Application of LP in Manufacturing and Project Management - The Mathematical Models in MATLAB - Engineering Assignment Help

Assignment Task:

OBJECTIVE:
This project provides students an opportunity to apply the knowledge and problem solving skills obtained in LP to a practical optimisation problem.

DESCRIPTION:
14 LP problems are listed in this document. A group of three or two students can either select three problems among the 14 problems (and will be allocated one from the three), or identify a potential/actual LP application based on their prior work experience or from some other personal experience. Please form a group of three or two students and add your names in one of the groups in Canvas “People – Groups – Project groups”. Note some groups are with three people (e.g. Group project – three persons A) and some groups are with two people (e.g. Group project – two persons A), if your group has three person, then select a three person group, if your group has two person, then select a two person group). After forming the group, please select three options from the list and post it within your group as soon as possible but no later than Friday 21 August 11:59pm, with the following format:

TASKS and REQUIREMENTS:
(1) Understand the LP problems
(2) Develop the mathematical models
(3) Applying the required optimisation techniques to solve the problem
(4) Write and submit a project report as a group, including problem description, mathematical model, a brief explanation of the applied optimisation technique, results and discussions. The code you developed and/or the spreadsheets used should also be attached.

Problem 1: (Difficulty level: 2)
During the next four quarters Dorian Auto must meet (on time) the following demands for cars: 4000 in quarter 1; 2000 in quarter 2; 5000 in quarter 3; 1000 in quarter 4. At the beginning of quarter 1, there are 300 cars in stock. The company has the capacity to produce at the most 3000 cars per quarter. At the beginning of each quarter, the company can increase (but not decrease) its production capacity. It costs $100 to increase production capacity by one unit. For example, it would cost $10,000 to increase production capacity from 3000 cars per quarter to 3100 cars per quarter. It also costs $60 per quarter to maintain each unit of production capacity (whether it is used or not). The variable cost of producing a car is $2000. A holding cost of $150 per car is assessed for each quarter’s ending inventory. It is required that at the end of quarter 4, plant capacity must be at least 4000 cars.

(1) Determine how to minimise the total cost incurred during the next four quarters.

There is a concern that due to rising material and labor costs the variable cost in the fourth quarter may increase by 10%.
(2) Should you change your production plan if you believe this increase will occur?
(3) What would you do if you believed that there would be a 20% increase in variable costs in the fourth quarter?
(4) Describe and formulate another similar problem and solve the problem again.

Use both MATLAB Optimisation Toolbox and Excel Solver to solve the problems.

Problem 2: (Difficulty level: 2)
Wivco produces two products: Product 1 and Product 2. The relevant data is shown below. Each week, up to 400 units of raw material can be purchased at a cost of $1.50 per unit. The company employs four workers, who work 40 hours per week (their salaries are considered a fixed cost). Workers can be asked to work overtime and are paid $6 per hour for overtime work. Each week 320 hours of machine time are available.

In the absence of advertising, 50 units of Product 1 and 60 units of Product 2 will be demanded each week. Advertising can be used to stimulate demand for each product. Each dollar spent on advertising product 1 increases its demand by 10 units, each dollar spent on product 2 increases its demand by 15 units. At most $14 can be spent on advertising.

Use the answer and the sensitivity analysis report to answer the following questions:
(b) If overtime were only $4/hour, would Wivco use it?
(c) If each unit of product 1 sold at $15.50, would the current solution still remain optimal? What would the new solution be?
(d) What is the most that Wivco would be willing to pay for another unit of raw material?
(e) How much would Wivco be willing to pay for another hour of machine time?
(f) If each worker were required (as part of their regular work week) to work for 45 hours per week (with their salary remaining the same as in the original problem, so in essence Wivco gets five extra hours from each employee for free), what would the company’s profit be?
(g) Wivco is considering producing a new product (product 3). Each unit sells for $17 and uses 2 hours of labor, 1 unit of raw material, and 2 hours of machine time. Should Wivco produce any of product 3?
(h) If each unit of product 2 sold for $10, would the current solution remain optimal?

Problem 3: (Difficulty level: 3)

The company cannot sell more than its demand. Also any unused crude oil is disposed off at no cost.
The decision of buying crude stock is taken well in advance and thus the company does not know what scenario holds at the time of buying crude stock. However at the time of production of oil the company knows the exact demand.
How much of crude stock must the company buy in order to maximize its expected profit?

Use either MATLAB Optimisation Toolbox or Excel Solver to solve the problem.

Problem 4: (Difficulty level: 3)
The Auto company of America (ACA) produces 4 types of cars: subcompact, compact, intermediate and luxury. ACA also produces trucks and vans. Vendor capacities limit total production capacity to at most 1,200,000 vehicles per year. Subcompacts and compacts are built together in a facility with a total annual capacity of 620,000 cars. Intermediate and luxury are built together in a facility with a total annual capacity of 400,000 cars; and the truck/ van facility has a capacity of 275,000. Profit margins and fuel efficiencies are summarized below:

If the company ends up making more cars than the demand, it can sell its excess stock through a discount channel and earn half the normal profit; for example surplus compact cars would yield profits of $112.5/ vehicle.
The average fleet fuel efficiency must be at least 27 MPG.
What should the production plan of the company be so that its:
a) Expected revenue is maximized
b) Minimum revenue is maximized

Problem 5: (Difficulty level: 1)
Short-Run Manufacturing Problems at DEC
In the fourth quarter of 1988, the corporate demand/supply group of Digital Equipment Corporation (DEC) was under pressure to come up with a manufacturing plan for dealing with a series of major supply shortfalls that were impacting on the production, revenue, and customer satisfaction of one of DEC's new family of general purpose computer systems and workstations. The critical components for this family of systems that were going to be in short supply in the first quarter of 1989 were CPU chip sets, 1-meg memory boards, 256K memory boards, and disk drives. Pertinent data for these components and their usage in the family of systems is given below:

Brian Shannahan then proceeded to analyze DEC's problem and to formulate a manufacturing strategy. What is the strategy?

Use both MATLAB Optimisation Toolbox and Excel Solver to solve the problem and perform sensitivity analysis. Then describe and formulate another two similar Linear Programming problems (with at least 5 design variables and the parameters should come from modern computers) and solve them using both MATLAB Optimisation Toolbox and Excel Solver.

Problem 6: (Difficulty level: 1)
New Bedford Steel Coking Coal Supply Problem
New Bedford Steel (NBS) is a small steel manufacturing company. Coking coal is a necessary raw material in the production of steel, and NBS procures 1.0 - 1.5 million tons of coking coal per year. It is now time to plan for the 1997 production, and Stephen Coggins, coal supply manager for NBS, has solicited and received bids from the following eight suppliers for next year. He has organized the information about the bids in the following table.

Based on market forecasts and 1996 production characteristics, NBS is planning to accept bids for 1,225 mtons (1,225 million tons) of coking coal. This coal must have average volatility of at least 19% (volatility is the percent of volatile or burnable matter in the coal). Also, as a hedge against adverse labor relations, NBS has decided to procure at least 60% of its coking coal from union mines (United Mine Workers). Finally, Steve Coggins needs to keep in mind that capacity for bringing in coal by rail is limited to roughly 650 mtons per year, and capacity for bring in coal by truck is limited to 730 mtons per year.
Questions:
1. How much coal should Coggins contract for from each supplier?
2. What will be NBS's total cost of supply?
3. What will be NBS's average cost of supply?
Use both MATLAB Optimisation Toolbox and Excel Solver to solve the problem and perform sensitivity analysis. Then formulate another two similar LP problems (with at least 5 design variables) by changing the products, suppliers, the requirements, and solve the problems again.

Problem 7: (Difficulty level: 1)
Dietary optimisation
There are six different foods: Bread, Milk, Cheese, Fish, Potato and Yogurt:

Problem 8: (Difficulty level: 2)
A company manufactures four products (1,2,3,4) on two machines (X and Y). The time (in minutes) to process one unit of each product on each machine is shown below:

Problem 9: (Difficulty level: 2)
A company assembles four products (1, 2, 3, 4) from delivered components. The profit per unit for each product (1, 2, 3, 4) is £11, £15, £22 and £17 respectively. The maximum demand in the next week for each product (1, 2, 3, 4) is 50, 60, 85 and 70 units respectively.
There are three stages (A, B, C) in the manual assembly of each product and the man-hours needed for each stage per unit of product are shown below:

The nominal time available in the next week for assembly at each stage (A, B, C) is 160, 180 and 80 man-hours respectively.
It is possible to vary the man-hours spent on assembly at each stage such that workers previously employed on stage B assembly could spend up to 20% of their time on stage A assembly and workers previously employed on stage C assembly could spend up to 30% of their time on stage A assembly.
Production constraints also require that the ratio (product 1 units assembled)/(product 4 units assembled) must lie between 0.9 and 1.15.
Formulate the problem of deciding how much to produce next week as a linear program and solve it.
Use both MATLAB Optimisation Toolbox and Excel Solver to solve the problem and perform sensitivity analysis. Then formulate another two similar LP problems (with at least 4 design variables) and solve the problems again.

Problem 10: (Difficulty level: 1)
A company makes three products and has available 4 workstations. The production time (in minutes) per unit produced varies from workstation to workstation (due to different manning levels) as shown below:

Problem 11: (Difficulty level: 1)
The owner of a shop producing automobile trailers wishes to determine the best mix for his three products: flat-bed trailers, economy trailers, and luxury trailers. His shop is limited to working 24 days/month on metalworking and 28 days/month on woodworking for these products. The following table indicates production data for the trailers.

Problem 12: (Difficulty level: 1)
Select 3 problems from (a) to (e); For each problem selected, apply the Simplex method, MATLAB Optimisation Toolbox, and Excel Solver to solve it and compare the results, then perform sensitivity analysis.
(a) A contractor hiring earth moving equipment has the choice of two machines. Type A costs 23$ per day to hire, needs one man to operate it and moves 30 tonnes of earth per day. Type B costs 10$ per day to hire, needs four men to operate it and moves 70 tonnes of earth per day. The contractor can spend up to 500$ per day, has a labour force of 64 men available and can use a maximum of 25 machines on the site.
Find the maximum weight of earth that the contractor can move in one day.
(b) A cycle manufacturer produces two types of mountain-bike: a basic Model X and a Super Model Y. Model X takes 7 man-hours to make per unit, while Model Y takes 10 man-hours per unit. There is a total of 460 man-hours available per week for the manufacture of the two models. Due to the difference in demand for the two models, handling and marketing costs 20$ per unit for Model X, but only 10$ per unit for Model Y. The total funds available for these purposes are 800$ per week. Profits per unit for Models X and Y are 20$ and 30$ respectively. The objective is to maximise weekly profits by optimising the numbers of each model produced.
(1) Assume the numbers of units of Model X and Model Y produced each week are x and y. Express the weekly profit in terms of x and y. Also write down inequalities representing the constraints on production.
(2) Find the maximum obtainable profit and the numbers of each model manufactured which give this profit.

(d) A maker of wooden furniture can produce three different types of furniture: sideboards, tables and chairs. Two machines are used in the production- a jigsaw and a lathe. The manufacture of a sideboard requires 1 hour on the jigsaw and 2 hours on the lathe. The manufacture of a table requires 4 hours on the jigsaw and none on the lathe. The manufacture of a chair requires 2 hours on the jigsaw and 8 hours on the lathe. The jigsaw can only operate 100 hours per week and the lathe for 40 hours per week. The profit made on a sideboard is 100$, 40$ on a table and 14$ on a chair. In order to determine how best to use the machines so as to maximise profits, formulate the problem as a linear programming problems and solve it.

(e) A diet-conscious housewife wishes to ensure her family’ daily intake of vitamins A, B and C does not fall below certain levels, say 24 units, 30 units and 18 units, respectively. For this she relies on two fresh foods which, respectively, provide 8, 6 and 3 units of vitamins per ounce of foodstuff and 3, 6 and 9 units per ounce. If the first foodstuff cost 3p per ounce and the second only 2p per ounce, find how many ounces of each foodstuff should be bought by the housewife daily in order to keep her food bill as low as possible.

Problem 13: (Difficulty level: 1)

In the following example given in the lecture slides in Lecture 1, the floor space constraint formula is not reasonable. Formulate three new LP problems by adding necessary information about the floor space and the dimensions of the machines to model the floor space constraint. Then solve the problem using both MATLAB Optimisation Toolbox and Excel Solver. After that, fix the number of machine A and obtain an LP problem with two variables and then solve it using the simplex method.

MyPC Company invests $12 million to buy 3 models of Component Placement Machines A,B,C to manufacture different kinds of motherboards for personal computers. Total number of operators available is 100. A floor space constraint needs to be satisfied. Determine how many of each kind is appropriate to maximise the number of boards manufactured per day

Problem 14: (Difficulty level: 3)
Formulate the following problem into LP and solve it using both MATLAB Optimisation Toolbox and Excel Solver, then perform sensitivity analysis. After that, formulate another similar LP problem on the same topic and solve it again.

Cash sales and accounts receivable produce immediate income to the checking account. Expenses are immediate deductions. Accounts payable amounts pt are not actually due until week t+3, but they are discounted by 2% if paid early in week t. The company can borrow at most $4 million in total from the bank but need to pay the bank 0.2% interest per week. In addition, the bank requires at least 20% of the current amount borrowed to be maintained (without earning interest) in the checking account. The other option is investment of excess cash in short-term money markets. The company can earn 0.1% interest per week on amounts invested in this way. The manage wishes to minimise net total cost in interest and lost discounts while maintaining at least a $20000 checking account safety balance. And, the company needs to pay off all the payable amount after the 8th week. Decide how to exercise the available options.

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