtime Subcontracting
Chisels
Weekly production range (units) 0 − 550 551 − 800 801 − ∞ 0 − 620 621 − 900 901 − ∞
Formulate the problem as a linear program.
Unit costs ($) 2.00 2.80 3.00 2.10 3.20 4.20
Production Models Solution
Let W r , C r be the number of wrenches and chisels manufactured using regular time; let W o , C o be the number of wrenches and chisels manufactured using overtime; finally, let W s , C s be the number of wrenches and chisels manufactured using subcontracting. The objective function is Max z = 2W r + 2.80W o + 3W s + 2.10C r + 3.20C o + 4.20C s . The proportion of chisels to wrenches gives (C r + C o + C s ) ≥ 2(W r + W o + W s ) Other constraints are W r ≤ 550,
C r ≤ 620
W o ≤ 250,
C o ≤ 280
W r + W o + W s ≥ 1500,
C r + C o + C s ≥ 1200
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 two years; 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 the end of 5 years? Propose a linear programming model for this problem.
Blending Model Taha, Section 2.3
HI-V produces three types of canned juice drinks, A, B, and C, using fresh strawberries, grapes and apples. The daily supply is limited to 200 tons of strawberries, 100 tons of grapes and 150 tons of apples. The cost per ton of strawberries, grapes and apples is $200, $100, and $90, respectively. Each ton makes 1500 lb of strawberry juice, 1200 lb of grape juice and 1000 lb of apple juice. Drink A is a 1:1 mix of strawberry and grape 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 in 16-oz (1 lb) cans. The price per can is $1.15, $1.25 and $1.20 for drinks A, B and C. Determine the optimal production mix of the three drinks.
Blending Model Solution–Objective Function
Define the following decision variables: x , x and x tons of strawberry, grapes and apples purchased daily, s g a respectively. x , x , x cans of drink A, B, C produced daily. (Each can holds 16 A B C oz.) x , x sA sB pounds of strawberries used in drink A, B ; x gA , x gB , x gC pounds of grapes used in drink A, B , C ; x aB , x aC pounds of apples used in drink B , C . (1 lb = 16 oz.) The objective function is Maximize z = 1.15x A + 1.25x B + 1.2x C − 200x s − 100x g − 90x a
Blending Model Solution–Constraints
We have three types of constraints: Raw material. Transformation from raw material to fruit juice. Mixing the different fruit juices to prepare the three types of drinks (proportion rates).
Blending Model Solution–Constraints
Raw Material x s ≤ 200,
x g ≤ 100,
x a ≤ 150
Transformation x sA + x sB = 1500x s , x A = x sA + x gA ,
x gA + x gB + x gC = 1200x g , x B = x sB + x gB + x aB ,
x aB + x aC = 1000x a x C = x gC + x aC ,
Proportion rates x sA = x gA ,
x sB = x gB ,
All variables are positive.
x gB = 0.5x aB ,
3x gC = 2x aC
Blending Model Taha, 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. The cans hold 355 ml. Modify the LP model for this new situation.
Multiperiod Financial Models Winston, Section 3.11
Finco Investment Corporation must determine the investment strategy for the firm during the next three years. At present (time 0) $100,000 is available for investment. Investments A, B, C, D, E are all available. The cash flow associated with investing $1 in each investment is given in the table below: A B C D E
0 –$1 $0 –$1 –$1 $0
1 +$0.50 –$1 +$1.20 $0 $0
2 +$1 +$0.50 $0 $0 –$1
3 $0 +$1 $0 +$1.9 +$1.50
For example, $1 invested in investment B requires $1 cash outflow at time 1 and returns $0.50 at time 2 and $1 at time 3.
Multiperiod Financial Models Winston, Section 3.11
To ensure that the company’s portfolio is diversified, Finco requires that at most $75,000 be placed in any single investment. 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 be immediately reinvested in investment B. Finco cannot borrow funds, so the cash available for investment at any time is limited to the cash on hand. Formulate a linear program that will maximize cash on hand at time 3.
Programming Human Resources Problem 2
Oxbridge University maintains a powerful mainframe computer for research use by its faculty, PhD students and research associates. During all work hours, an operator must be available to operate and maintain the computer, as well as to perform some programming services. Beryl Ingram, the director of the computer facility, oversees the operation. It is now the beginning of the semester and Beryl is confronted with the problem of assigning different working hours to her operators. Because all the operators are currently enrolled at the university, they are able to work only a limited number of hours each day, as shown in the table below.
Programming Human Resources
Table: Maximum Hours of Availability
Operator K.C. D.H. H.B. S.C. K.S. N.K.
Wage Rate $10.00/hour $10.10/hour $9.90/hour $9.80/hour $10.80/hour $11.30/hour
Mon 6 0 4 5 3 0
Tue 0 6 8 5 0 0
Wed 6 0 4 5 3 0
Thurs 0 6 0 0 8 6
Fri 6 0 4 5 0 2
There are six operators (four undergraduate students and two graduate students). They all have different wages because of their experience in with computers and in their programming ability. The above table shows their wage rates, together with the maximum number of hours that each can work each day.
Programming Human Resources
Each operator is guaranteed a minimum number of hours each week that will maintain an adequate knowledge of the operation. This level is set arbitrarily at 8 hours per week for the undergraduate students and 7 hours per week for the graduate students. The computer facility is to be open from 8AM to 10PM Monday through Friday with exactly one operator on duty between these hours. On Saturday and Sunday, the center is operated by other staff. Because of a tight budget, Beryl has to minimize cost. She wishes to determine the number of hours she should assign to each operator on each day. Formulate a linear programming model for this problem.
Programming Human Resources
A large department store operates 7 days a week. The manager estimates that the minimum number of salespersons required to provide prompt service is 12 for Monday, 18 for Tuesday, 20 for Wednesday, 28 for Thursday, 32 for Friday and 40 for each Saturday and Sunday. Each salesperson works 5 days a week, with the two consecutive off-days staggered throughout the week. For example, if 10 workers starts on Monday, then a possible allocation of off-days is that 2 salespersons can take their off-days on Tuesday and Wednesday, 5 on Wednesday and Thursday and 3 on Saturday and Sunday. How many salespersons should be contracted and how should their off-days be allocated?
Programming Human Resources
Let y ij the number starting their working day on day i and having their two days off on day j , j = i . The total number starting on day i (regardless of when they have their days off) is 7
x i =
i y ij , j =
j =1
The objective function is to minimize the total number of workers: Min z = x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 subject to . . . (Complete the LP model.)
Blending Model Shale Oil, located on the island of Aruba, has a capacity of 600,000 barrels of crude oil per day. The final products from the refinery include two types of unleaded gasoline: regular and premium. The refining process encompasses three stages: (1) a distillation tower that produces feedstock; (2) a cracker unit that produces gasoline stock by using a portion of the feedstock produced from the distillation tower; and (3) a blender unit that blends the gasoline stock from the cracker unit and the feedstock from the distillation tower. Both regular and premium gasoline can be blended from either the feedstock or the gasoline stock at different production costs. The company estimates that the net profit per barrel of regular gasoline is $7.70 and $5.20, depending on whether it is blended from feedstock or from gasoline stock. The corresponding profit values for the premium grade are $10.40 and $12.30.
Blending Model
Design specifications require 5 barrels of crude oil to produce 1 barrel of feedstock. The capacity of the cracker unit is 40,000 barrels of feedstock a day. All remaining feedstock is used directly in the blender unit to produce end-product gasoline. The demand limits for regular and premium gasoline is 80,000 and 50,000 barrels a day, respectively. Develop a linear program for determining the optimum production schedule for the refinery.
Blending Model Solution
Consider the following picture, which sums up the production process:
Blending Model Solution–LP Model
The objective is to maximize z = 7.70x 11 + 5.20x 21 + 10.40x 12 + 12.30x 22
subject to 5 (x 11 + x 12 + x 21 + x 22 ) ≤ 600, 000 x 21 + x 22 ≤ 40, 000 x 11 + x 21 ≤ 80, 000 x 12 + x 22 ≤ 50, 000 x ij ≥ 0, i , j = 1, 2.
Production Process Models Winston, Section 3.9
AmeriCo Oil has three different processes that can be used to manufacture various type of gasoline (G1, G2, G3). Each process involves blending oils in the company’s catalytic cracker. Running Process 1 for one hour costs $5 and require 2 barrels of C1 and 3 barrels of C2. The output from running process one for one hour is 2 barrels of G1and 1 barrel of G2. Running Process 2 for one hour costs $4 and require 1 barrel of C1 and 3 barrels of C2. The output from running process one for one hour is 3 barrels of G2. Running Process 3 for one hour costs $1 and require 2 barrels of C2 and 3 barrels of G2. The output from running process three for one hour is 2 barrels of G3.
Production Process Models Winston, 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 the following per-barrel prices: G1, $9; G2, $10; G3, $24. Formulate an LP whose solution will maximize total profit (revenue minus costs). Assume that only 100 hours are available at the catalytic cracker each week.
Production Process Models Solution–Objective Function
Define g i = barrels of gasoline i produced weekly, i = 1, 2, 3; o k = barrels of oil type k , k = 1, 2; x j = hours used in process j , j = 1, 2, 3. The objective function is maximize z = 9g 1 + 10g 2 + 24g 3 − 5x 1 − 4x 2 − x 3 − 2o 1 − 3o 2
Production Process Models Solution–Constraints
To better understand the constraints, let us consider the following table with the relevant information:
Blending Model Winston, Section 3.8
SunCo Oil manufactures three types of gasoline (G1, G2, G3). Each type is produced by blending three tys of crude oil (C1, C2, C3). The sales price per barrel (bbl) of gasoline and the purchase price per bbl of crude oil are given in the table below. SunCo can purchase 5000 bbl of each type 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 a second table below. The crude oil blended to form G1 must have an average (per volume) octane rating of at least 10 and contain at most 1% sulfur. The crude oil blended to form G2 must have an average octane rating of at least 8 and contain at most 2% sulfur. The crude oil blended to form G3 must have an average octane rating of at least 6 and contain at most 1% sulfur. It costs $4 to transform one barrel of oil into one barrel of gasoline, and SunCo’s refinery can produce up to 14,000 barrels of gasoline daily.
Blending Model Winston, Section 3.8
SunCo’s customers requires the following quantities of gasoline each day: G1–3000 barrels; G2–2000 barrels; G3–1000 barrels. The company considers an obligation to meet its customers’ demands. Formulate an LP that will enable SunCo to maximize daily profits. To simplify matters, consider that the gasoline cannot be stored, so it must be sold the day it is produced. G1 G2 G3
Sales Price/bbl $70 $60 $50
C1 C2 C3
Purchase Price/bbl $45 $35 $25
Table: Per Barrel of Crude
C1 C2
Octane Rating 12 6
Sulfur Content 0.5% 2 0%
Winston, Section 3.8 Solution–Decision Variables
Let x ij = the number of barrels of crude i used to produce gasoline j (i , j = 1, 2, 3). Thus, x i 1 + x i 2 + x i 3 , i = 1, 2, 3 is the number of barrels of crude i used daily, and x 1 j + x 2 j + x 3 j ,
j = 1, 2, 3
is the number of barrels of gasoline j produced daily.
Winston, Section 3.8 Solution–Objective Function
The objective function is the daily total profit, which is calculated as total revenues from selling gasoline minus total purchasing costs of crude oil minus total transformation costs. Max z =70(x 11 + x 21 + x 31 ) + 60 (x 12 + x 22 + x 32 ) + 50 (x 13 + x 23 + x 33 ) − 45 (x 11 + x 12 + x 13 ) − 35 (x 21 + x 22 + x 23 ) − 25 (x 31 + x 32 + x 33 ) − 4 (x 11 + x 12 + x 13 + x 21 + x 22 + x 23 + x 31 + x 32 + x 33 ) Thus, the objective is to maximize z = 21x 11 + 11 x 12 + x 13 + 31 x 21 + 21 x 22 + 11 x 23 + 41 x 31 + 31 x 32 + 21 x 33
Winston, Section 3.8 Solution–Demand Constraints
The gasoline j , j = 1, 2, 3, produced daily should equal its demand: x 11 + x 21 + x 31 = 3000 x 12 + x 22 + x 32 = 2000 x 13 + x 23 + x 33 = 3000
Demand G1 Demand G2 Demand G3
Winston, Section 3.8 Solution–Crude Supply Constraints
At most a certain quantity of crude i , i = 1, 2, 3, can be purchased daily: x 11 + x 12 + x 13 ≤ 5000 x 21 + x 22 + x 23 ≤ 5000 x 31 + x 32 + x 33 ≤ 5000
Supply C1 Supply C2 Supply C3
Winston, Section 3.8 Solution–Limited Refinery Capacity
At most 14,000 barrels of gasoline can be produced daily: x 11 + x 21 + x 31 + x 12 + x 22 + x 32 + x 13 + x 23 + x 33 ≤ 14, 000
Winston, Section 3.8 Solution–Octane Specifications
G1 must have an average octane rating of at least 10: (12x 11 + 6x 21 + 8x 31 ) ≥ 10 (x 11 + x 21 + x 31 ) G2 must have an average octane rating of at least 8: (12x 12 + 6x 22 + 8x 32 ) ≥ 8 (x 12 + x 22 + x 32 ) G3 must have an average octane rating of at least 62 : (12x 13 + 6x 23 + 8x 33 ) ≥ 6 (x 13 + x 23 + x 33 )
Winston, Section 3.8 Solution–Sulfur Specifications
G1 must contain at most 1% of sulfur: (0.005x 11 + 0.02x 21 + 0.03x 31 ) ≤ 0.01 (x 11 + x 21 + x 31 ) G2 must contain at most 2% of sulfur: (0.005x 12 + 0.02x 22 + 0.03x 32 ) ≤ 0.02 (x 12 + x 22 + x 32 ) G3 must contain at most 1% of sulfur: (0.005x 13 + 0.02x 23 + 0.03x 33 ) ≤ 0.01 (x 13 + x 23 + x 33 )
