Question Orange Juice Production Planning - Mathematical Programming using XPress-IVE

Q This piece of work will count for 30% of the overall mark for MATH6002. It is intended

that you carry out the analysis using the Xpress-IVE modeling and optimisation software system. Completed work should be handed in to the Student Office Building 58 (Room 2127) by Wednesday, January 10th, 2015, at 2:00PM. An Assignment Hand In Cover

must be completed and stapled to the front of your work prior to submission. Make sure that you are given a receipt when you submit your coursework. You should make sure that you are given a receipt. The Student Office is open to

students between 9.00 a.m. and 3.00 p.m. Students will not be able to submit coursework outside these hours. The deadline is strict.

Please refer to the MSc OR [and Finance] Blue [Pink] book for the penalties applied to late assignments. Your report should be word processed. Ensure that you take frequent and multiple

backups of your work, since excuses concerning lost or corrupted files will not be treated sympathetically. As a very rough guide, I would like about six to ten A4 sides of description, with

any extra material (e.g., diagrams, tables or computer output) attached at the end. Your report should include: • an introduction;

• the formulation of your model with a statement of any assumptions made; • a brief description of how you solved it; • a presentation of the results;

• a summary of recommended policies that should be followed (and possible suggestions for further investigations); • printouts of the XPress-IVE model(s) and corresponding output in one or more

appendices. It should be written from the point of view of you, the OR analyst, describing some work that you have done to an OR manager (who will know the mathematical programming terminology).

Marks will be awarded on the basis of 1• quality of the analysis; • clarity of the presentation; • originality.

Your work must be written up independently. The model and report

must be your own work. You are reminded of the University policy on plagiarism. Your report must acknowledge clearly all people with whom you

have discussed any part of the coursework, as well as any references you might have used.

You are expected to complete the coursework without help from staff. Limited assistance will be available from a postgraduate tutor (you will be told when the tutor is available). No other assistance will be provided.

Orange Juice Production Planning You have been assigned to provide Operational Research experience to the company

TropiJuice. TropiJuice is a major producer of frozen concentrated orange juice in Brazil. Three orange varieties are used: Hamlin, Pera, and Valencia. The company buys the production from selected orange plantations, picks the fruits, processes it into juices,

and blends them into two products called Standard and Dairy. The Standard juice is primarily used by beverage industries all over the world. The slightly more expensive Dairy has to satisfy certain conditions on flavour, acidity, and colour, among others.

The planning horizon is three months. The forecasted demand (in tonnes) over the planning horizon is as follows: product Demand in month selling price 1 2 3 (£/tonne)

Standard 1500 1500 700 1000 Dairy 200 100 100 1100

One tonne equals 1000kg. There is no penalty for unfulfilled demand. The different orange varieties have different maturation periods. The expected availabilities of different varieties during the planning horizon are as follows: month expected availability (in 1000s of boxes)

Hamlin Pera Valencia 1 300 250 0 2 350 350 50

3 100 500 100 Juice (kg/box)

for each variety 3.5 3.7 3.4 2There are limits on the percentage of juice of each variety allowed in each product.

These are as follows. • Standard can have at most 30% Hamlin juice, must have at least 60% Pera juice, and at most 40% Valencia juice.

• Dairy can have at most 35% Hamlin juice, must have at least 50% Pera juice, and between 15% and 50% Valencia juice. Juice made in a month can be sold in that month itself, or later. Unsold juices may

be stored in refrigerated tanks at a cost of £10/tonne/month for all varieties. The cost of purchasing, picking, and transporting the fruit to the juice plant is estimated to be £1/box for each variety. The cost of running the juice plant is a fixed

amount of £0.5 million/month, and the plant has adequate capacity and workforce to handle all the fruit expected to be available in each month of the planning horizon. Your tasks are as follows: 1. Juice inventories are negligible at the beginning of the planning horizon. The company would like to have no juice inventories at the end of the planning horizon.

It is required to determine a production plan that maximises the company's net profit during the planning horizon. Formulate a corresponding mathematical model and solve it using XPRESS-IVE. 2. In part 1, possible capacity limitations to the juice plant have been ignored. The

equipment at the plant can process 0.4 million boxes of Pera oranges per month

if this is the only variety processed. Hamlin oranges use 10% more capacity, and Valencia oranges 10% less capacity. The effective plant capacity will therefore depend on the proportion of the different varieties processed. Revise your model

from part 1 to take into account these processing capacity constraints and solve the resulting model using XPRESS-IVE.

3. Orange growers in northern Brazil produce a lot of Valencia oranges. A truckload of Valencia oranges consists of 1000 boxes. Let λ denote the cost for purchasing and transporting a truckload of oranges from the north to the juice plant. Evaluate the effect of increasing the availability of Valencia oranges on the optimal production plan, treating λ as an optimal parameter.

4. Reconsider part (1) with the planning horizon being 12 months with the demands and the expected availabilities in the first three months are still unchanged. However, for months 4-12, the demands and the expected availabilities are the same with those in month 3. Please compare the answers among three cases: (a) The factory must be opened every month, (b) There is option to open/close the factory in each month and c) The factory must be opened for at least 10 months throughout the year.