Just need a little direction on linear programming models. The problem has been attached
1. Betty Malloy, owner of the Eagle Tavern in Pittsburgh, is preparing for Super Bowl Sunday, and she must determine how much beer to stock. Betty stocks three brands of beer – Yodel, Shotz, and Rainwater. The cost per gallon (to the tavern owner) of each brand is as follows:
Brand Cost/gallon
Yodel $1.50
Shotz $0.90
Rainwater $0.50
The tavern has a budget of $2,000 for beer for Super Bowl Sunday. Betty sells Yodel at a rate of $3.00 per gallon, Shotz at $2.50 per gallon, and Rainwater at $1.75 per gallon. Based on past football games, Betty has determined the maximum customer demand to be 400 gallons of Yodel, 500 gallons of Shotz, and 300 gallons of Rainwater. The tavern has a capacity to stock 1,000 gallons of beer; Betty wants to stock up completely. Betty wants to determine the number of gallons of each brand of beer to order so as to maximize profit.
2. A transportation problem involves the following costs, supply and demand.
To
From 1 2 3 4 Supply
1 $500 $750 $300 $450 12
2 650 800 400 600 17
3 400 700 500 550 11
Demand 10 10 10 10
a. Formulate a linear programming model for these problems written in a format similar to:
Maximize Z = 0X1+0X2+0X3
Subject to: …
Using Excel, please solve for the following:
The Elixer Company produces a drug from two ingredients. Each ingredient contains the same three antibiotics in different proportions. Ine gram of ingredient 1 contributes 3 units, and ingredient 2 contributes 1 unit of antibiotic 1; the drug requires 6 units. At least 4 units of antibiotic 2 are required; and the ingredients each contribute 1 unit per gram. At least 12 units of antibiotic 3 are required; a gram of ingredient 1 contributes 2 units, and a gram of ingredient 2 contributes 6 units. The cost for a gram of ingredient 1 is $80, and the cost for a gram of ingredient 2 is $50. The company wants to formulate a linear programming model to determine the number of grams of each ingredient that must go into the drug in order to meet the antibiotic requirements at the minimum cost.
b. Solve this model by using graphical analysis
Use the geometric method of linear programming to maximize the objective function f(x,y)=3x-6y subject to the constraints.
x>= 1
x-y<= 3
2x+y>= 6
2x+y<= 8
6. Consider the linear programming problem:
max 4x − 3y
x + y <= 5
6x − 3y <= 12
x, y >= 0
The graph of the constraints is given below with the feasible region shown in grey.
y-axis
x-axis
(a) Determine the coordinates of all four corner points of the feasible region and label them on the
graph.
(b) Find the optimal solution to the linear programming problem. Show your work.
I’m not sure how to go about finding the optimal solution for part a and b ( please see the attached file it includes my partial solution to the question)
2)
Describe the effect on the minimum transportation cost when capacity at each factory
or warehouse is altered by adding or subtracting one ton. What are the minimum
capacity changes at Glasgow that will alter the optimum set of routes and what will
those alterations be? Explain how you arrive at each one of your answers.
8. Embassy Motorcycles (EM) manufactures two motorcycles designed for easy handling and safety. The EZ-Rider model has a new engine and a low profile that make it easy to balance. The Lady-Sport model is slightly larger, uses a more traditional engine, and is specifically designed to appeal to women riders. EM produces the engines for both models at is Des Moines, Iowa plant. Each EZ-Rider engine requires 6 hours of manufacturing time and each Lady-Sport engine requires 3 hours of manufacturing time. The Des Moines plant has 2100 hours of engine manufacturing time available for the next production period. EM’s motorcycle frame supplier can supply as many EZ-Rider frames as needed. However, the Lady-Sport frame is more complex and the supplier can only provide up to 280 Lady-Sport frames for the next production period. Final assembly and testing requires 2 hours for each EZ-Rider model and 2.5 hours for each Lady-Sport model. A maximum of 1000 hours of assembly and testing time are available for the next production period. The company’s accounting department projects a profit contribution of $2,400 for each EZ-Rider produced and $1,800 for each Lady-Sport produced.
a. Formulate a linear programming model that can be used to determine the number of units of each model that should be produced to maximize the total contribution profit.
b. Solve the problem graphically. What is the optimal solution?
An office manager needs to buy new filing cabinets. Cabinet A costs $7, and takes up 6 square feet of floor space, and holds 9 cubic feet of files. Cabinet B costs $8, takes up 8 square feet, and holds 13 cubic feet. He has only $66 to spend and the office has room for no more than 60 square feet of cabinets. (Find the Objective Function and the Constraints; Let Cabinet A be “x”, and Cabinet B be “y”) How many of each can be bought to maximize storage capacity? What is the maximum storage capacity?
A bank has $650,000 in assets to allocate among investments in bonds, home mortgages, car loans, and personal loans. Bonds are expected to produce a return of 10%, mortgages 8.5%, car loans 9.5%, and personal loans 12.5%. To make sure the portfolio is not too risky, the bank wants to restrict personal loans to no more than 25% of the total portfolio. The bank also wants to ensure that more money is invested in mortgages than in personal loans. And it wants to invest more in bonds than personal loans.
Formulate this as an LP in a spreadsheet and solve with Solver.
How much should the bank invest in each of the asset classes to maximize total expected return?
Answer the question in the spreadsheet
How would you define in linear programming format that an employee has to work 5 consecutive days, and then has 2 days off?
12. Durham Designs manufactures home furnishings for department stores. Planning is underway for the production of items in the “Wildflower” fabric pattern during the next production period.
Bedspread Curtains Dust Ruffle
Fabric required (yds) 7 4 9
Time required (hrs) 1.5 2 .5
Packaging material 3 2 1
Profit 12 10 8
Inventory of the Wildflower fabric is 3000 yards. Five hundred hours of production time have been scheduled. Four hundred units of packaging material are available.
Each of these values can be adjusted through overtime or extra purchases.
Durham has the following goals:
Priority 1:
 Achieve a profit of $3200,
Priority 2:
 Avoid purchasing any more fabric or packaging material than is available, and
 Use all of the hours scheduled.
Give the goal programming model, then solve and report the solution.
Formulation:
Just need constraint for “each zone is covered by at least two boxes”
Your express package courier company is drawing up new zones for the location of drop boxes for customers. The city has been divided into the seven zones shown below. You have targeted six possible locations for drop boxes. The list of which drop boxes could be reached easily from each zone is listed below.
Zone Can Be Served By Locations:
Downtown Financial 1, 2, 5, 6
Downtown Legal 2, 4, 5
Retail South 1, 2, 4, 6
Retail East 3, 4, 5
Manufacturing North 1, 2, 5
Manufacturing East 3, 4
Corporate West 1, 2, 6
Let xi = 1 if drop box location i is used, 0 otherwise.
Formulate a model to provide the smallest number of locations yet make sure that each zone is covered by at least two boxes. Then solve and report answers in spaces provided.
Formulation: (define variables, give objective function, give all constraints):
These two problems should be done in excel, and must be done using the solver application, which would make it a lot simpler. Need to provide the formulas.
13-3
The Seaboard Trucking Company has expanded its shipping capacity by purchasing 120 trucks and trailers from a competitor that went bankrupt. The company subsequently located 40 of the purchased trucks at each of its shipping warehouses in Charlotte, Memphis, and Louisville. The company makes shipments from each of these warehouses to terminals in St. Louis, Atlanta and New York. Each truck is capable of making one shipment per week. The terminal managers have each indicated their capacity for extra shipments. The manager at St. Louis can accommodate 40 additional per week, the manager at Atlanta can accommodate 60 additional trucks, and the manager at New York can accommodate 50 additional trucks. The company makes the following profit per truckload shipment from each warehouse to each terminal. The profits differ as a result of differences in products shipped, shipping costs, and transport rates.
Warehouse St. Louis Atlanta New York
Charlotte $1,800 2100 1600
Memphis 1000 700 900
Louisville 1400 800 2200
The company wants to know how many trucks to assign to each route to maximize profit. Formulate a linear programming model for this problem and solve it.
13-6
A small metal-parts shop contains three machines- a drill press, a lathe, and a grinder- and has three operators, each certified to work on all three machines. However, each operator performs better on some machines than on others. The shop has contracted to do a big job that requires all three machines. The times required by the various operators to perform the required operations on each machine are summarized as follows:
Operator Drill press Lathe Grinder
1 22 18 35
2 29 30 28
3 25 36 18
(time in minutes). The shop manager wants to assign one operator to each machine so that the total operating time for all three operators is minimized. Formulate and solve a linear programming model for this problem.
Missouri Mineral Products (MMP) purchases two unprocessed ores from Bolivia Mining, which it uses in the production of various compounds. Its current needs are for 800 pounds copper, 600 pounds of zinc, and 500 pounds of iron. The amount of each mineral found in each 100 pounds of the unprocessed ores and MMP’s cost per 100 pounds are given in the following table.
ORE copper per zinc per iron per waste per cost per
100 lbs 100 lbs 100 lbs 100 lbs 100 lbs
La Paz Ore 20 20 20 40 $100
Sucre Ore 40 25 10 25 $140
The objective is to determine the amount of each ore that should be purchased in order to minimize the total purchasing cost.
a) Formulate the linear programming model for the problem.
b) Use the Graphical method to find the optimal solution. Show all steps.
c) Use Excel Solver to find the optimal solution. Copy and paste your spreadsheet and the Answer report in its entirety from Excel. Remember to not delete/modify any part of the Answer Report.
4. Solve the following mixed integer linear programming model by using the computer
Maximize Z = 5X1 +6X2 + 4X3
Subject to
5X1 +3X2 + 6X3 ≤ 20
X1 + 3X2 + ≤12
X1, X3 ≥0
X2 ≥0 and integer
6. Brooks City has three consolidated high schools, each with a capacity of 1,200 students. The school board has partitioned the city into five busing districts – north, south, east, west, and central – each with different high school student populations. The three schools are located in the central, west and south districts. Some students must be bused outside their districts, and the school board wants to minimize the total bus distance traveled by these students. The average distances from each district to the three schools and the total student population in each district are as follows:
District Distance (miles) Student
Population Central West South Population
North 8 11 14 700
South 12 9 – 300
East 9 16 10 900
West 8 – 9 600
Central – 8 12 500
The school board wants to determine the number of student to bus from each district to each school to minimize the total busing miles traveled.
a) formulate a linear programming model for this problem
b) solve the model by using the computer
(See attached file)
I need some help solving these linear programming problems using ecel solver or QM for windows by computer. Please provide a detailed and easy to foowing solution. Thanks.
See attached file for proper format of tables.
Formulate the problem described here as a linear programming model. Do not use Solver to find an optimal solution. You can use WORD to type your answer and print it.
You are planning production of trucks and cars in a plant over next two months (M1 and M2). Use X1 and X2 to denote number of trucks produced in M1 and M2; UseY1 and Y2 for cars produced. At the end of each month, there may be inventory of vehicles. Use U1 and U2 for truck inventory and V1, V2 for car inventory. Note that there are 8 variables in the problem. Technically, variable names make use of subscripts; you can use these without subscripts (makes it easier to type).
Ignore integer constraints. Make sure that all variables in the constraints appear on the LHS and only numbers appear on the RHS. Here is the information you need.
Production costs M1 M2
Truck 15000 15200
Car 12500 12600
Inventory holding costs M1 M2
Truck 150 155
Car 120 125
Demand M1 M2
Truck 450 300
Car 400 600
Demand must be met (i.e. no shortages at the end of any month).
Steel (Tons/unit) M1 M2
Truck 2 2
Car 1 1
Available steel (tons) M1 M2
2500 2200
Initial inventory (units) Trucks Cars
100 150
Miles per gallon (mpg) Trucks Cars
10 20
Number of trucks and cars produced in any month should be such that the average consumption exceeds 16 mpg.
Explain how you obtained mpg constraint for M1 (you can write the constraint directly for M2). Explain how you obtained one inventory constraints.
Hints:
– Objective function should contain production costs and inventory holding costs.
– Inventory at the end of a month (for a product) is equal to the inventory at the end of the previous month plus production minus demand for that product.
– There are two constraint for mpg, two constraints for steel availability and four constraints for inventory.
The cost of processing the rock is $22 per ton at Plant 1 and $18 per ton at Plant 2.
a) Formulate a linear program to minimize the total processing and transportation cost.
b) Define the decision variables.
c) What are the constraints?
d) What is the objective function?
e) What is the optimal solution?
See the attached file.
Acme estimates it costs $1.50 per month for each unit of this appliance carried in inventory (estimated by averaging the beginning and ending inventory levels each month). Currently, Acme has 120 units in inventory on hand for the product. To maintain a level workforce, the company wants to produce at least 400 units per month. They also want to maintain a safety stock of at least 50 units per month. Acme wants to determine how many of each appliance to manufacture during each of the next four months to meet the expected demand at the lowest possible total cost… see attachment for remainder of problem.
I am working to define constraints and am having problems in doing so. I would like a detalied explanation of how this is done and how Lindo will interrprut the information.
(See attached file for full problem description)
—
A company needs to lease warehouse storage space for five months at the start of the year. The space requirements (in square feet) and the leasing costs of each type of lease are given in the two tables below:
Month Required Space (sq. feet)
Jan 15,000
Feb 10,000
Mar 20,000
Apr 5,000
May 25,000
Lease Term
(months) Cost per 1,000 sq. feet leased
1 $280
2 $450
3 $600
4 $730
5 $820
Leases for different terms can begin at the same time. In addition, a lease should not be entered into if its term extends past May.
(a) Create and solve a linear programming model for determining the leasing schedule that provides the required amounts of space at minimum cost.
(b) What would the impact of a 1,000 square foot increase in the space required for each of the months from January to May?
—
What advantages do computer applications that solve LP problems have? Find at least one computer application, other than Excel’s Solver of course, that solves LP problems. Present it’s web address, and then discuss whether it is free or purchased and then discuss the purpose of that computer application.
Find the complete optimal solution to this linear programming problem.
Min 3X + 3Y
s.t. 12X + 4Y > 48
10X + 5Y > 50
4X + 8Y > 32
X , Y > 0
4) ABC Corporation, produces two types of drinks. For each of them should be made by 3 processes, which are: Distilling, Bottling and Packing. In the drink X is used in each process the next time: 6, 3 and 4 hours respectively. As for the drink and is used in the above processes 6, 6 and 2 hours respectively.
The distillation plant capacity is 420 hours, the bottling is 300 hours and the Pack is 240 hours. In the sale of each unit of beverage X, earn $ 3 in each unit of beverage and earn $ 2.
1)Set Bottle Restriction
2) ABC Corporation, produces two types of drinks. For each of them should be made by 3 processes, which are: Distilling, Bottling and Packing. In the drink X is used in each process the next time: 6, 3 and 4 hours respectively. As for the drink and is used in the above processes 6, 6 and 2 hours respectively.
The distillation plant capacity is 420 hours, the bottling is 300 hours and the Pack is 240 hours. In the sale of each unit of beverage X, earn $ 3 in each unit of beverage and earn $ 2.
1)set the objective function
The Lannon Lock Company manufactures commercial security locks at plants in Atlanta. Louisville, Detroit and Phoenix. The unit cost of production at each plant is $35.50, $37.50, $37.25 and $36.25, respectively; the annual capacities are 18,000, 15,000, 25,000, and 20,000, respectively. The locks are sold through wholesale distributors in seven locations around the country. The unit shipping cost for each plant-distributor combination is shown in the following table, along with the demand forecast from each distributor for the coming year:
Tacoma San Diego Dallas Denver St. Louis Tampa Baltimore
Atlanta 2.5 2.75 1.75 2 2.1 1.8 1.65
Louisville 1.85 1.9 1.5 1.6 1 1.9 1.85
Detroit 2.3 2.25 1.85 1.25 1.5 2.25 2
Phoenix 1.9 0.9 1.6 1.75 2 2.5 2.65
Demand 5,500 11,500 10,500 9,600 15,400 12,500 6,600
a. Determine the least costly way of producing and shipping locks from plants to distributors
b. Suppose that the unit cost at each plant were $10 higher than the original figure. What change in the optimal distribution plan would result? What general conclusions can you draw for transportation models with no identical plant related costs?
An ad campaign for a new snack chip willbe conducted in a limited geographical area and can use TV time, radio time, and newspaper ads. Information about each medium is shown below.
Medium Cost per ad #reached exposure quality
TV 500 10000 30
Radio 200 3000 40
Newspaper 400 5000 25
If the number of TV ads cannot exceed the number of radio ads by more than 4, and if the advertising budget is $10000, develop the model that will maximize the number reached and achieve an exposure quality of at least 1000.
A) set up the problem (provide the objective function and set of constraints)
B) find the optimal solution
C) list the values of the objective function and the descision variables in the optimal soultion you’ve found
The linear programming problem whose output follows is used to determine how many bottles of fire red nail polish (x1), bright red nail polish (x2), basil green nail polish (x3), and basic pink nail polish (x4) a beauty salon should stock. The objective function measures profit; it is assumed that every piece stocked will be sold. Constraint 1 measures display space in units. Constraint 2 measures time to set up the display in minutes. Note that green nail polish does not require any time to prepare its display. Constraints 3 and 4 are marketing restrictions. Constraint 3 indicates that the maximum demand for fire red and green polish is 25 bottles, while constraint 4 specifies that the minimum demand combined for bright red, green, and pink nail polish bottles is at least 50 bottles.
MAX 100×1 + 120×2 + 150×3 + 125×4
Subject to: 1. x1 + 2×2 + 2×3 + 2×4 <= 108
2. 3×1 + 5×2 + x4 <= 120
3. x1 + x2 <= 25
4. x2 + x3 + x4 >= 50
x1, x2 , x3, x4 >= 0
Optimal Solution:
Objective Function Value = 7475.000
Variable Value Reduced Costs
X1 8 0
X2 0 5
X3 17 0
X4 33 0
Constraint Slack / Surplus Dual Prices
1 0 75
2 63 0
3 0 25
4 0 -25
Objective Coefficient Ranges
Variable Lower Limit Current Value Upper Limit
X1 87.5 100 none
X2 none 120 125
X3 125 150 162
X4 120 125 150
Right Hand Side Ranges
Constraint Lower Limit Current Value Upper Limit
1 100 108 123.75
2 57 120 none
3 8 25 58
4 41.5 50 54
By how much can the per bottle profit on green basil nail polish increase before the solution (product mix) would change?
The linear programming problem whose output follows is used to determine how many bottles of fire red nail polish (x1), bright red nail polish (x2), basic green nail polish (x3), and basic pink nail polish (x4) a beauty salon should stock. The objective function measures profit; it is assumed that every piece stocked will be sold. Constraint 1 measures display space in units. Constraint 2 measures time to set up the display in minutes. Note that green nail polish does not require any time to prepare its display. Constraints 3 and 4 are marketing restrictions. Constraint 3 indicates that the maximum demand for fire red and bright red polish is 25 bottles, while constraint 4 specifies that the minimum demand combined for bright red, green, and pink nail polish bottles is at least 50 bottles.
MAX 100×1 + 120×2 + 150×3 + 125×4
Subject to: 1. x1 + 2×2 + 2×3 + 2×4 <= 108
2. 3×1 + 5×2 + x4 <= 120
3. x1 + x2 <= 25
4. x2 + x3 + x4 >= 50
x1, x2 , x3, x4 >= 0
Optimal Solution:
Objective Function Value = 7475.000
Variable Value Reduced Costs
X1 8 0
X2 0 5
X3 17 0
X4 33 0
Constraint Slack / Surplus Dual Prices
1 0 75
2 63 0
3 0 25
4 0 -25
Objective Coefficient Ranges
Variable Lower Limit Current Value Upper Limit
X1 87.5 100 none
X2 none 120 125
X3 125 150 162
X4 120 125 150
Right Hand Side Ranges
Constraint Lower Limit Current Value Upper Limit
1 100 108 123.75
2 57 120 none
3 8 25 58
4 41.5 50 54
What is the lowest value for the amount of time available to setup the display before the solution (product mix) would change?
Please see attached file.
Each day, workers at the Gotham City Police Department work two 6-hour shifts chosen from midnight to 6AM, 6AM to noon, noon to 6PM and 6PM to midnight. The following numbers of workers are needed during each shift:
1. 15 from midnight to 6AM
2. 5 from 6AM to noon
3. 12 from noon to 6PM
4. 6 from 6PM to midnight
Workers whose two shifts are consecutive are paid $12 per hour, whereas workers whose shifts are not consecutive are paid $18 per hour. Determine how to minimize the cost of meeting the daily workforce demands of Gotham City police department.
Simplon Manufacturing must decide on the processes to use to produce 1650 units. If machine 1 is used, its production will be between 300 and 1500 units. Machine 2 and/or machine 3 can be used only if machine 1’s production is at least 1000 units. Machine 4 can be used with no restrictions.
Machine Fixed
cost Variable
cost Minimum
Production Maximum
Production
1 500 2.00 300 1500
2 800 0.50 500 1200
3 200 3.00 100 800
4 50 5.00 any any
HINT: Use an additional 0-1 variable to indicate when machines 2 and 3 can be used.
Formulate and solve this problem. Report answers below in spaces provided:
Problem Formulation (define variables, give objective function, give all constraints):
Optimal solution, AND objective function value:
Need tutorial assistance involving Excel Solver.
The city of Spring View is taking bids from six bus companies on the eight routes that must be driven in the surround school district. Each company enters a bid on how much it will charge to drive selected routes, although not all companies bit on all routes. The data are contained in the Excel file attached. If the company does not bid on a route, the corresponding cell is blank. The city must decide which companies to assign to which routes with the specifications that (1) if a company does not bid on a route, it cannot be assigned to that route, (2) exactly one company must be assigned to each route, and (3) a company can be assigned to at most two routes. The objective is to minimize the total cost of covering all routes.
in the optimal solution to the bus route assignment problem (provided in the Excel file), company 2 is assigned to bus routes 6 and 7. Support these two routes are far enough apart that is infeasible for one company to service both of them.
(a) Write down algebraic constraint in terms of the decision variables defined during lecture to accommodate this restriction.
(b) Add the constraint to the existing model and comment on the new optimal solution.
Question 3:
Blair & Rosen, Inc (B&R), is a brokerage firm that specializes in portfolios designed to meet the specific risk tolerances of its clients. A client who contacted B&R this past week has a maximum of $50,000 to invest. B&R’s investment advisor decides to recommend a portfolio consisting of two investment funds: an internet fund and a Blue Chip fund. The Internet fund has a projected annual return of 12%, while the Blue Chip fund has a projected annual return of 9%. The investment advisor requires that at most $35,000 of the client’s funds should be invested in the Internet fund. B&R services include a risk rating for each investment alternative. The Internet fund, which is the more risky of the two investment alternatives, has a risk rating of 6 per thousand dollars invested. The Blue Chip fund has a risk rating of 4 per thousand dollars invested. For example, if $10,000 is invested in each of the two investment funds, B&R’s risk rating for the portfolio would be 6(10) + 4(10) = 100. Finally, B&R developed a questionnaire to measure each client’s risk tolerance. Based on the responses, each client is classified as a conservative, moderate, or aggressive investor. Suppose that the questionnaire results classified the current client as a moderate investor. B&R recommends that a client who is a moderate investor limit his or her portfolio to a maximum risk rating of 240.
A. What is the recommended investment portfolio for this client? What is the annual return for the portfolio?
B. Suppose that a second client with $50,000 to invest has been classified as an aggressive investor. B&R recommends that the maximum portfolio risk rating for an aggressive investor is 320. What is the recommended investment portfolio for this aggressive investor? Discuss what happens to the portfolio under the aggressive investor strategy.
C. Suppose that a third client with $50,000 to invest has been classified as a conservative investor. B&R recommends that the maximum portfolio risk rating for a conservative investor is 160. Develop the recommended investment portfolio for the conservative investor. Discuss the interpretation of the slack variable for the total investment fund constraint.
Note: “Discussion” comments asked in this question can be answered very generally – I’m more concerned about the math side, I can figure out how to interpret the results myself.
The problem is in the attachment file. Thank you.
Optimization
The Sentry Lock Corporation manufactures a popular commercial security lock at plans in Macon, Louisville, Detroit, and Phoenix. The per unit cost of production at each plant is $35.5, $37.5, $39, and $36.25, respectively, while the annual production capacity at each plant is 18,000, 15,000, 25,000, and 20,000, respectively. Sentry’s locks are sold to retailers through wholesale distributors in 7 cities across a nation. The unit cost of shipping from each plant to each distributor is summarized in the following table along with the forecasted demand from each distributor for the coming year.
Unit Shipping Cost to Distributor in
Plants Tacoma San Diego Dallas Denver St. Louis Tampa Baltimore
Macon $2.5 $2.75 $1.75 $2.00 $2.1 $1.8 $1.65
Louisville $1.85 $1.9 $1.5 $1.6 $1.00 $1.9 $1.85
Detroit $2.3 $2.25 $1.85 $1.25 $1.5 $2.25 $2.00
Phoenix $1.9 $0.9 $1.6 $1.75 $2.00 $2.5 $2.65
Demand 8,500 14,500 13,500 12,600 18,000 15,000 9,000
Sentry wants to determine the least expensive way of manufacturing and shipping locks from their plants to the distributors. Because the total demand from distributors exceeds the total production capacity for all the plants, Sentry realizes they will not be able to satisfy all the demand for their product, but wants to make sure each distributor will have the opportunity to fill at least 80% of the orders they receive.
1.Build an appropriate Linear Programming Model for the Sentry Lock problem.
2.Solve your Linear Programming model by EXCEL. Attach a copy of your EXCEL model.
3.Give a brief description of the meaning of your Linear Programming solution.
The Bluegrass Distillery produces custom blended whiskey. A particular blend consists of rye and bourbon whiskey. The company has received an order for a minimum of 400 gallons of the custom blend. The customer specified that the order must contain at least 40% rye and not more than 250 gallons of bourbon. The customer also specified that the blend should be mixed in the ratio of two parts rye to one part bourbon. The distillery can produce 500 gallons per week regardless of the blend. The production manager wants to complete the order in one week. The blend is sold for $5 per gallon.
The distillery company’s cost per gallon is $2 for rye and $1 for bourbon. The company wants to determine the blend mix that will meet customer requirements and maximize profits.
On the computer, formulate a linear programming model for this problem using excel solver.
A) Identify the sensitivity ranges for the objective function coefficient and explain what the upper and middle limits are.
B) How much would it be worth to the distillery to obtain additional production capacity?
C) if the customer decided to change the blend requirements for its custom-made whiskey to a mix of three parts rye to one part bourbon, how would this change the optimal solution?
Why should decision makers who are primarily concerned with marketing or finance or production know about linear programming?
1. Linear Programming Properties
Which of the following statements is not true?
a) An infeasible solution violates all constraints.
b) A feasible solution point does not have to lie on the boundary of the feasible solution.
c) A feasible solution satisfies all constraints.
d) An optimal solution satisfies all constraints.
2. Minimization Graphical Solution
Solve the following linear model graphically and select the set of extreme points that make up the possible feasible solutions.
a) (x1=12, x2=0, z=120), (x1=6, x2=5, x3=160), (x1=0, x2=8, z=160)
b) (x1=0, x2=12, z=240), (x1=6, x2=5, x3=160), (x1=20, x2=0, z=200)
c) (x1=0, x2=12, z=240), (x1=20/3, x2=16/3, x3=520/3), (x1=20, x2=0, z=200)
d) (x1=12, x2=0, z=120), (x1=20/3, x2=16/3, x3=520/3), (x1=0, x2=8, z=160)
3. Minimization Graph Surplus Variables
Based on the optimal solution from the previous problem, which of the following statements is true if s1 represents the slack from the first constraint and s2 represents the slack from the second constraint?
a) Constraint 1 has no slack
Constraint 2 does have slack
b) Constraint 1 does have slack
Constraint 2 has no slack
c) Both constraints 1 and 2 have no slack.
4. Maximization Feasible Solutions
Given the following maximization linear programming model, which of the possible solutions provided below is NOT feasible?
a) x1 = 0 and x2 = 120
b) x1 = 75 and x2 = 90
c) x1 = 90 and x2 = 75
d) x1 = 135 and x2 = 0
5. Maximization Graphical Solution
Graphically solve the linear programming model from the previous problem and determine the set of extreme points that make up the set of feasible solutions.
a) (x1=0, x2=120, z=240), (x1=90, x2=75, z=420), (x1=240, x2=0, z=720)
b) (x1=0, x2=120, z=240), (x1=90, x2=75, z=420), (x1=135, x2=0, z=405)
c) (x1=0, x2=225, z=450), (x1=90, x2=75, z=420), (x1=135, x2=0, z=405)
d) (x1=0, x2=225, z=450), (x1=90, x2=75, z=420), (x1=240, x2=0, z=720)
NOTE: The linear programming model and accompanying Excel sensitivity report in problem 6 are also to be used in problems 7, 8, and 9.
6. Excel Sensitivity Analysis 1
The following model was solved using Excel
Excel the produced the following sensitivity report.
What is the maximum profit that can be made on Product 1 WITHOUT affecting the optimal product mix?
a) $36.71 b) $50.00 c) $61.53 d) $65.31
7. Excel Sensitivity Analysis 2
Using the model and the sensitivity report from the previous problem, a manager is offered the opportunity to make a bulk purchase of 150 additional hours for Process 1 at a cost of $1050. Which of the following should the manager do without impacting the current product mix?
a) Refuse because the purchase price per additional hour is greater than the shadow price.
b) Accept because the purchase price for each additional hour is less than the shadow price and this does not impact the current product mix?
c) Refuse because this would impact his current optimal product mix.
8. Excel Sensitivity Analysis 3
Using the model and sensitivity report from the previous two problems, how many pounds of Material A are left over from the optimal solution? NOTE: If you calculations are off by + or – 0.10 from any of these choices, then select the choice that is closest to your solution.
a) 0 b) 70.92 c) 111.45 d) 149.67
9. Excel Sensitivity Analysis 4
Use the model and sensitivity report from the previous three problems.
A manager elects to purchase 50 additional pounds of Material B at $2.00 per pound, how much additional profit can be made from this purchase?
a) $30 b) $50 c) $100 d) $130
NOTE: The linear programming model and accompanying sensitivity report from QM for Windows are to be used for problems 10, 11, and 12.
Please solve this problem using excel solver.
The Pittsburg Chemical Company of Akron, Ohio, produces a variety of products within its four divisions for agricultural and industrial customers. Division II will produce only two products this week for sale to wholesale distributors and all production will be sold. Product 1 yields a contribution margin (contribution to profit) of $25 per ton and product 2 yields a profit contribution of $10 per ton. Both products are manufactured by mixing raw materials from inventory. There are 12,000 tons of material 1, 4,000 tons of material 2, and 6,000 tons of material 3 in inventory in Division II this week. Product 1 is manufactured by mixing materials 1 and 2 in ratios of 60% and 40% respectively. Product 2 is manufactured by mixing 50% of material 1, 10% of material 2, and 40% of material 3. Any unused materials are maintained in inventory at insignificant cost for future production.
a. Formulate a linear programming model that can be used to determine the optimal production scheduling mix that yields the best contribution solution while meeting the inventory capacity restrictions of the division for the week.
b. Determine the optimal production scheduling mix for this division of Pittsburg Chemical Company for the week using the Management Scientist software, including the quantity of each product line manufactured, the quantity of unused materials (if any) remaining, and the total profit contribution for the week. Provide a narrative that explains the Management Scientist solution used. Include all information available relative to division resource utilization, and resources remaining, total production, and profit contribution.
See attached file for proper format.
Linear programming: Analytical methods of supply chain management
Problem 1
You have six projects from which to select (P1, P2, P3, P4, P5, P6). The net present value (NPV) per project is given below. You are given the following constraints also:
(a) Due to limitations on the hours a project manager will be available, a maximum of four out of six projects could be selected.
(b)If Projects 2, 3, and 4 all are selected, then Project 5 must be selected.
(c)If Project 2 is selected, then, and only then, Project 3 could be selected. However, Project 2’s selection is not dependent on Project 3’s selection.
NPV
($millions)
$100
$150
$20
$380
$40
$80
Project
1
2
3
4
5
6
Question: Which projects would you select and what is the Net Present Value?
Define the decision variables, the objective function, and the constraints within your answer to this question. Then, solve the model using Excel Solver. Finally, copy your spreadsheet into the appendix of your Word report and submit your Excel file to the dropbox.
Steelco manufactures two types of steel at three different steel mills. During a given month, each steel mill has 200 hours of blast furnace time available. Because of the differences in the furnaces at each mill, the time and cost to produce a ton of steel differ for each mill, as listed in the file P04_62.xls. Each month Steelco must manufacture at least 500 tons of steel 1 and 600 tons of steel 2. Determine how Steelco can minimize the cost of manufacturing the desired steel.
I have been able to figure out previous questions, but this has a restriction where it has to produce ATLEAST 500 and 600 tons. When I do the constraints, I am not sure if these numbers would be the constraints, and also if I should put in a time frame of MONTH 1 etc.
Please see attachment
I have one last work problem where my solution just doesn’t to be correct
Set up the objective function and constraints and then solve for the following:
A company makes a single product on two separate production lines, A and B. The company’s labor force is equivalent to 1,000 hours per week, and it has $3,000 outlay weekly for operating costs. It takes 1 hour and 4 hours to produce a single item on lines A and B, respectively. The cost of producing a single item on A is $5 and on B is $4. How many items should be produced on each line to maximize the total output?
See attached
Stannic Metals wishes to produce at the lowest cost
a new alloy that is 40 percent tin, 35 percent zinc, and
25 percent lead from their current allow stocks:
Alloy Stocks
Alloy 1 2 3 4 5
% Tin 60 25 45 20 50
% Zinc 10 15 45 50 40
%Lead 30 60 10 30 10
Cost/lb 22 20 25 24 27
What weight of each Alloy Stock is required to make a pound of the new alloy?
To the extent permitted by local law, each Acme Home Improvements store, including AMC, is open from 7 am – 11 pm every day. Acme Mexico City planners have provided the following table, which identifies the minimum number of customer service employees estimated to be needed on the floor of the store each hour of the day:
Customer Service Employees
Time Period
Minimum number needed on the floor
7 am – 8 am
10
8 am – 9 am
12
9 am – 10 am
18
10 am – 11 am
22
11 am – 12 pm
22
12 pm – 1 pm
26
1 pm – 2 pm
26
2 pm – 3 pm
26
3 pm – 4 pm
26
4 pm – 5 pm
26
5 pm – 6 pm
28
6 pm – 7 pm
28
7 pm – 8 pm
24
8 pm – 9 pm
22
9 pm – 10 pm
14
10 pm – 11 pm
12
Full-time customer service employees at AMC work a 9 hour shift (8 hours of work plus a 1 hour meal break) either from 7 am to 4 pm or from 2 pm to 11 pm. Workers on the 7-4 shift are assigned an hour-long lunch break at either 11 am or 12 noon. Workers on the 2-11 shift are assigned an hour-long dinner break at either 5 pm or 6 pm.
Part-time customer service employees work four consecutive hours per day and their shifts can start any hour between 7 am and 7 pm.
By corporate policy, which is consistent with Mexican labor law, the company limits part-time customer service employees hours to 50% of the day’s total required hours.
Further, in the interest of proper supervision of employees and employee safety, management has a policy that the actual number of customer service employees on the floor in the store should never exceed 30 during any given time period.
Part-time customer service employees earn $500 per day, and full-time customer service employees earn $1100 per day in salary and benefits
Assignment Taskings
Each student prepares a customer service employee daily assignment schedule for Acme Mexico City that addresses the following tasks:
An executive summary
An optimal schedule for full-time and part-time customer service employees that minimizes personnel costs to the extent congruent with the qualitative factors that you determine are relevant
A discussion of the process used to determine the schedule, to include a discussion of all model assumptions and of your qualitative factors
A manufacturer of excercise equipment will begin production of two types of machines: Body Plus 100 and Body Plus 200.
The Body Plus 100 consists of a frame unit, a press station, and a pec-dec station. each frame produced uses 4 hours of machining and welding time and 2 hours of finishing and painting time. Each press station requires 2 hours of machining and welding time and 1 hour of painting and finishing time, and each pec-dec station requires 2 hours of machining and welding and 2 hours of painting and finishing time. In addition, 2 hours are spent assembling, testing and packaging each body Plus 100. The raw material costs are $450 for each frame, $300 for each press station, and $250 for each pec-dec station. Packaging costs are expected to be $50 per unit.
The Body Plus 200 consists of a frame unit, a press station, a pec-dec station, and a leg-press station. Each frame produced uses 5 hours of machining and welding time and 4 hours of finishing and painting time. Each press station requires 3 hours of machining and welding time and 2 hours of painting and finishing time. Each leg-press station requires 2 hours of machining and welding time and 2 hours of finishing and painting time. In addition, 2 hours are spent assembling, testing and packaging each. The raw materials costs are $650 for each frame, $400 for each press station, $250 for each pec-dec station, and $200 for each leg-press station. Packaging cost is estimated to be $75 per unit.
For the next production period, management estimates that 600 hours of machining and welding time, 450 hours of painting and finishing time, and 140 hours of assembly, testing and packaging time will be available. Current labor costs are $20 per hour for machining and welding, $15 pre finishing and painting time, and $12 for assembly, testing and packaging time.
A retail price of $2400 for each Body Plus 100 and $3500 for each Body Plus 200 are suggested. However, authorized dealers can purchase the machines at 70% of the suggested retail price. Management also believes that the Body Plus 200 can help position the company as one of the leaders in high-end exercise equipment and ordered that the number of Body Plus 200 units produced be at least 25% of the total production.
Question: Make a recommendation to management how many units of each Body Plus 100 and Body Plus 200 to produce. (Use Linear Programming/Optimal Solution to get an answer). Please show all the work as to how you came to the answer. Also please use a table in your work.
Thank You for all of your help.
Allstate Oil produces two grades of gasoline: reguler and premium. The profit contributions are $0.30 per gallon for regular gasoline, and 0.50 per gallon for premium gasoline.
Each gallon of regular gasoline contains 0.3 gallons of grade A crude oil and each gallon of premium contains 0.6 gallons of grade A crude oil.
For the next production period, Allstate has 18,000 gallons of grade A crude oil available. The refinery used to produce the gasoline has a production capacity of 50,000 gallons for the next production period. Allstate distributors indicated that the demand for the premium gasoline for the next production period would be at most 20,000 gallons.
1) Formulate a linear programming model that can be used to determine the number of gallons of regular gasoline and the number of gallons of premium gasoline that should be produced in order to maximize total profit contribution.
2) Write the linear programming in standard form
3) What is the optimal solution
4) What are the values and interpretations of the slack variables
5) What are the binding constraints
Baseball Inc produces Regular gloves and Catcher?s mitt. The linear programming problem is listed below:
Max 5R + 8C
s.t.
R + 3C < or equal to 1800 Cutting dept
3R + 2C < or equal to 1800 Finishing dept
R + 2C < or equal to 800 Packaging dept
R, C, > or equal to 0
The computer solution obtained using the Management Scientist is attached (see Figure A). Use the computer solution to answer the following questions:
1) what is the optimal solution
2) What is the value of the total profit contribution
3) Which constraints are binding
4) What are the dual prices for the resources. Interpret each.
5) If overtime can be scheduled in one of the departments, where would you recommend doing so.
6) What does the 100% Rule in relation to the Allowable Increases and Allowable decreases mean in this example. Explain the concept with an example.
See attached file for proper format.
Linear Programming – Analytical Methods of Supply Chain Management
Problem 2
The Tots Toys Company is trying to schedule production of two very popular toys for the
next three months: a rocking horse and a scooter. Information about both toys is given
below.
Beg. Inv.
June 1
30
10
Scooter
Plastic
Available
3500
5000
4800
Time
Available
2100
4000
2500
Monthly Demand
Horse
220
350
600
Monthly Demand
Scooter
450
700
520
Required
Plastic/Unit
5
4
Required
Time/Unit
2
3
Production
Cost/Unit
10
12
Holding
Cost/Unit/Month
1.50
1.20
Toy
Rocking
Horse
Summer Schedule
June
July
August
Develop a linear programming model that would tell the company how many of each toy
to produce during each month. You are to minimize total cost (production cost + holding
cost). Inventory holding cost will be levied on any items in ending inventory on June 30,
July 31, or August 31 after demand for the month has been satisfied. The company wants
to end the summer with 85 rocking horses and 50 scooters as beginning inventory for
Sept. 1.
Questions: How many of each toy will be produced each month? What will the total
cost equal? Be sure to specify that all variables are integer.
Define the decision variables, the objective function, and the constraints within your
answer to this question. Then, solve the model using Excel Solver. Finally, copy your
spreadsheet into the appendix of your Word report and submit your Excel file to the
dropbox.
2
See attached file for proper format.
Linear Programming – Analytical Methods of Supply Chain Management
Problem 4
Hansen Controls has been awarded a contract for a large number of control panels. To
meet this demand, it will use its existing plants in San Diego and Houston, and consider
new plants in Tulsa, St. Louis, and Portland. Finished control panels are to be shipped to
Seattle, Denver, and Kansas City. Pertinent information is given in the table.
Shipping Cost/Unit to
Destination:
Seattle Denver Kansas city
5 7 8
10 8 6
9 4 3
12 6 2
4 10 11
3,000 8,000 9,000
Sources
San Diego
Houston
Tulsa
St. Louis
Portland
Construction
Cost
0
0
350,000
200,000
480,000
Demand
Capacity
of Source
2,500
2,500
10,000
10,000
10,000
Questions: Which plants will be used, how many units will be shipped from each
plant to each destination, and what is the value of the objective function? Be sure to
specify that all variables are integer.
Define the decision variables, the objective function, and the constraints within your
answer to this question. Then, solve the model using Excel Solver. Finally, copy your
spreadsheet into the appendix of your report and submit your Excel file to the dropbox.
4
Case 2-3: Staffing a call center
I am having trouble setting up the problem; i.e. identifying the changing and output cells, the constraints, and the target cell. With all the info/data listed in the attached problem/case I have gotten lost. I am looking for a demonstration of how to set up the info in Excel so that the case questions can then be addressed.
I wanted to make sure I came up with the correct answer. I need to find the minimum cost for the attached problem. I came up with 13,250 but not sure if that is correct.
The contract calls for 10,000 hoses.
This solution offers assistance with the attached problem.
The Skimmer Boat Company manufactures the Water Skimmer bass fishing boat. The company purchased the engines it installs in its boats from Mar-gine Company, which specialized in marine engines. Skimmer has the following production schedule for April, May, June, and July:
Month Production
April 60
May 85
June 100
July 120
Mar-gine usually …………….see attachment
Subject: Linear Programming
Details: The programs that can be used are Excel Slover, Data analyis, spreed sheets, linear Programming and PERT/CPM models.
=
19. A company that makes garments is unionized. Each full time employee works 40 hours per week,earns $13 per hour, and the union contract says that the number of full time employees can never drop below 20. Nonunion part time employees can be hired, but there must be at least 2 full time employees for each part time employee, and part time employees can work no more than 20 hours per week. They make $10 per hour. The company has 5,000 square feet of material available each week. The cost of this material is reflected in the Gross Profit per garment (but the cost of labor is not included in that cost). Fabrication requirements are as follow:
Product Material Required Labor Required Gross Profit per garment
A 2 square feet 30 minutes $8
B 1.5 square feet 45 minutes $10
C 1 square feet 40 minutes $6
Maximize the company’s net profit: gross profit minus the cost of labor.
20. A Computer Center must have the following minimum number of consultants on duty during each 4-hour period:
8 AM to Noon 6 Consultants
Noon to 4 PM 8 Consultants
4 PM to 8 PM 12 Consultants
8 PM to Midnight 6 Consultants
Full time consultants are paid $14 per hour and they work any of the following 8 consecutive hour shifts:
8 AM to 4 PM
Noon to 8 PM
4 PM to Midnight
Part time consultants are paid $12 per hour and they work four-hour shifts.
At all times, there must be at least 2 full time consultants for each part time consultant on duty. Determine how many full time and part time consultants to employ during each work period to produce the lowest possible cost.
5) The owner of The East End Technology Company has collected statistics on previous demand:
Daily Demand 0 10 20 30 40 or more
# of days 5 8 15 12 10
a. Using the expected monetary value model, what is the best development alternative?
b. What is the value of perfect information?
c. Draw the decision tree for this problem.
d. Using the maximum likelihood criteria, determine the best alternative.
The Dub-Dub and Dub Company produces and markets three lines of WEB page designs: A, B, and C; A is a standard WEB page design and B and C are professional WEB page designs. The manufacturing process for the WEB page designs is such that two development operations are required – all WEB page designs pass through both operations. Each WEB page design requires 3 hours of development time in Operation 1. In Operation 2 WEB page design A requires 2 hours of development time; WEB page design B requires 4 hours; and WEB page design C requires 5 hours. Operation 1 has 50 hours of development time per week and Operation 2 has sufficient manpower to support 80 hours of development per week. The market group for Dub-Dub and Dub has projected that the demand for the standard WEB page design will be no more than 25 per week. Because WEB page designs B and C are similar in quality, the combined demand for those WEB page designs has been forecast – the total demand is ten or more, but not more than 30 per week. The sale of WEB page design A results in $7 profit while WEB page design B and C provide $8 and $8.5 profits respectively. How many of WEB page designs A, B, and C should be produced weekly if the company seeks to maximize profits? Formulate the problem as a standard LP model.
See attachment
a. Graph the problem.
b. What is the optimal solution?
c. What would the solution be if the third constraint were removed from the problem?
To the extent permitted by local law, each XX store, is open from 7 am – 11 pm every day. Planners have provided the following table, which identifies the minimum number of customer service employees needed on the floor each hour of the day:
Time Period Minimum number of employees
7 am – 8 am 15
8 am – 9 am 15
9 am – 10 am 15
10 am – 11 am 20
11 am – 12 pm 20
12 pm – 1 pm 20
1 pm – 2 pm 20
2 pm – 3 pm 20
3 pm – 4 pm 25
4 pm – 5 pm 25
5 pm – 6 pm 25
6 pm – 7 pm 20
7 pm – 8 pm 20
8 pm – 9 pm 15
9 pm – 10 pm 15
10 pm – 11 pm 15
Full-time employees at XX work a 9 hour shift (8 hours of work plus a 1 hour meal break) either from 7 am to 4 pm or from 2 pm to 11 pm. Workers on the 7-4 shift are assigned an hour-long lunch break at either 10 am, 11 am or 12 noon. Workers on the 2-11 shift are assigned an hour-long dinner break at either 5 pm, 6 pm or 7 pm.
Part-time employees work four consecutive hours per day and their shifts can start any time between 7 am and 7 pm.
By corporate policy, which is consistent with local labor law, the company limits part-time hours to 50% of the day’s total hours (part-time hours plus full-time hours) worked.
Part-time employees earn $300 per day, and full-time employees earn $750 per day in salary and benefits ($ = Moneda Nacional, ie, the Mexican peso).
Using Excel, Lindo or comparable software, each student prepares an employee assignment schedule for XX store that addresses the following tasks:
1. An optimal schedule for full-time and part-time customer service employees that minimizes personnel costs.
Note: attached sample excel to use the Solver function–similar templates to solve this problem.
Use graphical methods to solve the linear programming problem (attached)
Give an evaluation of how to use one of the quantitative tool (Linear programing )and in which situations they work best.
A candidate for mayor in a small town has allocated $40,000 for last minute advertising in the days preceding the election. Two types of ads will be used: radio and television. Each radio ad costs $200 and reaches an estimated 3,000 people. Each television ad cost $500 and reaches an estimated 7,000 people. In planning the advertising campaign, the campaign manager would like to reach as many people as possible, but she has stipulated that at least 10 ads of each type must be used. Also, the number of radio ads must be at least as great as the number of television ads. How many ads of each type should be used? How many people will this reach?
Solve the following problems: Find the optimal solution and graph. Please show all work:
1. MINC .5x + .3y
x + 2y is > or = to 10a
x + y is > or = to 8b
2. change the constraint B to < or = to.
See attached file.
A. Minimization Graphical Solution
Solve the following linear programming model graphically and select the set of extreme points that make up the solution:
Minimize Z = 20X1 + 10X2
subject to:
X1 + X2 < 12
2X1 + 5X2 > 40
X2 < 13
Note: The triplets are in the form of (X1 = ,X2 = , Z = )
a) (0, 12, 120), (0, 8, 80), (20/3, 16/3, 560/3)
b) (0, 12, 240), (6, 5, 160), (20, 0, 200)
c) (12, 0, 120), (6, 5, 160), (0, 8, 160)
d) (12, 0, 120), (6, 5, 160), (20/3, 16/3, 560/3)
Answer: _____
B. Minimization Graph Surplus Variables
Based on the optimal solution from the previous problem, which of the following statements is true if s1 represents the slack from the first constraint and s2 represents the slack from the second constraint?
a) Constraint 1 has no slack
Constraint 2 has no slack
b) Constraint 2 has no slack
Constraint 3 has no slack
c) Constraint 1 has no slack
Constraint 3 has no slack
Answer: _____
This is a problem related to linear programming and sensitivity analysis.
Please solve this and explain how you arrived at the solution.
Please do this in Excel by using “Solver” or management science software…which ever is convinient for you.
The problem is attached.
Thank you.
Need help with following using Excel:
Attached please find three problems that need resolution. Please provide your responses using Excel (not Lindo). I need to know the variables, constraints, max as well as a detailed answer to the questions in each problem. For each question, show the steps to resolution as well as the final answer.
See attached file for full problem description.
See attached
1.)
Without graphing, determine which of the following three points:
P1 = (8,6)
P2 = (2,5)
P3 = (4,1)
are part of the graph of the following system:
y – 10x <= 0
2y – 3x >= 0
y + x <= 15
2.)
Maximize and minimize the quantity z = 15x + 20y subject to the constraints:
x <= 6
y <= 6
3x + 2y >= 6
x >= 0
y >= 0
3.)
Maximize:
z = 4x + y
subject to the constraints:
0 <= x <= 7
0 <= y <= 8
x + y >= 2
4.) Transpose this augmented matrix:
[ 1 8 7 9 ]
[ 0 2 5 1 ]
[ -4 -1 0 17 ]
5.)
Maximize:
P = 300×1 + 200×2 + 450×3
Subject to :
4×1 + 3×2 + 5×3 <= 140
x1 + x2 + x3 = 30
6.)
Minimize
P = 2×1 + x2
subject to:
2×1 + 2×2 >= 8
x1 – x2 >= 2
7.)
Maximize:
P = x1 + 2×2 + x3
Subject to the constraints:
3×1 + x2 + x3 <= 3
x1 – 10×2 – 4×3 <= 20
x1 >= 0
x2 >= 0
x3 >= 0
Consider the following linear programming problem:
Min x1 + 2×2
s.t.
x1 + 4×2 ≤ 21
2×1 + x2 ≥ 7
3×1 +1.5×2 ≤ 21
-2×1 + 6×2 ≥ 0
x1, x2 ≥ 0
a. Find the optimal solution using the graphical solution procedure and the value of the objective function.
b. Determine the amount of slack or surplus for each constraint.
c. Suppose the objective function is changed to max 5×1 + 2×2. Find the optimal solution and the value of the objective function.
Need help with the following using Excel,
Need help with the following problem, using Excel.
A company produces two products: Product A and Product B. Each product must go through two processes. Each Product A produced requires 2 hours in Process 1 and 5 hours in Process 2. Each Product B produced requires 6 hours in Process 1 and 3 hours in Process 2. There are 80 hours of capacity available each week in each process. Each unit of Product A produced generates $6.00 in profit for the company. Each unit of Product B produced generates $9.00 in profit for the company. If the company produces 6 units of Product A and 9 units of Product B then the company’s objective function would be equal to…?
Southern Air is considering purchase of new aircraft and has set aside $1.5B dollars for this.
They have enough pilots to crew 30 new planes and enough maintence personnel to crew
40 new short range planes.
Maint for long range planes is 1.67 times
that for short range planes
Maint for med range planes is 1.33 times
that for short range planes
New Planes Cost Profit
long range 67M 4.2M
med. range 50M 3M
short range 35M 2.3M
How many planes of what types would you recommend Southern to buy?
Using Excel, please solve the following:
The Hickory Cabinet and Furniture Company produces sofas, tables, and chairs at its plant in Greensboro, N.C.. The plant uses three main resources to make furniture- wood, upholstery, and labor. The resource requirements for each piece are as follows:
(See chart in attached file)
The furniture is produced weekly and is stored in a warehouse until the end of the week, when it is shipped out. The warehouse has a total capacity of 650 pieces of furniture. Each sofa earns $400 profit, each table $275, and each chair $190. The company wants to know how many pieces of each type of furniture to make each week to maximize profit.
a) Formulate a linear programming model for this problem.
b) Solve the model using the computer.
Triumph Trumpet Company makes two styles each of both trumpets and comets deluxe and professional models. Its unit profit on deluxe trumpets is $80 and on deluxe comets is $60. The professional models realize twice the profit of the deluxe models.
Trumpets and comets are made basically from two mixtures of two different brass alloys. The amount of each alloy (in pounds) required to produce each type of horn is summarized in the following table along with the monthly availability to Triumph of the alloys.
Trumpets Comets Monthly Availability
Deluxe Pro Deluxe Pro
Alloy 1 2 1.5 1.5 1 2000
Alloy 2 1 1.5 1 1.5 1800
Triumph must fulfill contracts calling for at least 500 deluxe trumpets and 300 deluxe comets monthly. Monthly demand for professional trumpets is not expected to exceed 150 and for professional comets is not expected to exceed 100. Production set-ups are such that the company will produce exactly twice as many trumpets as comets.
Formulate this problem as a linear program.
Just formuate this problem and explain
The “Mill” produces five different fabrics. Each fabric can be woven
on one or more of the mill’s 38 looms. The sales department has
forecast demand for the next month. The demand data are shown in Table
1.0, along with data on the selling price per yard, variable cost per
yard, and purchase price per yard. The mill operates 24 hours a day
and is scheduled for 30 days during the coming month.
The Mill has two types of looms: dobbie and regular. The dobbie looms
are more versatile and can be used for all five fabrics. The regular
looms can produce only three of the fabrics. The Mill has a total of
38 looms: 8 are dobbie and 30 are regular. The rate of production for
each fabric on each type of loom is given in Table 1.1. The time to
change over from producing one fabric to another is negligible and
does not have to be considered.
The Mill satisfies all demand with either its own fabric or fabric
purchased from another mill. That is, fabrics that cannot be woven at
The Mill because of limited loom capacity will be purchased from
another mill. The purchase price of each fabric is also shown in Table
1.0.
Question
Develop a model that can be used to schedule production for The Mill,
and at the same time, determine how many yards of each fabric must be
purchased from another mill. Include a discussion and analysis of the
following items in your answer:
1. The final production schedule and loom assignments for each fabric
2. The projected total contribution to profit
3. A discussion of the value of additional loom time (The Mill is
considering purchasing a ninth dobbie loom. What is your estimate of
the monthly profit contribution of this additional loom?)
4. A discussion of the objective coefficient ranges
5. A discussion of how the objective of minimizing total costs would
provide a different model than the objective of maximizing total
profit contribution: How would the interpretation of the objective
coefficients ranges differ for these two models?
Table 1.0
Monthly Demand, Selling Price, Variable Cost, and Purchase Price Data
for The Mill
Demand Selling Price Variable Cost Purchase Price
Fabric (yards) ($/yard) ($/yard) ($/yard)
1 16,500 0.99 0.66 0.80
2 22,000 0.86 0.55 0.70
3 62,000 1.10 0.49 0.60
4 7,500 1.24 0.51 0.70
5 62,000 0.70 0.50 0.70
Table 1.1
Loom Production Rates for The Mill
Loom Rate (yards/hour)
Fabric Dobbie Regular
1 4.63 *
2 4.63 *
3 5.23 5.23
4 5.23 5.23
5 4.17 4.17
* Fabrics 1 and 2 can be manufactured only on the dobbie loom.
Already Tried:
To define the variables and constraints Do not know how to set out in excel and where and how to use the formulas
and create the objective function
Green Valley Mills produces carpet at plants in St Louis and Richmond. The plants ship the carpet to two outlets in Chicago and Atlanta. The cost per ton of shipping carpet from each of the two plants to the two warehoused is as follows:
From Chicago Atlanta
St Louis $40 $65
Richmond $70 $30
The plant at St Louis can supply 250 tons of carpet per week, and the plant at Richmond can pull 400 tons per week. The Chicago outlet has a demand of 300 tons per week; the outlet at Atlanta demands 350 tons per week. Company managers want to determine the number of tons of carpet to ship from each plant to each outlet in order to minimize the total shipping cost.
a. solve the model using the computer.
Please help with solving model using the computer.
The linear programming problem whose output follows is used to determine how many bottles of fire red nail polish (x1), bright red nail polish (x2), basil green nail polish(x3), and basic pink nail polish(x4) a beauty salon should stock. The objective function measures profit; it is assumed that every piece stocked will be sold. Constraint 1 measures display space in units, constraint 2 measures time to set up the display in minutes. Note that green nail polish does not require any time to prepare its display. Constraints 3 and 4 are marketing restrictions. Constraint 3 indicates that the maximum demand for fire red and green polish is 25 bottles, while constraint 4 specifies that the minimum demand for bright red, green and pink nail polish bottles combined is at least 50 bottles.
MAX 100×1 + 120×2 + 150×3 + 125×4
Subject to 1. x1 + 2×2 + 2×3 + 2×4 108
2. 3×1 + 5×2 + x4 120
3. x1 + x3 25
4. x2 + x3 + x4 50
x1, x2 , x3, x4 0
Optimal Solution:
Objective Function Value = 7475.000
Variable Value Reduced Costs
X1 8 0
X2 0 5
X3 17 0
X4 33 0
Constraint Slack / Surplus Dual Prices
1 0 75
2 63 0
3 0 25
4 0 -25
Objective Coefficient Ranges
Variable Lower Limit Current Value Upper Limit
X1 87.5 100 none
X2 none 120 125
X3 125 150 162
X4 120 125 150
Right Hand Side Ranges
Constraint Lower Limit Current Value Upper Limit
1 100 108 123.75
2 57 120 none
3 8 25 58
4 41.5 50 54
a) To what value can the per bottle profit on fire red nail polish drop before the solution (product mix) would change?
b) By how much can the per bottle profit on green basil nail polish increase before the solution (product mix) would change?
Tots Toys makes a plastic tricycle that is composed of three major components: a handlebar-front wheel-pedal assembly, a seat and frame unit, and rear wheels. The company has orders for 12,000 of these trikes.
As indicated in the table below, the company obviously does not have the resources available to manufacture everything needed for the completion of 12000 tricycles, so it has arranged to purchase additional components, as necessary.
Develop a linear programming model to tell the company how many of each component should be manufactured and how many should be purchased in order to provide 12000 fully completed tricycles at the minimum cost.
(Hints: (1) there are three components, each of which can either be manufactured or purchased – this tells you how many decision variables there are. (2) There are three resources that are utilized when components are produced (not when they are purchased), which determines the set of resource constraints. (3) finally, we need constraints to ensure that we have adequate supplies of each of the three components, and as indicated, each component can be purchased or manufactured. Bear in mind that it may be cost effective to manufacture different percentages of each component….
I need help with the attached problem. I took a shot at part of it, cannot finish the rest and am not sure if I even got that part correct. Please assist!
Claims company processes insurance claims, their perm operators can process 16 claims/day and temp process 12/day and the average for the company is at least 450/day. They want to limit claims error to 25 per day total, and the perm generate .5 errors/day and temp generate 1.4 error per day. The perm operators are paid $465/day and temp/$42/day. Need to determine the number of permanent and temp operators to hire to minimize costs:
1. Need to formulate a linear programming model for this problem
2. use graphical analysis to solve this model.
The production manager for the Whoppy soft drink company is considering the production of 2 kinds of soft drinks: regular and diet. The company operates one “8 hour” shift per day. Therefore, the production time is 480 minutes per day. During the production process, one of the main ingredients, syrup is limited to maximum production capacity of 675 gallons per day. Production of a regular case requires 2 minutes and 5 gallons of syrup, while production of a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case. What is the optimal daily profit?
a. $220
b. $270
c. $320
d. $420
e. $520
Mallory furniture buys 2 products for resale: big shelves (B) and medium shelves (M). Each big shelf costs $500 and requires 100 cubic feet of storage space, and each medium shelf costs $300 and requires 90 cubic feet of storage space. The company has $75,000 to invest in shelves this week, and the warehouse has 18,000 cubic feet available for storage. Profit for each big shelf is $300 and for each medium shelf is $150. What is the objective function?
a. Z = $300B + $150M
b. Z = $300M + $150B
c. Z = $300B + $500M
d. Z = $500B + $300M
Mallory furniture buys 2 products for resale: big shelves (B) and medium shelves (M). Each big shelf costs $500 and requires 100 cubic feet of storage space, and each medium shelf costs $300 and requires 90 cubic feet of storage space. The company has $75000 to invest in shelves this week, and the warehouse has 18000 cubic feet available for storage. Profit for each big shelf is $300 and for each medium shelf is $150. What is the storage space constraint?
a. 90B + 100M 18000
b. 100B + 90M ≤ 18000
c. 90B + 100M 18000
c. 500B + 300M 18000
d. 300B + 500M 18000
Mallory furniture buys 2 products for resale: big shelves (b) and medium shelves (m). Each big shelf costs $500 and requires 100 cubic feet of storage space, and each medium shelf costs $300 and requires 90 cubic feet of storage space. The company has $75000 to invest in shelves this week, and the warehouse has 18000 cubic feet available for storage. Profit for each big shelf is $300 and for each medium shelf is $150. What is the weekly maximum profit?
a. $25000
b. $35000
c. $45000
d. $55000
e. $65000
For a linear programming problem, assume that a given resource has not been fully used. In other words, the slack value associated with the resource constraint is positive. We can conclude that the shadow price associated with that constraint:
a. will have a positive value
b. will have a negative value
c. will have a value of zero
d. could have a positive, negative or a value of zero. (no sign restrictions)
For a maximization problem, assume that a constraint is binding. If the original amount of a resource is 4 lbs., and the range of feasibility (sensitivity range) for this constraint is from
3 lbs. to 6 lbs., increasing the amount of this resource by 1 lb. will result in the:
a. same product mix, different total profit
b. different product mix, same total profit as before
c. same product mix, same total profit
d. different product mix, different total profit
The production manager for Beer etc. produces 2 kinds of beer: light (L) and dark (D). Two resources used to produce beer are malt and wheat. He can obtain at most 4800 oz of malt per week and at most 3200 oz of wheat per week respectively. Each bottle of light beer requires 12 oz of malt and 4 oz of wheat, while a bottle of dark beer uses 8 oz of malt and 8 oz of wheat. Profits for light beer are $2 per bottle, and profits for dark beer are $1 per bottle. If the production manager decides to produce of 0 bottles of light beer and 400 bottles of dark beer, it will result in slack of
a. malt only
b. wheat only
c. both malt and wheat
d. neither malt nor wheat
Mallory Furniture buys 2 products for resale: big shelves (B) and medium shelves (M). Each big shelf costs $500 and requires 100 cubic feet of storage space, and each medium shelf costs $300 and requires 90 cubic feet of storage space. The company has $75000 to invest in shelves this week, and the warehouse has 18000 cubic feet available for storage. Profit for each big shelf is $300 and for each medium shelf is $150. Which of the following is not a feasible purchase combination?
a. 0 big shelves and 200 medium shelves
b. 0 big shelves and 0 medium shelves
c. 150 big shelves and 0 medium shelves
d. 100 big shelves and 100 medium shelves
Formulate and solve the following problem. Use method of your choice.
The Marketing Club at your college has decided to raise funds by selling three types of T-shirts: one with single-color “ordinary” design, one with a two-color “fancy” design, and one with a three-color “very fancy” design. The club feels that it can sell up to 300 T-shirts. “Ordinary” T-shirts will cost the club $6 each, “fancy” T-shirts $8 each, and “very fancy” T-shirts $10 each, and the club has a total purchasing budget of $3,000. It will sell “ordinary” T-shirts at profit of $4 each, “fancy” T-shirts at profit of $5 each, and “very fancy” T-shirts at a profit of $4 each. How many of each kind of T-shirt should the club order to maximize profit?
(See attached file for full problem description)
—
Problem 5
The Southern Sporting Goods Company makes basketballs and footballs. Each product is produced from two resources- rubber and leather. The resource requirements for each product and the total resources available are as follows.
Resource Requirements per Unit
Product Rubber (lb) Leather ft^2)
Basketball 3 4
Football 2 5
Total resources available 500 lb 800 ft^2
Each basketball produced results in a profit of $12, and each football earns $16 in profit.
A. Formulate a linear programming model to determine the number of basketballs, and footballs to produce in order to maximize profit.
B. Transform this model into standard form.
1. Problem 6 –
For this problem, you will need to do problem 5. The solution is provided below.
8. Consider the following minimization problem.
Min z = x1 + 2×2
s.t. x1 + x2 300
2×1 + x2 400
2×1 + 5×2 750
x1, x2 0
Which constraints are satisfied at the optimal solution (x1 = 250, x2 = 50)?
9. Consider the following minimization problem.
Min z = 1.5×1 + 2×2
s.t. x1 + x2 300
2×1 + x2 400
2×1 + 5×2 750
x1, x2 0
What are the optimal values of x1, x2, and z ?
10. Consider the following linear programming problem:
Max Z = $15x + $20y
Subject to : 8x + 5y 40
0.4x + y 4
x, y
Determine the values for x and y that will maximize revenue. Given this optimal revenue, what is the amount of slack associated with the first constraint?
A business dedicates to the import and sale of worn out coffee. The company concerns three types of coffee of select quality.: Colombian coffee, Honduran coffee and Dominican coffee. The company prepares three different mixtures in which it uses those three types of coffee. These mixtures are sold in bags of one pound. The three mixtures are:
Mix Mix Composition Price
Great Select At least 35% of Colombian coffee and at least 25% of Honduran coffee $8.75/lb
Select At least 60% of Honduran coffee 7.25/lb
Regular No more than 60% of Dominican coffee and at least 25% of Colombian coffee 6.25lb
The company counts with the amounts of coffee of each type that occur ahead, reason why has paid the prices indicated by pound:
Type of Coffee Quantity Available Cost/lb
Colombian 105 lb 3.25
Honduran 180 lb 3.00
Dominican 130lb 2.75
The company wishes to determine the amount of pounds that need to prepare of each one of the three mixtures to maximize its gain, under the assumption that he could sell the totality of each prepared mixture.
Formulates the model of lineal programming to solve that situation.
1. A company must meet on time the following demands: quarter 1, 3000 units; quarter 2, 2000 units; quarter 3, 4000 units. Each quarter, up to 2700 units can be produced with regular-time labor, at a cost of $40 per unit. During each quarter, an unlimited number of units can be made with overtime labor, at a cost of $60 per unit. Of all units produced, 20% are unsuitable for sale and cannot be used for demand. Also, at the end of each quarter, 10% of all units on hand spoil and cannot be used to meet any future demands. After each quarterâ??s demand is satisfied and spoilage is accounted for, a cost of $15 per unit is assessed against the quarterâ??s ending inventory. Assume 1000 units are available initially.
a. Formulate this model
b. Solve the model you have formulated using Solver. Report the important decisions that the company must make. How much money will the solution cost the company?
Harvard Business Case:
9-189-163
April 18, 1990
Anirudh Dhebar
Merton Truck Company
Question 1, parts a – d are addressed in the response.
Please formulate the constraints for this problem as well as solutions to parts c and d.
—
Round Tree Manor is a hotel that has two types of rooms with three rental classes: Super Saver, Deluxe, and Business. The profit contribution per night for each type of room and rental class is as follows:
(see attached file for chart)
Type I rooms do not have Internet access and are not available for the Business rental class. Round Tree’s management makes a forecast of the demand by rental class for each night in the future. A linear programming model developed to maximize profit could be used to determine the number of reservations to accept for each rental class. The demand forecast for a particular night is 130 rentals in the Super Saver class, 60 rentals in the Deluxe class, and 50 rentals in the Business class. Round Tree has 100 Type I rooms and 120 Type II rooms.
a. Formulate a linear programming model that can be used to determine the number of reservations to accept in each rental class and how these reservations should be allocated to room types.
b. Determine the number of reservations that can be accommodated in each rental class and whether the demand by any rental class is not satisfied.
c. Management is considering offering a free breakfast to anyone upgrading from a Super Saver reservation to Deluxe class. If the cost to Round Tree Manor of providing the breakfast is $5, determine if this incentive should be offered.
d. With a little work, an unused office area could be converted to a rental room. Assuming that the conversion cost is the same for both types of rooms, provide a recommendation and explanation for converting the office to a Type I or a Type II room.
—
Innis Investments manages funds for a number of companies and wealthy clients. The investment strategy is tailored to each clientâ??s needs. For a new client, Innis has been authorized to invest up to $1.2 million in two investment funds: a stock fund and a money market fund. Each unit of the stock fund costs $50 and provides an annual rate of return of 10%; each unit of the money market fund costs $100 and provides an annual rate of return of 4%.
The client wants to minimize risk subject to the requirement that the annual income from the investment be at least $60,000. According to Innisâ??s risk measurement system, each unit invested in the stock fund has a risk index of 8, and each unit invested in the money market fund has a risk index of 3; the higher risk index associated with the stock fund simply indicates that it is the riskier investment. Innisâ??s client has also specified that at least $300,000 be invested in the money market fund.
Letting S = units purchased in the stock fund
M = units purchased in the money market fund
leads to the following formulation:
Min z = 8S + 3M
Subject to:
50S + 100M â?¤ 1,200,000 Funds available
5S + 4M > 60,000 Annual income
M â?¥ 3,000 Units in money market
S, M â?¥ 0
a. Determine how many units of each fund Innis should purchase for the client to minimize the total risk index for the portfolio.
b. How much annual income will this investment strategy generate.
c. Suppose the client desires to maximize annual return, How should the funds be invested.
***please show all work and output of any programming used*****
What are the conditions causing linear programming problems to have multiple solutions?
The Heinlien and Krampf Brokerage has just been instructed by one its clients to invest $250,000 for her money obtained recently through the sale of land holdings in Ohio. The client has a good deal of trust in the investment house, but also has her own ideas about the distribution of the funds being invested. In particular, she requests that the firm select whatever stocks and bonds they believe are wel rated, but within the following guidelines:
(a) Municipals bonds should constitute at least 20% of the investment.
(b) At least 40% of the funds should be placed in a combination of electronic firms, aerospace firms, and drug manufacturers.
(c) No more than 50% of the amount invested in municpal bonds should be placed in a high-risk high-yield nursing home stock.
Subject to this restraints, the client’s goal is to maximize project return of investments. The analysts at Heinlien and Krampf, aware of these guidelines, prepare a list of high-quality stocks and their corresponding rates of return:
Los Angeles municipal bonds – 5.3%
Thompson Electronics Inc – 6.8%
United Aerospace Corp – 4.9%
Palmer Drugs – 8.4%
Happy Days Nursing Homes – 11.8%
1. Formulate this portfolio selection problem using Linear Programming.
2. Solve this problem.
See attached file.
Please use Excel.
Following please find two problems for which I need answers. Use Excel in formulating your linear calculations. Please also show your work in detail and explain each step of each problem.
1) Problem #1 Atlantic Seafood Company (20 points)
The Atlantic Seafood Company (ASC) is a buyer and distributor of seafood products that are sold to restaurants and specialty seafood outlets throughout the Northeast. ASC has a frozen storage facility in New York City that serves as the primary distribution point for all products. One of the ASC products is frozen large black tiger shrimp, which are sized at 16-20 pieces per pound. Each Saturday ASC can purchase more tiger shrimp or sell the tiger shrimp at the existing New York City warehouse market price. The ASC goal is to buy tiger shrimp at a low weekly price and sell it later at a higher price. ASC currently has 20,000 pounds of tiger shrimp in storage. Space is available to store a maximum of 100,000 pounds of tiger shrimp each week. In addition, ASC developed the following estimates of tiger shrimp prices for the next four weeks:
Week Price/Ib.
1 $6.00
2 $6.20
3 $6.65
4 $5.55
ASC would like to determine the optimal buying-storing-selling strategy for the next four weeks. The cost to store a pound of shrimp for one week is $0.15, and to account for unforeseen changes in supply or demand, management also indicated that 25,000 pounds of tiger shrimp must be in storage at the end of week 4.
Determine the optimal buying-storing-selling strategy for ASC including the projected four-week profit.
2) EZ-Windows, Inc., manufactures replacement windows for the home remodeling business. In January, the company produced 15,000 windows and ended the month with 9000 windows in inventory. EZ-Windows management team would like to develop a production schedule for the next three months. A smooth production schedule is obviously desirable because it maintains the current workforce and provides a similar month-to-month operation. However, given the sales forecasts, the production capacities, and the storage capabilities as shown, the management team does not think a smooth production schedule with the same production quantity each month possible.
February March April
Sales forecast 15,000 16,500 20,000
Production capacity 14,000 14,000 18,000
Storage capacity 6,000 6,000 6,000
The company’s cost accounting department estimates that increasing production by one window from one month to the next will increase total cost by $1.00 for each unit increase in the production level. In addition, decreasing production by one unit from one month to the next will increase total costs by $0.65 for each unit decrease in the production level.
Ignoring production and inventory carrying costs, formulate and solve a linear programming model that will minimize the cost of changing production levels while still satisfying the monthly sales forecasts.
1. An investor is considering three types of investments: a high-risk venture into oil leases with a potential return of 15%, a medium-risk investment in stocks with a 9% return, and a relatively safe bond investmentwiht a 5% return. He has $50,000 to invest. Because of the risk, he will limit his investment in oil lease and stocks to 30% and his investment in oil leases and bonds to 50%. How much should be invest in each to maximize his return, assuming investment returns are as expected?
Provide an appropriate response.
1) Explain the result if the simplex tableau is solved using a quotient other than the smallest non-nega five quotient.
2) Explain why a different slack variable must be used for each constraint when converting
constraints to linear equations.
3) When would the simplex method be used instead of the graphical method?
4) Each solution of a simplex tableau corresponds to
5) What happens if an indicator other than the most negative one is chosen to solve a simplex tableau?
6) A negative number in the rightmost column of a simplex tableau tells you that you have made what kind of error when pivoting?
7) No unique optimtum solution found from a simplex tableau corresponds to
8) You are given the following linear programming problem (P):
Minimize
subject to:
zx1+x2
?4xi +4x2s1
xl -3x2s2
xl O,x2O
1)
2)
3)
4)
5)
6)
7)
8)
The dual of (P) is (D). Which of the following statements are true? a. (P) has no feasible solution and the objective ftmction of (D) is unbounded. b. (D) has no feasible solution and the objective function of (P) is unbounded. Cl The objective functions of both (1′) and (D) are unbounded.
d. Both (P) and(D) have optimal soJutions.
e. Neither (P) nor (D) has feasible solutions.
MULTIPLE CHOJCE.
Choose the one alternative that best completes the statement or answers the question.
The initial tableau of a linear programming problem is given. Use the simplex method to solve the problem.
9)
X1X2 X3 12
001
A)Maximumat9forxi 8,x2=2
C) Maximum at 32 for xz = 8, sj =2
B)Maximumat18forx2=8,x32
D) Maximum at 36 for x ? 2, si = 8
Based on your previous communication, attached please find a problem that I need to solve using linear programming.
Please explain, step by step, how to resolve the problem with linear programming including the variables, the min. or max formula as well as any and all constraints so that I can, thereafter, input said information into Scientific Management Software. In providing your response, please do not use Lindo. Word and/or Excel are fine.
Problem 3: Linear Programming
I have a six acre farm that I can use for the productionof rice (X1) or corn (X2). I have also
27 days of labor and 32 tons of fertilizer.
One acre of rice requires three days of labor and eight tons of fertilizer, while one acre of corn
requires nine days of labor and four tons of fertilizer.
One acre of rice produces 50 tons which can be sold for $4.00 per ton. One ton of corn produces 100 tons ton. One acre of corn produces 20 tons
which can be sold for $10 per ton.
Questions 7. state mathematically the objective function
8. state mathematically the constraint function
9. How many acres of rice and corn do you plant to maximize your profit
10. What is the maximum profit
You can use the graphic analysis or the linear programming software in the management scientist.
See attached
For all linear programming problems, the implied non-negativity constraint is assumed. Don’t forget to include this constraint if you are using Excel to solve any of these problems.
1. Linear Programming Properties
Which of the following statements is not true?
a) An infeasible solution violates all constraints.
b) A feasible solution point does not have to lie on the boundary of the feasible solution.
c) A feasible solution satisfies all constraints.
d) An optimal solution satisfies all constraints.
Answer: _____
2. Minimization Graphical Solution
Solve the following linear model graphically and select the set of extreme points that make up the possible feasible solutions.
a) (x1=12, x2=0, z=120), (x1=6, x2=5, x3=160), (x1=0, x2=8, z=160)
b) (x1=0, x2=12, z=240), (x1=6, x2=5, x3=160), (x1=20, x2=0, z=200)
c) (x1=0, x2=12, z=240), (x1=20/3, x2=16/3, x3=520/3), (x1=20, x2=0, z=200)
d) (x1=12, x2=0, z=120), (x1=20/3, x2=16/3, x3=520/3), (x1=0, x2=8, z=160)
The Kodiak Oil Company owns a pipeline network that is used to convey oil from its source (Node 1) to several storage locations. The network flow is shown on the attachment. Due to varying pipe sizes, the flow capacities vary (shown below in â??000s of gallons per hour). By selectively opening and closing sections of the pipeline network, the firm can supply any of the storage locations.
a. If the firm wants to fully utilize system capacity to supply storage location 7, how
long will it take to satisfy a location 7 demand of 1 million gallons?
b. What is the impact on the time to meet location 7 demand if pipeline 3-6 gradually reduces its flow to a maximum of 5000, 4000, and finally only 3000 gallons per hour?
Evaluate all parts using Excel Solver including the three scenarios in part b.
NOTE: The network flow diagram is attached
AntiFam, a hunger-relief organization has earmarked between $2 and $2.5 million (inclusive) for aid to two African countries, country A and country B. Country A is to receive between $1 and 1.5 million (inclusive) and country B is to receive at least $0.75 million. It has been estimated that each dollar spent in country A will yield an effective return of $.60, where a dollar spent in country B will yield an effective return of $.80. How should the aid be allocated if the money is to be utilized most effectively according to the these criteria.
if x and y denote the amount of money to be given to country A and country B, respectively, then the objective function to be maximized is P=.6x +.8y.
a Identify your variables.
b Set up the objective and constraints.
c Graph the feasible region.
d Determine the corner points. Show the algebra if solving for the intersection of lines.
e Determine the optimal solution and report you results in terms of the names of the variables.
We are given the following linear programming problem:
Mallory furniture buys 2 products for resale: big shelves (B) and medium shelves (M). Each big shelf costs $500 and requires 100 cubic feet of storage space, and each medium shelf costs $300 and requires 90 cubic feet of storage space. The company has $75000 to invest in shelves this week, and the warehouse has 18000 cubic feet available for storage. Profit for each big shelf is $300 and for each medium shelf is $200.
The linear programming formulation is
Max 300B + 200M
Subject to
500B + 300 M < 75000
100B + 90M < 18000
B, M > 0
I have solved the problem by using QM for Windows and the output is given below. I don’t know what it mean though. Can you help me with this. I only need to answer one of the question below. Can you tell me how to go about figuring that information out?
The Original Problem w/answers:
B M RHS Dual
Maximize 300 200
Cost Constraint 500 300 <= 75,000 .4667
Storage Space Constraint 100 90 <= 18,000 .6667
Solution-> 90 100 Optimal Z-> 47,000
Ranging Result:
Variable Value Reduced Cost Original Val Lower Bound Upper Bound
B 90. 0 300. 222.22 333.33
M 100. 0 200. 180. 270.
Constraint Dual Value Slack/Surplus Original Val Lower Bound Upper Bound
Cost Constraint 0.4667 0 75000 60000 90000
Storage Space Constraint 0.6667 0 18000 15000 22500
1. Determine and interpret the optimal solution and optimal objective function value from the output given above.
2. Find the range of optimality for the profit contribution of a big shelf from the output given above and interpret its meaning.
3. Find the range of optimality for the profit contribution of a medium shelf from the output given above and interpret its meaning.
4. Find the range of feasibility for the right hand side value (availability) of money constraint from the output given above and interpret its meaning.
5. Find the range of feasibility for the right hand side value (availability) of storage space constraint from the output given above and interpret its meaning.
6. Determine and interpret the shadow (dual) prices of the two resources.
Please find the problem statement in the attachment.
A company producing a standard line and a deluxe line of dishwasher has the following time requirements (in minutes) in departments where either model can be processed
Standard Deluxe
Stamping 3 6
Motor Installation 10 10
Wiring 10 15
The standard mode contributes $20 each and the deluxe $30 each to profits. Because the company produces other items that shares resources used to make the dishwasher, the stamping machine is available only 30 minutes per hour, on average. The motor installation production line has 60 minutes available each hour. There are two lines for wiring, so the time available is 90 minutes per hour.
Let X = Number of standard dishwashers produced per hour
Y = Number of deluxe dishwasher produced per hour
Write the formulation for this linear Program.
Valley Fruit Products Company has contracted with apple growers in Ohio, Pennsylvania, and New York to purchase apples that the company then ships to its plants in Indiana and Georgia, where they are processed into apple juice. Each bushel of apples produces 2 gallons of apple juice. The juice is canned and bottled at the plants and shipped by rail and truck to warehouse/ distribution centers in Virginia, Kentucky, and Louisiana. The shipping cost per bushel from the farms to the plants and shipping costs per gallon from the plants to the distribution centers are summarized in the following tables:
Plant
Farm 4. Indiana 5. Georgia Supply (bushels)
1. Ohio .41 .57 24,000
2. Pennsylvania .37 .48 18,000
3. New York .51 .60 32,000
Plant Capacity 48,000 35,000
Distribution Center
Plant 6. Virginia 7. Kentucky 8. Louisiana
4. Indiana .22 .10 .20
5. Georgia .15 .16 .18
Demand (gal) 9,000 12,000 15,000
Formulate and solve a linear programming model to determine the optimal shipments from the farms to the plants and from the plants to the distribution centers in order to minimize total shipping costs.
Please include what program is used to solve this.
Using Excel, please solve the following:
A company produces two products that are processed on two assembly lines. Assembly line 1 has 100 available hours, and assembly line 2 has 42 available hours. Each product requires 10 hours of processing time on line 1, while on line 2 product 1 requires 7 hours and product 2 requires 3 hours. The profit for product 1 is $6 per unit, and the profit for product 2 is $4 per unit.
Formulate a linear programming model for this problem.
1.
Which of the following mathematical relationships could be found in a linear programming model? And which could not (why)?
a. -1A + 2B ≤ 70
b. 2A – 2B = 50
c. 1A – 2B^2 ≤ 10
d. 3 √ A + 2B ≥ 15
e. 1A + 1B = 6
f. 2A + 5B + 1AB ≤ 25
2.
Find the solutions that satisfy the following constraints:
a. 4A + 2B ≤ 16
b. 4A + 2B ≥ 16
c. 4A + 2B = 16
Please see the attached file for the complete problem.
AutoIgnite produces electronic ignition systems for automobiles at a plant in Cleveland, Ohio. Each ignition system is assembled from two components produced at AutoIgnite’s plants in Buffalo, New York, and Dayton, Ohio. The Buffalo plant can produce 2000 units of component 1, 1000 units of component 2, or any combination of the two components
each day. For instance, 60% of Buffalo’s production time could be used to produce component 1 and 40% of Buffalo’s production time could be used to produce component 2; in this case, the Buffalo plant would be able to produce 0.6(2000) = 1200 units of component 1 each day and 0.4(1000) = 400 units of component 2 each day. The Dayton plant can produce 600 units of component 1, 1400 units of component 2, or any combination of the two components each day. At the end of each day, the component production at Buffalo and Dayton is sent to Cleveland for assembly of the ignition systems on the following work day.
a. Formulate a linear programming model that can be used to develop a daily production schedule for the Buffalo and Dayton plants that will maximize daily production of ignition systems at Cleveland.
b. Find the optimal solution.
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