Here’s a little help with Week 2 homework; The first three problems are from Tuesday’s Session (3). See the Excel examples on tabs 5,6,7. Remember you can delete middle rows and columns from the examples to fit these problems. Do make sure they are from the middle of the array, not the ends.
First is a Make/Buy problem. It is a 6 by 6 problem. See pages 153-156. Use SOLVER here.
Second is a Production Scheduling problem. It is a 10 by 12 problem. See pages 157-163. us SOLVER here.
Third is a product mix. It is a 3 by 3 problem. problem. See pages 163-164. Write Equations only.
Instructions
The way that you write your answers matters! Syntax matters! Before you take this homework, be sure that you have successfully completed the “Getting Started in Web Assign for Mathematics” quiz this week. This quiz provides 2 bonus points that can be used during the course, and, most importantly, makes it clear how to provide answers that the homework and quizzes understand.
Description
Assignment Submission & Scoring
Assignment Submission
For this assignment, you submit answers by question parts. The number of submissions remaining for each question part only changes if you submit or change the answer.
Assignment Scoring
Your last submission is used for your score.
1.
A linear programming computer package is needed.
Frandec Company manufactures, assembles, and rebuilds material-handling equipment used in warehouses and distribution centers. One product, called a Liftmaster, is assembled from four components: a frame, a motor, two supports, and a metal strap. Frandec’s production schedule calls for 5,500 Liftmasters to be made next month. Frandec purchases the motors from an outside supplier, but the frames, supports, and straps may be either manufactured by the company or purchased from an outside supplier. Manufacturing and purchase costs per unit are shown.
Component Manufacturing Cost Purchase Cost
Frame $39.00 $52.00
Support $12.50 $16.00
Strap $7.50 $8.50
Three departments are involved in the production of these components. The time (in minutes per unit) required to process each component in each department and the available capacity (in hours) for the three departments are as follows.
Component Department
Cutting Milling Shaping
Frame 3.5 2.2 3.1
Support 1.3 1.7 2.6
Strap 0.8 — 1.7
Capacity (hours) 350 420 680
(a)
Formulate and solve a linear programming model for this make-or-buy application. (Let FM = number of frames manufactured, FP = number of frames purchased, SM = number of supports manufactured, SP = number of supports purchased, TM = number of straps manufactured, and TP = number of straps purchased. Express time in minutes per unit.)
Min
Cutting constraint
Milling constraint
Shaping constraint
Frame constraint
Support constraint
Strap constraint
FM, FP, SM, SP, TM, TP ≥ 0
How many of each component should be manufactured and how many should be purchased? (Round your answers to the nearest whole number.)
(FM, FP, SM, SP, TM, TP) =
(b)
What is the total cost (in $) of the manufacturing and purchasing plan?
$
(c)
How many hours of production time are used in each department? (Round your answers to two decimal places.)
Cutting hrs
Milling hrs
Shaping hrs
(d)
How much (in $) should Frandec be willing to pay for an additional hour of time in the shaping department?
$
(e)
Another manufacturer has offered to sell frames to Frandec for $45 each. Could Frandec improve its position by pursuing this opportunity? Why or why not? (Round your answer to three decimal places.)
. The reduced cost of indicates that the solution be improved.
2.
A linear programming computer package is needed.
EZ-Windows, Inc., manufactures replacement windows for the home remodeling business. In January, the company produced 15,000 windows and ended the month with 9,000 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 costs 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 a linear programming model that will minimize the cost of changing production levels while still satisfying the monthly sales forecasts. (Let F = number of windows manufactured in February, M = number of windows manufactured in March, A = number of windows manufactured in April, I1 = increase in production level necessary during month 1, I2 = increase in production level necessary during month 2, I3 = increase in production level necessary during month 3, D1 = decrease in production level necessary during month 1, D2 = decrease in production level necessary during month 2, D3 = decrease in production level necessary during month 3, s1 = ending inventory in month 1, s2 = ending inventory in month 2, and s3 = ending inventory in month 3.)
Min
s.t.February Demand
March Demand
April Demand
Change in February Production
Change in March Production
Change in April Production
February Production Capacity
March Production Capacity
April Production Capacity
February Storage Capacity
March Storage Capacity
April Storage Capacity
Find the optimal solution.
(F, M, A, I1, I2, I3, D1, D2, D3, s1, s2, s3) =
Cost = $
________________________________________
o
________________________________________
________________________________________
o
3.
The Ace Manufacturing Company has orders for three similar products.
Product Orders
(units)
AA 1,800
BB 400
CC 1,400
Three machines are available for the manufacturing operations. All three machines can produce all the products at the same production rate. However, due to varying defect percentages of each product on each machine, the unit costs of the products vary depending on the machine used. Machine capacities for the next week and the unit costs are shown below.
Machine Capacity
(units)
11 1,700
22 1,400
33 800
Machine
Product 1 2 3
A $1.00 $1.30 $1.10
B $1.20 $1.40 $1.00
C $0.90 $1.20 $1.20
Use the transportation model to develop the minimum cost production schedule for the products and machines.
(a)
Show the linear programming formulation. (Let xA1 be the number of units of product A produced by machine 1,
xij
be the number of units of product i produced by machine j, etc.)
Min
s.t.Machine 1 Capacity
Machine 2 Capacity
Machine 3 Capacity
Product A Orders
Product B Orders
Product C Orders
xij ≥ 0 for all i, j.
(b)
Show the production schedule.
(xA1, xA2, xA3, xB1, xB2, xB3, xC1, xC2, xC3) =
(c)
Determine the cost (in dollars) of the production schedule.
Total = $
4.
Scott and Associates, Inc., is an accounting firm that has three new clients. Project leaders will be assigned to the three clients. Based on the different backgrounds and experiences of the leaders, the various leader-client assignments differ in terms of projected completion times. The possible assignments and the estimated completion times in days are as follows.
Project Leader Client
1 2 3
Jackson1 10 16 32
Ellis2 14 22 40
Smith3 22 24 34
(a)
Develop a network representation of this problem. (Submit a file with a maximum size of 1 MB.)
This answer has not been graded yet.
(b)
Formulate the problem as a linear program. (Express your answers in the form xij where xij represents the completion time from leader i to client j.)
Min
s.t.Jackson
Ellis
Smith
Client 1
Client 2
Client 3
xij ≥ 0 for all i, j.
Solve.
Relationship Assignment Time (days)
x11
x12
x13
x21
x22
x23
x31
x32
x33
What is the total time required (in days)?
days
5.
Consider the following network representation of a transportation problem.
A network diagram between five locations is shown. The left-hand side of the graph is labeled “Supplies” and lists Jefferson City and Omaha. The right-hand side of the graph is labeled “Demands” and lists Des Moines, Kansas City, and St. Louis. Lines are shown between various locations. The following list contains the numbers placed on the graph.
• Jefferson City: 45
• Omaha: 40
• Des Moines: 20
• Kansas City: 15
• St. Louis: 50
• Jefferson City–Des Moines: 16
• Jefferson City–Kansas City: 9
• Jefferson City–St. Louis: 8
• Omaha–Des Moines: 8
• Omaha–Kansas City: 10
• Omaha–St. Louis: 23
The supplies, demands, and transportation costs per unit are shown on the network.
(a)
Develop a linear programming model for this problem; be sure to define the variables in your model.
Let
• x11 = amount shipped from Jefferson City to Des Moines
• x12 = amount shipped from Jefferson City to Kansas City
• x13 = amount shipped from Jefferson City to St. Louis
• x21 = amount shipped from Omaha to Des Moines
• x22 = amount shipped from Omaha to Kansas City
• x23 = amount shipped from Omaha to St. Louis
Min x11
+ x12
+ x13
+ x21
+ x22
+ x23
s.t.
x11 + x12 + x13 ≤
x21 + x22 + x23 ≤
x11 + x21 =
x12 + x22 =
x13 + x23 =
x11, x12, x13, x21, x22, x23 ≥ 0
(b)
Solve the linear program to determine the optimal solution.
Amount Cost
Jefferson City–Des Moines
Jefferson City–Kansas City
Jefferson City–St. Louis
Omaha–Des Moines
Omaha–Kansas City
Omaha–St. Louis
Total
6.
Cleveland Area Rapid Delivery (CARD) operates a delivery service in the Cleveland metropolitan area. Most of CARD’s business involves rapid delivery of documents and parcels between offices during the business day. CARD promotes its ability to make fast and on-time deliveries anywhere in the metropolitan area. When a customer calls with a delivery request, CARD quotes a guaranteed delivery time. The following network shows the street routes available. The numbers above each arc indicate the travel time in minutes between the two locations.
A network diagram between six parties, distributed across four columns, is shown. The left-most column contains node 1. The second column contains nodes 2 and 3. The third column contains nodes 4 and 5. The right-most column contains node 6. Lines are shown between various locations. The following list contains the numbers placed on the graph.
• 1–2: 35
• 1–3: 30
• 2–1: 35
• 2–3: 12
• 2–4: 18
• 2–6: 39
• 3–1: 30
• 3–2: 12
• 3–5: 15
• 4–2: 18
• 4–5: 12
• 4–6: 16
• 5–3: 15
• 5–4: 12
• 5–6: 30
• 6–2: 39
• 6–4: 16
• 6–5: 30
(a)
Develop a linear programming model that can be used to find the minimum time required to make a delivery from location 1 to location 6. (Express your answers in the form xij, where each xij represents the arc from node i to node j as either 1 or 0.)
Min
s.t.Node 1 Flows
Node 2 Flows
Node 3 Flows
Node 4 Flows
Node 5 Flows
Node 6 Flows
For all xij = 0, 1.
(b)
How long (in minutes) does it take to make a delivery from location 1 to location 6? (Round your answer to the nearest whole number.)
min
(c)
Assume that it is now 1:00 p.m. and that CARD just received a request for a pickup at location 1. The closest CARD courier is 8 minutes away from location 1. If CARD provides a 20% safety margin in guaranteeing a delivery time, what is the minimum guaranteed delivery time if the package picked up at location 1 is to be delivered to location 6? (Enter your answer in standard time. Round your answer to the nearest minute.)
