Oefeningen: Besliskunde
formuleren van problemen
oefening 1
• Farmer Jones bakes two types of cake (chocolate and vanilla) to supplement his income. Each chocolate cake can be sold for $1, and each vanilla cake can be sold for 50¢. Each chocolate cake requires 20 minutes of baking time and uses 4 eggs. Each vanilla cake requires 40 minutes of baking time and uses 1 egg. Eight hours of baking time and 30 eggs are available. Formulate an LP to maximize farmer Jones’s revenue. Then graphically solve the LP, check your solution using Lindo. (A fractional number of cakes is okay.)
oefening 2
• Finco has the following investments available :
Investment A. For each dollar invested at time 0, we receive $0.10 at time 1 and $1.30 at time 2.
(Time 0 = now; time 1 is one year from now; and so on.)
Investment B. For each dollar invested at time 1, we receive $1.60 at time 2.
Investment C. For each dollar invested at time 2, we receive $1.20 at time 3.
At any time, leftover cash may be invested in T-bills, which pay 10% per year. At time 0, we have
$100. At most $50 can be invested in each of investments A, B, and C. Formulate an LP that can be used to maximize Finco’s cash on hand at time 3, solve the problem using Lindo and discuss the solution.
oefening 3
• A farm family owns 125 acres of land and has $40,000 in funds available for investment. Its members can produce a total of 3,500 person-hours worth of labor during the winter months (mid- September to mid-May) and 4,000 person-hours during the summer. If any of these person-hours are not needed, younger members of the family will use them to work on a neighboring farm for
$5/hour during the winter months and $6/hour during the summer.
Cash income may be obtained from three crops and two types of livestock: dairy cows and laying hens. No investment funds are needed for the crops. However, each cow will require an investment outlay of $1,200 and each hen will cost $9.
Each cow will require 1.5 acres of land, 100 person-hours of work during the winter months, and another 50 person-hours during the summer. Each cow will produce a net annual cash income of
$1,000 for the family. The corresponding figures for each hen are : no acreage, 0.6 person-hour during the winter, 0.3 more person-hour during the summer, and an annual net cash income of $5.
The chicken house can accomodate a maximum of 3,000 hens, and the size of the barn limits the herd to a maximum of 32 cows.
Estimated person-hours and income per acre planted in each of the three crops are :
Soybeans Corn Oats Winter person-hours
Summer person-hours Net annual cash income ($)
20 50 600
35 75 900
10 40 450
The family wishes to determine how much acreage should be planted in each of the crops and how many cows and hens should be kept to maximize its net cash income. Formulate the linear programming model for this problem and solve using Lindo.
oefening 4
• A cargo plane has three compartments for storing cargo : front, center and back. These compartments have capacity limits on both weight and space, as summarized below :
Compartment Weight Capacity
(Tons)
Space Capacity (Cubic Feet) Front
Center Back
12 18 10
7,000 9,000 5,000
Furthermore, the weight of the cargo in the respective compartments must be the same proportion of that compartment's weight capacity to maintain the balance of the airplane.
The following four cargoes have been offered for shipment on an upcoming flight as space is available :
Cargo Weight (Tons)
Volume (Cubic Feet/Ton)
Profit ($/Ton) 1
2 3 4
20 16 25 13
500 700 600 400
320 400 360 290
Any portion of these cargoes can be accepted. The objective is to determine how much (if any) of each cargo should be accepted and how to distribute each among the compartments to maximize total profit for the flight.
Formulate the linear programming model for this problem and solve using Lindo.
oefening 5
• The Birdeyes Real Estate Co. owns 800 acres of prime, but undeveloped, land on a scenic lake in the heart of the Ozark Mountains. In the past, little or no regulation was applied to new developments around the lake. The lake shores are now lined with clustered vacation homes.
Because of the lack of sewage service, septic tanks, mostly improperly installed, are in extensive use. Over the years, seepage from the septic tanks has resulted in a severe water pollution problem.
To curb further degradation in the quality of water, county officials introduced and approved some stringent ordinances applicable to all future developments.
1. Only single-, double-, and triple-family homes can be constructed, with the single-family homes accounting for at least 50 % of the total.
2. To limit the number of septic tanks, minimum lot sizes of 2, 3, and 4, acres are required for single-, double-, and triple-family homes.
3. Recreation areas of 1 acre each must be established at the rate of one area per 200 families.
4. To preserve the ecology of the lake, underground water may not be pumped for house or garden use.
The president of Birdeyes Real Estate is studying the possibility of developing the company’s 800 acres on the lake. The new development will include single-, double-, and triple-family homes. He estimates that 15 % of the acreage will be consumed in the opening of streets and easements for utilities. He also estimates his returns from the different housing units :
Housing Units Single Double Triple
Net Return per Unit ($) 10,000 12,000 15,000
The cost of connecting water service to the area is proportionate to the number of units constructed.
However, the county stipulates that a minimum of $100,000 must be collected for the project to be economically feasible. Additionally, the expansion of the water system beyond its present capacity is limited to 200,000 gallons per day during peak periods. The following data summarize the cost of connecting water service as well as the water consumption assuming an average size family :
Housing Unit Single Double Triple Recreation Water service cost per unit ($)
Water cons. per unit (gal/day)
1000 400
1200 600
1400 840
800 450
oefening 6
• An international relief agency, the Food and Agriculture Organization, is sending agricultural experts to two underdeveloped countries whose greatest need is to increase their food production by improving their agricultural techniques. Therefore, the experts will be used to develop pilot projects and training programs to demonstrate and teach these techniques. However, the number of such projects that can be undertaken is restricted by the limited availability of three required
resources : equipment, experts, and money. The question is how many projects should be undertaken in each country in order to make the best possible use of the resources.
It has been estimated that each full project undertaken in country 1 eventually would increase the food production in this country sufficiently to feed 2,000 additional people. The corresponding estimate for country 2 is for an increase that would feed an additional 3,000 people. The two countries differ in the mix of resources needed for projects. These data are summarized in the table. It is feasible to consider projects at fractional levels as well as whole projects. We assume that fractions of projects will affect the data of the table proportionally.
Resource
Amount used Country 1
per project Country 2
Amount available Equipment
Experts Money
0 1 60,000
5 2 20,000
20 10 300,000
People fed 2,000 3,000
Because both countries are in desperate need, the Food and Agriculture Organization is determined to increase the food production in both countries as much as possible. Therefore, it has chosen the overall objective of maximizing the minimum increase in food production in the two countries.
Formulate and solve the problem using LINDO.
oefening 7
• During the next three months Airco must meet (on time) the following demands for air conditioners : month 1, 300; month 2, 400; month 3, 500. Air conditioners can be produced in either New York or Los Angeles. It takes 1.5 hours of skilled labor to produce an air conditioner in Los Angeles, and 2 hours in New York. It costs $400 to produce an air conditioner in Los Angeles, and $350 in New York. During each month each city has 420 hours of skilled labor available. It costs $100 to hold an air conditioner in inventory for a month. At the beginning of month 1 Airco has 200 air conditioners in stock. Formulate an LP whose solution will tell Airco how to minimize the cost of meeting air conditioner demands for the next three months, discuss the solution.
Simplexmethode en sensitiviteitsanalyse
oefening 1
• Widgetco produces two products: 1 and 2. Each requires the amounts of raw material and labor, and sells for the price given in the table. Up to 350 units of raw material can be purchased at $2.00 per unit, while up to 400 hours of labor can be purchased at $1.50 per hour. To maximize profit, Widgetco must solve an LP-problem.
product 1 product 2
Raw Material Labor Sales price
1 unit 2 hours
$7.00
2 units 1 hour
$8.00
Formulate the LP-problem, solve it graphically. Demonstrate the correspondence between corner points and basic feasible solutions. Solve the LP-problem with the simplex-algorithm and check your answer with the LINDO-output and the graphical representation.
oefening 2
• Use the simplex algorithm to find the optimal solution of the following LP-problem.
max z = 4x1 + x2
s.t. 2x1 + 3x2 ≤ 4 x1 + x2 ≤ 1 4x1 + x2 ≤ 2
x1, x2 ≥ 0
Check your solution with LINDO, and discuss the solution.
oefening 3
• Consider a maximization problem with the optimal tableau given. The optimal solution to this LP is z = 10, x3 = 3, x4 = 5, x1 = x2 = 0. Determine the second best basic feasible solution to this LP.
z x1 x2 x3 x4 rhs
1 0 0
2 3 4
1 2 3
0 1 0
0 0 1
10 3 5
oefening 4
• Solve the following LP-problem (graphically, with simplex-algorithm, with LINDO) : max z = x1 + x2
s.t. 2x1 + x2 ≥ 3 3x1 + x2 ≤ 3.5
x1 + x2 ≤ 1 x1, x2 ≥ 0
oefening 5
• A camper is considering taking two types of items on a trip. Item 1 weighs a1 lb, and item 2 weighs a2 lb. Each type 1 item earns the camper a benefit of c1 units, and each type 2 item earns the camper c2 units. The knapsack can hold items weighing at most b lb.
- Assuming that the camper can carry a fractional number of items along on the trip, formulate an LP to maximize benefit.
- Show that if c a
c a
2 2
1 1
≥ then the camper can maximize benefit by filling a knapsack with
b
a2 type 2 items.
oefening 6
• Ghostbusters, Inc. exorcises ghosts. During each of the next months, they will receive the following number of calls from people who want their ghosts exorcised : January, 100 calls;
February, 300 calls; March, 200 calls. Ghostbusters is paid $800 for each ghost exorcised during the month they are made, but if a call is responded to one month after it is made, Ghostbusters loses
$100 in future goodwill, and if a call is responded to two months after it is made, Ghostbusters loses $200 in goodwill. Each employee of Ghostbusters can exorcise 10 ghosts during a month. At the beginning of January, the company has 8 workers. Workers can be hired and trained (in 0 time) at a cost of $5000 per worker. Workers can be fired at a cost of $4000 per worker. Formulate an LP to maximize Ghostbusters’ profit (revenue - costs) over the next three months. Assume that all calls must be handled by the end of March.
Solve the problem using LINDO and discuss the solution.
oefening 7
• Write down the dual of the following LP-problem : max z = -4x1 - x2
s.t. 4x1 + 3x2 ≥ 6 x1 + 2x2 ≤ 3 3x1 + x2 = 3
x1, x2 ≥ 0
oefening 8
• For the Giapetto problem : x1, x2, and s3 are the basic variables in the optimal tableau. Reconstruct the optimal tableau without performing any simplex iterations. Check your solution with the LINDO-output.
The LP-problem is : max z = 3x1 + 2x2
s.t. 2x1 + x2 ≤ 100 (finishing constraint) x1 + x2 ≤ 80 (carpentry constraint) x1 ≤ 40 (demand constraint)
x1 ≥ 0, x2 ≥ 0 (x1 = # soldiers, x2 = # trains)
Adjust the optimal tableau when the capacity of the finishing department increases from 100 to 125 hours. Indicate this new solution on a graphical representation. Find the new optimal solution using the appropriate technique and indicate it also on the graphical representation.
oefening 9
• For the Giapetto-problem, make a graphical representation of the problem. Enumerate all solutions (feasible and infeasible ones). Find the corresponding dual solutions (use complementary slackness).
oefening 10
• Consider the following LP and its optimal tableau (see table).
max z = 4x1 + x2 + 2x3
s.t. 8x1 + 3x2 + x3 ≤ 12 6x1 + x2 + x3 ≤ 8
x1, x2 ≥ 0
Find the dual to this LP and its optimal solution.
Find the range of values of the objective function coefficient of x3 for which the current basis remains optimal.
Find the range of values of the objective function coefficient of x1 for whicht the current basis remains optimal.
z x1 x2 x3 s1 s2 rhs z=16
s1=4 x3=8
1 0 0
8 2 6
1 2 1
0 0 1
0 1 0
2 -1
1
16 4 8
oefening 11
• Start from the optimal solution and tableau of the (original) Giapetto-problem and comment on the following suggestions.
The LP-problem is : max z = 3x1 + 2x2
s.t. 2x1 + x2 ≤ 100 (finishing constraint) x1 + x2 ≤ 80 (carpentry constraint) x1 ≤ 40 (demand constraint)
x1 ≥ 0, x2 ≥ 0 (x1 = # soldiers, x2 = # trains)
a) Giapetto wants to make wooden aircrafts. The numbers of hours required is 3 hours and 2 hours in the finishing department and the carpentry department respectively. What is the minimum profit for the new product to be competitive compared with the soldiers and trains ? What is the new production plan if the profit is 20 % higher than the minimum profit ?
b) In the (original) Giapetto-problem a constraint, i.e. budget constraint, was left out. How does this budget constraint, 3x1 + 2x2 ≤ 165, affect the solution (i.e. x1=20, x2=60, and s3=20 with z=180).
oefening 12
• Zales Jewelers uses rubies and sapphires to produce two types of rings. A type 1 ring requires 2 rubies, 3 sapphiers, and 1 hour of jeweler’s labor. A type 2 ring requires 3 rubies, 2 sapphires, and 2 hours of jeweler’s labor. Each type 1 ring sells for $400, and each type 2 ring sells for $500. All rings produced by Zales can be sold. A present, Zales has 100 rubies, 120 sapphires, and 70 hours of jeweler’s labor. Extra rubies can be purchased at a cost of $100 per ruby. Market demand requires that the company produces at least 20 type 1 rings and at least 25 type 2 rings. The company wants to maximize its revenue.
Formulate the problem and solve with LINDO.
Answer the following questions :
- Suppose that instead of $100, each ruby costs $190. Would Zales still purchase rubies ? What would be the new optimal solution to the problem ?
- Suppose that Zales were only required to produce at least 23 type 2 rings. What would Zales’ profit now be ?
- What is the most that Zales would be willing to pay for another hour of jeweler’s labor ? - What is the most that Zales would be willing to pay for another sapphire ?
- Zales is considering producing type 3 rings. Each type 3 ring can be sold for $550 and requires 4 rubies, 2 sapphires, and 1 hour of jeweler’s labor. Should Zales produce any type 3 rings ?
oplossen van problemen m.i.v. sensitiviteitsanalyse
oefening 1
• Ballco manufactures large softballs, regular softballs, and hardballs. Each type of ball requires time in three departments : cutting, sewing, and packaging, as shown in the following table (time in minutes). Because of marketing considerations, at least 1000 regular softballs must be produced.
Cutting time Sewing time Packaging time Regular softballs
Large softballs Hardballs
15 10 8
15 15 4
3 4 2
Each regular softball can be sold for $3, each large softball, for $5; and each hardball, for $4. A total of 18,000 minutes of cutting time, 18,000 minutes of sewing time, and 9000 minutes of packaging time are available. Ballco wants to maximize sales revenue. If we define
RS = number of regular softballs produced LS = number of large softballs produced HB = number of hardballs produced then the appropriate LP is
max z = 3 RS + 5 LS + 4 HB
s.t. 15 RS + 10 LS + 8 HB ≤ 18,000 15 RS + 15 LS + 4 HB ≤ 18,000 3 RS + 4 LS + 2 HB ≤ 9000 RS ≥ 1000
RS, LS, HB ≥ 0
The optimal tableau for this LP is :
z RS LS HB s1 s2 s3 e4 a4 rhs
1 0 0 0 0
0 0 0 0 1
0 0 1 0 0
0 1 0 0 0
0.500 0.188 -0.050 -0.175
0
0 -0.125
0.100 -0.150 0
0 0 0 1 0
4.500 0.938 0.750 -1.875
-1
M-4.5 -0.938 -0.750 1.875
-1
4500 187.5 150 5025 1000
? Find the dual of the Ballco problem and its optimal solution.
? Show that the Ballco problem has an alternative optimal solution. Find it. How many minutes of sewing time are used by the alternative optimal solution ?
? By how much would an increase of 1 minute in the amount of available sewing time increase Ballco’s revenue ? How can this answer be reconciled with the fact that the sewing constraint is binding ? (Hint : Look at the answer to question 2)
? Assuming the current basis remains optimal, how would an increase of 100 in regular softball requirement affect Ballco’s revenue ?
oefening 2
• Sailco Corporation must determine how many sailboats should be produced during each of the next four quarters (one quarter = three months). 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. At 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 cost of $400 per sailboat. 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 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.
? If quarter 1 demand decreased to 35 sailboats, what would be the total cost of satisfying the demands during the next four quarters ?
? If the cost of producing a sailboat with regular-time labor during quarter 1 were $420, what would be the new optimal solution ?
? Suppose a new customer is willing to pay $425 for a sailboat. If his demand must be met during quarter 1, should Sailco fill the order ? How about if his demand must be met during quarter 4 ?
oefening 3
• Consider the following two Lps : max z = c1 x1 + c2 x2
s.t. a1 1 x1 + a1 2 x2 ≤ b1 LP 1 a2 1 x1 + a2 2 x2 ≤ b2
x1, x2 ≥ 0
max z = 100 c1 x1 + 100 c2 x2
s.t. 100 a1 1 x1 + 100 a1 2 x2 ≤ b1 LP 2 100 a2 1 x1 + 100 a2 2 x2 ≤ b2
x1, x2 ≥ 0
Suppose that BV = {x1, x2} is an optimal basis for both LPs, and the optimal solution to LP 1 is x1
= 50, x2 = 500, z = 550. Also suppose that for LP 1, the shadow prices of both constraint 1 and constraint 2 = 100/3. Find the optimal solution to LP 2 and the optimal solution to the dual of LP 2. (Hint : If we multiply each number in a matrix by 100, what happens to B-1 ?)
oefening 4
• Consider a farmer having 60 ha of land, for which he has to decide, the proportion of land which he will plant with wheat and sugar-beet. Due to production limitations (e.g. resulting from EC agriculture policies), he may have as a maximum 30 ha of wheat and 40 ha of sugar-beet. The return of sugar-beet over the last three years was 58,000 BEF/ha, 63,000 BEF/ha, 59,000 BEF/ha and he has not yet an idea on the expected return. However, he expects a return of 40,000 of BEF/ha for wheat and his objective is to maximize the return. The labour required for one hectare of sugar-beet is twice the amount of wheat (one labour day per ha), and the maximum available labour is 85 working days. Besides, the cost for fertilizer needed for one hectar of wheat is twice as much for one hectare of sugar-beet (one unit per ha). Buying more than 70 units of it is thought not to be economical.
? Formulate the foregoing problem as a mathematical programming problem and solve it using Lindo.
? Your neighbour has 10 ha of free land and you are allowed to use it. How much (BEF/ha) are you prepaired to pay for it ?
? How sensitive is the solution to the return of sugar-beet?
? You want to consider cultivating maize. 1.5 labor hours are required per ha and 1.5 units of fertilizer per ha. There is a limit of 40 ha for maize. What is the minimum expected return required before you start cultivating maize? Add 10 % to that return, what is the new solution?
? Due to stronger EC regulations, you are only allowed to use 60 units of fertilizer. What is the effect on your solution (total return, proportion of land used for sugar-beat and wheat)?
(all questions should be solved starting from the original problem formulation)
oplossen van transportproblemen en aanverwante problemen
oefening 5
• The Metro Water District is an agency that administers the distribution of water in a certain large geographic region. The region is fairly arid, so the District must purchase and bring water from outside the region. The sources of this imported water are the Colombo, Sacron, and Calorie Rivers. The District then resells the water to users in its region. Its main customers are the water departments of the cities of Berdoo, Los Devils, San Go, and Hollyglass.
It is possible to supply any of these cities with water brought in from any of the three rivers, with the exception that no provision has been made to supply Hollyglass with Calorie River water.
However, because of the geographic layouts of the viaducts and the cities in the region, the cost to the District of supplying water depends upon both source of the water and the city being supplied.
The variable cost per acre foot of water (in dollars) for each combination of river and city is given in the next table. Despite these variations, the price per acre foot charged by the District is independent of the source of the water and is the same for all cities
The management of the District is now faced with the problem of how to allocate the available water during the upcoming summer season. Using units of 1 million acre feet, the amounts available from the three rivers are given in the right-hand column of the table. The District is committed to providing a certain minimum amount to meet the essential needs of each city (with the exception of San Go, which has an independent source of water), as shown in the Min. needed row of the table. The Requested row indicates that Los Devils desires no more than the minimum amount, but that Berdoo would like to buy as much as 20 more, San Go would buy up to 30 more, and Hollyglass will take as much as it can get.
? Formulate the problem as a transportation problem, that allocates all the available water from the three rivers to the four cities in such a way as to at least meet the essential needs of each city while minimizing the total cost of the District.
City Cost ($) per acre foot
River Berdoo Los Devils San Go Hollyglass Supply
Colombo River Sacron River Calorie River
16 14 19
13 13 20
22 19 23
17 15 --
50 60 50 Min. needed
Requested
30 50
70 70
0 30
10
∞
(million acre feet)
oefening 6
• Sailco Corporation must determine how many sailboats should be produced during each of the next four quarters (one quarter = three months). 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. At 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 cost of $400 per sailboat. 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 demand has been satisfied), a carrying or holding cost of $20 per sailboat is incurred.
? Formulate the problem as a transportation problem.
oefening 7
• An owner of a plantation has three sites or sources of wood and three markets or lumber companies to be supplied. Site 1 can produce up to 50 m3 per week; site 2, up to 100 m3 per week; and site 3, up to 50 m3 of wood per week. The trees are shipped to three lumber companies or customers. The profit earned per m3 wood depends on the site where the trees were cultivated and on the lumber company who purchases the wood (see table). Customer 1 is willing to purchase up to 80 m3 per week; customer 2, up to 90; and customer 3, up to 100. The owner wants to find the shipping plan that will maximize profits.
To From Customer 1 Customer 2 Customer 3
Site 1 Site 2 Site 3
$75
$79
$85
$60
$73
$76
$69
$68
$70
? formulate a balanced transportation problem that can be used to maximize the profits.
? use the NW-corner method to find a basic feasible solution to the problem.
? use the transportation simplex to find an optimal solution to the problem.
oefening 8
• The Better Fruits Company has decided to initiate the cultivation and sale of four new fruit varieties, using three conservatories that currently have excess capacity. The fruit varieties require a comparable cultivation effort per plant, so available cultivation capacity of the conservatories is measured by the number of plants of any fruit variety that can be cultivated, as given in the last column of the following table. The bottom row gives the required cultivation rate to meet projected sales. Each conservatory can cultivate any of these fruits, except that conservatory 2 lacks the means to cultivate fruit variety 3. However, the variable costs per unit of each fruit differ from conservatory to conservatory, as shown in the main body of the table.
Fruit Variety Unit Cost
Conservatory 1 2 3 4 Capacity
available 1
2 3
41 40 37
27 29 30
28 -- 27
24 23 21
75 75 45
Cultivation rate 20 30 30 40
Management now needs to make a decision on how to allocate the fruit varieties to the different conservatories and has opted for no “product splitting” (i.e., each fruit variety must be assigned to just one plant). Formulate and solve the problem.
oefening 9
• Canned Vegies has two plants, two warehouses, and three customers. The locations of these are as follows :
Plants : Detroit and Atlanta
Warehouses : Denver and New York
Customers : Los Angeles, Chicago, and Philadelphia
Canned vegetables are produced at the two plants, then shipped to warehouses, and finally shipped to customers. Detroit can produce 150 pallets of cans per week and Atlanta can produce 100 pallets per week. Los Angeles requires 80 pallets per week, Chicago, 70; and Philadelphia, 60. It costs
$10,000 to produce a pallet at each plant and the cost of shipping a pallet between two cities is given in the tables. Model the problem as a transportation problem and determine how to meet Canned Vegetables’ weekly demands at minimum cost (using LINDO and LINGO).
To
From Denver New York
Detroit $1253 $637
Atlanta $1398 $841
To From Los Angeles Chicago Philadelphia Denver
New York
$1059
$2786
$996
$802
$1691
$100
oplossen van geheeltallige problemen oefening 1
• The Cubs are trying to determine which of the following free agent pitchers should be signed : Rick Sutcliffe (RS), Bruce Sutter (BS), Dennis Eckersley (DE), Steve Trout (ST), Tim Stoddard (TS).
The cost of signing each pitcher and the number of victories each pitcher will add to the Cubs are shown in the table.
Cost of Signing Pitcher (millions)
Victories Added to
Cubs RS
BS DE ST TS
$6
$4
$3
$2
$2
6 (righty) 5 (righty) 3 (righty) 3 (lefty) 2 (righty)
Subject to the following restrictions, the Cubs want to sign the pitchers who will add the most victories to the team.
? At most $12 million can be spent.
? If DE and ST are signed, then BS cannot be signed.
? At most two right-handed pitchers can be signed.
? The Cubs cannot sign both BS and RS.
Formulate an IP to help the Cubs determine who they should sign. Solve using LINDO.
oefening 2
• Dorian Auto is considering manufacturing three types of autos : compact, midsize, and large. The resources required for, and the profits yielded by, each type of car are shown in the table. At present, 6000 tons of steel and 60,000 hours of labor are available. For production of a type of car to be economically feasible, at least 1000 cars of that type must be produced. Formulate an IP to maximize Dorian’s profit and solve using LINDO.
Compact Midsize Large Steel required
Labor required Profit yielded
1.5 tons 30 hours
$2000
3 tons 25 hours
$3000
5 tons 40 hours
$4000
oefening 3
• Eastinghouse ships 12,000 capacitors per month to their customers. The capacitors may be produced at three different plants. The production capacity, fixed monthly cost of operation, and variable cost of producing a capacitor at each plant are given in the table. The fixed cost for a plant is incurred only if the plant is used to make any capacitors. Formulate an integer programming model whose solution will tell Eastinghouse how to minimize their monthly costs of meeting their customers’ demands. Solve the problem using LINDO.
Fixed Cost (in thousands)
Variable Cost
Production Capacity Plant 1
Plant 2 Plant 3
$80
$40
$30
$20
$25
$30
6000 7000 6000
oefening 4
• Use the branch and bound technique to solve : min z = 50 x1 + 100 x2
s.t. 7 x1 + 2 x2 ≥ 28 2 x1 + 12 x2 ≥ 24 x1, x2 ≥ 0 and integer
oefening 5
• Four jobs must be processed on a single machine. The time required to perform each job, the due date, and the penalty (in dollars) per day the job is late are given in the table. Use branch and bound to determine the order of performing the jobs that will minimize the total penalty costs due to delayed jobs.
Time Due Date Penalty
Job 1 Job 2 Job 3 Job 4
4 days 5 days 2 days 3 days
day 4 day 2 day 13
day 8
4 5 7 2
herhaling
(zie eveneens hoofdstukken 1-9)
oefening 1
• Three professors must teach six sections of finance. Each professor must teach two sections of finance and has ranked the six time periods during which finance is thaught, as shown in the table.
A ranking of 10 means that the professor wants to teach that time, and a ranking of 1 means that he or she does not want to teach at that time. Maximize the total satisfaction of the three professors.
9A.M. 10A.M. 11A.M. 1P.M. 2P.M. 3P.M.
Prof1 Prof2 Prof3
8 9 7
7 9 6
6 8 9
5 8 6
7 4 9
6 4 9
oefening 2
• Farmer Leary grows wheat and corn on his 45-acre farm. He can sell at most 140 bushels of wheat and 120 bushels of corn. Each planted acre yields either 5 bushels of wheat or 4 bushels of corn.
Wheat sells for $30 per bushel, and corn sells for $50 per bushel. Six hours of labor are needed to harvest an acre of wheat, and 10 hours are needed to harvest an acre of corn. Up to 350 hours of labor can be purchased at $10 per hour.
? Maximize profit for farmer Leary.
? What is the most that Leary should be willing to pay for an additional hour of labor ?
? What is the most that Leary should be willing to pay for an additional acre of land ?
? If only 40 acres of land were available, what would be Leary’s profit ?
? If the price of wheat dropped to $26, what would be the new optimal solution ?
? Farmer Leary is considering growing barley. Demand for barley is unlimited. An acre of yields 4 bushels of barley and requires 3 hours of labor. If barley sells for $30 per bushel, should Leary produce any barley ?
oefening 3
• A company must meet the following demands for cash at the beginning of each of the next six months: month 1, $200; month 2, $100; month 3, $50; month 4, $80; month 5, $160; month 6, $140.
At the beginning of month 1, the company has $150 in cash and $200 worth of bond 1, $100 worth in bond 2, and $400 worth of bond 3. The company will have to sell some bonds to meet demands, but a penalty will be charged for any bonds sold before the end of month 6. The penalties for selling $1 worth of each bond are as shown in the table.
? Assuming that all bills must be paid on time, minimize the cost of meeting the cash demands for the next six months.
Month of sale
Bond 1 2 3 4 5 6
1 2 3
$0.21
$0.50
$1.00
$0.19
$0.50
$1.00
$0.17
$0.50
$1.00
$0.13
$0.33
$1.00
$0.09
$0.00
$1.00
$0.05
$0.00
$0.00
oefening 4
• A court decision has stated that the enrollment of each high school in Metropolis must be at least 20 percent black. The number of black and white high school students in each of the city’s five school districts are shown in the table. The distance (in miles) that a student in each district must travel is also shown in the following table. School board policy requries that all the students in a given district attend the same school. Assuming that each school must have an enrollment of at least 150 students. Minimize the total distance that Metropolis students must travel to high school.
Whites Blacks High School 1 High School 2 District 1
District 2 District 3 District 4 District 5
80 70 90 50 60
30 5 10 40 30
1 0.5 0.8 1.3 1.5
2 1.7 0.8 0.4 0.6
oefening 5
• An American professor will be spending a short sabbatical leave at the University of Iceland. She wishes to bring all needed items with her on the airplane. After collecting together the professional items that she must have, she finds that airline regulations on space and weight for checked luggage will severely limit the clothes she can take. (She plans to carry on a warm coat, and then purchase a warm Icelandic sweater upon arriving in Iceland.) Clothes under consideration for checked luggage include 3 skirts, 3 slacks, 4 tops, and 3 dresses. The professor wants to maximize the number of outfits she will have in Iceland (including the special dress she will wear on the airplane). Each dress constitutes an outfit. Other oufits consists of combination of a top and either a skirt or slacks.
However, certain combinations are not fashionable and so will not qualify as an outfit.
In the following table, the combinations that will make an outfit are marked with an x.
Top
1 2 3 4 Icelandic Sweater
Skirt
1 2 3
x x
x
x x
x x
x
x
1 x x
Slacks 2 3
x x
x
x x
x x
The weight (in grams) and volume (in cubic centimeters) of each item is shown in the following table :
Weight Volume
Skirt
1 2 3
600 450 700
5,000 3,500 3,000
Slacks
1 2 3
600 550 500
3,500 6,000 4,000
Top
1 2 3 4
350 300 300 450
4,000 3,500 3,000 5,000
Dress
1 2 3
600 700 800
6,000 5,000 4,000
Total allowed 4,000 32,000
? Formulate a model to choose which items of clothing to take and solve the problem using Lindo or Lingo.
