Essay A Steel Manufacturer Produces Four Sizes of I-Beams - Management Assignment Help

Assignment Task:

Task:
Problem #1
A steel manufacturer produces four sizes of I-beams: small, medium, large, and extra-large. These beams can be produced on any one of the three machine types: A, B, C. The length (in feet) of the I-beams that can be produced on the machines per hour is summarized.
BEAM MACHINE
A B C
Small 300 600 800
Medium 250 400 700
Large 200 350 600
Extra-large 100 200 300
Assume that each machine can be used up to 50 hours/week, and that the hourly operating costs of these machines are $30, $50, and $80, respectively. Further suppose that 10000, 8000, 6000, and 6000 feet of the different size I-beams are respectively required weekly.
Formulate this machine scheduling problem as a linear program to minimize operating costs.
Solve your formulation from part (a) and determine the optimal solution.
Problem #2
A trucking company with $2,000,000 to spend on new equipment is contemplating purchasing three types of vehicles: A, B, and C. Vehicle A has a 10-ton payload and is expected to cover 55 mi/hr, and costs $40,000. Vehicle B has a 20-ton payload and is expected to cover 50 mi/hr, and costs $65000. Vehicle C is a modified form of B; it carries sleeping quarters for one driver, and this reduces its capacity to 18-tons and raises the cost to $75000.
Vehicle A requires a crew of one man, and if driven on three shifts per day, could be run for an average of 18 hours/day. Vehicles B and C require a crew of two men each, but whereas B can be driven 18 hours/day with three shifts, C can be run an average of 21 hours/day. The company has 150 drivers available each day and would find it difficult to obtain further crews. Maintenance facilities are such that the total number of vehicles must not exceed 30. Formulate a linear program and solve to determine how many vehicles of each type should be purchased if the company wishes to maximize its capacity in ton-mi per day.

Problem #3
Larry Edison is the director of the Computer Center at Buckley College. He now needs to schedule the staffing of the center. The center is open from 8:00 a.m. until midnight. Based on previous usage of the center, Larry has determined that the following number of computer consultants are required to monitor the center at various times of the day:
Time of Day Minimum Number of Consultants
Required to be on Duty
8:00 a.m. – Noon 4
Noon – 4:00 p.m. 8
4:00 p.m. – 8:00 p.m. 10
8:00 p.m. – Midnight 6
Two types of computer consultants can be hired: full-time and part-time. Full-time consultants work for 8 consecutive hours in any of the following shifts: morning (8:00 a.m. – 4:00 p.m.), afternoon (Noon – 8:00 p.m.), or evening (4:00 p.m. – Midnight), and are paid $14/hour. Part-time consultants can be hired to work in any of the four-hour shifts listed above, and they are paid $12/hour. An additional requirement is that there must be at least 2 full-time consultants on duty for every part-time consultant on duty. Determine the number of full-time and part-time consultants that must work each shift to meet the above demand at a minimum cost.
Problem #4
Solve the problem algebraically (without using MS Excel).

Rangaswamy owns 50 acres of land in the Hassan district of Karnataka on which he grows bananas and papayas. The total cost of producing bananas is Rs. 1,05,000 per acre while the total cost for papayas is Rs. 2,10,000 per acre. For the coming year Rangaswamy has a budget of Rs. 63,00,000. He does not want to use any more than 15 acres for bananas and plans on using at least 20 acres for papayas. The profit from each acre of bananas is Rs. 80,000 and from papayas is Rs. 1,40,000.
105b + 210p <= 6300 b + 2p <= 60 , b<=15 & p>=20 and b+p <= 50 , b = 50-p
Formulate a linear programming problem to help Rangaswamy decide how many acres of each fruit to produce in order to maximize his total profit. Clearly define all the decision variables, objective function, and constraints.
Use the graphical method to find the optimal solution and optimum objective function value.
How many acres of land should Rangaswamy plan on cultivating in the coming year?
It is expected that the profit per acre from bananas will drop in the coming year due to over-production of bananas in other regions. To what level can the profit per acre from bananas drop without affecting Rangaswamy’s optimal production plan? How many acres of bananas should he produce if the profit drops below this level?
Due to a shortage of papayas, the profit will increase to Rs. 1,60,000 per acre. What are the various production plans that will maximize Rangaswamy’s total profit?
Does Rangaswamy stand to make a higher profit if he relaxes the maximum limit of 15 acres that he plans on using for producing bananas? Why or why not? If yes, what would he gain per additional acre that he allows for producing bananas? For how many additional acres will this additional gain per acre be valid?
Problem #5
Deep Train, Inc., a major logistics operator, has been contracted to transport finished goods from three depots to retail outlets in four regions. The per unit shipping cost matrix from each depot to each region is provided below.
To
From ¯ Region 1 Region 2 Region 3 Region 4 Available supply
Depot 1 141.00 228.00 260.00 122.00 450
Depot 2 250.00 116.00 263.00 278.00 700
Depot 3 182.00 132.00 120.00 185.00 500
Required Demand 450 200 300 300 Formulate and solve the problem to determine the optimal shipping policy and the optimum logistics cost.
With a new expressway being built, the per unit transportation costs from Depot 3 to all regions reduces by 10%, i.e., the new costs have now become 0.9×C3j. Based on the sensitivity report from part (a), is the optimal transportation policy likely to change due to the reduction in costs?
Roadrunner, Inc. is a ‘small-time’ logistics company that wishes to make inroads into retail deliveries. The CEO of this company makes the following two offers to Deep Train Logistics, Inc.
Offer #1: Roadrunner offers to ship items to Region 1 at a price of INR 200/item, i.e., Deep Train will have to pay Roadrunner a per unit cost of INR 200 for every item that they deliver to Region 1.
Offer #2: Roadrunner will buy each item from Depot 3 at its cost price and, pay an additional fee of INR 70 for each item purchased to Deep Train. Thus, if the cost price of the item is INR 100, then Roadrunner will pay INR 170. (Roadrunner uses these purchased items to be delivered to a different customer.)
Is Offer #1 worthwhile? Why or why not? If yes, then how many items should Roadrunner be allowed to deliver to Region 1?
Is Offer #2 worthwhile? Why or Why not? If yes, then how many items should be ‘sold’ to Roadrunner from Depot 3?
Between the two offers, which one should Deep Train accept, if any? Why?
Problem #6
Ranchoddas Shamaldas Chanchad (Rancho) established CNG (Cleanliness is Next to Godliness), a manufacturing firm that specialized in manufacturing hand sanitizers in 2020. Sanitizers manufactured by CNG are made of three main ingredients: (i) Benzalkonium Chloride (BC); (ii) Ethyl alcohol (EA); and (iii) Isopropyl alcohol (IS). CNG makes three different varieties of hand sanitizers namely: Regular, Premium, and OCD. The ingredients used (in ml) to produce one litre of sanitizer is shown in the following table, along with demand for different sanitizers and availability of ingredients.
Sanitizer Ingredient in ml for 1 litre of Sanitizer Minimum Demand for Sanitizer (in litres) Profit from Sanitizer (Rs/litre)
BC EA IS Regular A11 600 200 75 40
Premium 100 300 450 80 60
OCD 100 400 A33 20 100
Availability of ingredients (in ml) 20000 80000 100000 The corresponding LP which maximizes profit subject to resource availability and demand constraints is formulated below:
Maximize 40×1 + 60×2 + 100x3subject to:
A11*x1 + 100 x2 + 100×3 ? 20000 (constraint for BC)
600×1 + 300×2 + 400×3 ? 80000 (constraint for EA)
200×1 + 450×2 + A33*x3? 100000 (constraint for IS)
x1 ? 75 (constraint for Regular Sanitizer)
x2 ? 80 (constraint for Premium Sanitizer)
x3 ? 20 (constraint for OCD Sanitizer)
xi ? 0 for i = 1, 2, and 3.
The Excel Solver sensitivity output is given below:
Variable Cells Final Reduced Objective Allowable Allowable
Cell Name Value Cost Coefficient Increase Decrease
$A$2 X1 75 0 40 110 1E+30
$B$2 X2 80 0 60 15 1E+30
$C$2 X3 27.5 0 100 1E+30 20
Constraints Final Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease
$A$4 BC 14500 20000 1E+30 5500
$A$5 EA 80000 0.25 80000 22000 3000
$A$6 IS 64750 100000 1E+30 35250
$A$7 Regular 75 75 5 55
$A$8 Premium 80 80 10 80
$A$9 OCD 27.5 20 7.5 1E+30
A11*75+ 100*80 + 100*27.5=14500A11 = (14500 – 8000 – 2750)/75 = 50
200*75 + 450*80 + A33*27.5=64750A33 = (64750 – 15000 – 36000)/27.5 = 500
Determine the values of A11 and A33 in the LP formulation. – 50 & 500
What is the shadow price for the Benzalkonium Chloride constraint? 0
Write the dual to the LP primal problem formulation provided above. Using the complementary slackness theorem, calculate the shadow price values of constraints, Regular and Premium.
Due to Covid-19, the profit earned from regular sanitizer has increased from 40 to 100, what will be the impact of this change on the optimal solution and the profit?
CNG was informed that one of the suppliers of Ethyl Alcohol can supply additional 20000 ml of EA. Should CNG buy this additional quantity of EA? What will be the impact of this additional quantity on the profit?
Due to increased demand for sanitizers, CNG has increased the price such that the profit of regular is 60, premium is 90 and OCD is 150. What will be the impact of this increase on the profit?
There is a demand for 5000ml BC in the open market at INR 2 per ml. Should CNG sell BC directly in the market?
CNG would like to introduce a new Luxury Sanitizer, which requires 200ml of BC, 500ml of EA and 400ml of IS. The profit from this new Sanitizer is 140. Should CNG introduce the new Sanitizer?

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