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Linear Programming
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You have taken over the management of a paper production firm.

INSTRUCTIONS TO CANDIDATES
ANSWER ALL QUESTIONS

1. You have taken over the management of a paper production firm. You find that the firm is contracted to deliver 3,000, 6,000, 5,000, and 2,000 reams of paper over the next four

schedule to deliver the paper over the next four periods. There 0156999 01565s. At the moment, there are two 

An efficient machine that can produce paper at the cost of $1 per ream. Currently, this machine is calibrated to produce 4,000 reams of paper each period.

It can be recalibrated from one period to the next to produce more paper at a cost of $3 per extra ream. (For example, if the current production level is 4,000 per period, and you wish to use the machine to produce 4,100 in the next period, this will cost an extra $300.)

per ream. (For example, if the current production level is 4,100 per period, and you wish to mester 1

50015699 can also be recalibrated from one period to the next to produce less paper at a cost of $0.80 An emergency machine that can produce paper at the cost of $1.40 per ream. This 50016999

the machine to produce 4,000 in the next period, this will cost an extra $80.)

produce any amount of paper in a period without any recalibration costs. Any paper that is left over at the end of a period incurs $0.05 per ream storage cost. You would like

US210 reduce the output of the efficient machine to 3,000 reams per period at the end of the fourth period.

Furthermore, you need at least 500 reams of paper in inventory by the e pay an additional penalty of $0.15 per ream over 500.

the fourth period, 3 Se (a) (14 points) Formulate the optimal production schedule problem as a linear program. 500 (6 points) Please refer to the Jupyter Notebook for this question. Implement the model in c

Make sure that the model can take into account data of different sizes.

2. We consider a one-period financial market with n assets.

At the start of the period, each asset j € [n] has a known price ; ≥0. The vector of prices is denoted

3 Seme asset j in scenario i.

It is unknown what the prices of the assets will be at the end of the period. However, market research has produced N scenarios for the price vectors: PPN ER

To clarify, at the end of the period, one, and only one, scenamese, Pj denotes the end price of iEN will take place. So the pester 1 50015 for the assets will be represented by one, and only one, vector pi, ie., the prices will be pi, P999

for assets 1,...,n. However, we do not know which scenario will take place, and we do not have any probabilities on the scenarios taking place.

• A portfolio is a vector ЄR" where each entry a, tells us how many units of asset j we own. Note that we assume the number of units owned may be fractional.

US2310 Furthermore, the portfolio may have negative amounts: 2,

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