# How to translate OPL to mathematics program

*label*Mathematics

*timer*Asked: Jan 21st, 2019

**Question description**

1. This question will test your ability to understand OPL code.

2. An oil renery obtains crude oil from 3 dierent sources. The renery distills up to

100,000 barrels of crude oil per day in its fractionater. The fractionater heats up the

crude oil until various products called DN (distillation naphtha), DHO (distillation

heating oil), and DGO (distillation gas oil), are given o in vapour form and collected

at various levels. Other non-usable byproducts are also produced and discarded. It

costs $0.75 to run the fractionater on a barrel of crude oil. The table below illustrates

the outputs of the fractionater for each type of crude.

3. A company supplies toner for oce photocopiers. They have m clients C = f1; : : : ;mg

and n distribution centers D = f1; : : : ; ng. Each client i has di units of toner they

demand and must be satised by the distribution centers. Note: Multiple distribution

centers can supply any one client.

Depending on which distribution center supplies any particular client, a certain amount

of prot is incurred. The prot function for the total amount of demand x that center

j satises is a concave function profitj(x) where

profitj(x) =

0:5x if x 40

cjx 40(cj 0:5) otherwise

and cj is a specic constant for center j that is always bigger that 0.5.

The company transports the toner using delivery trucks where each truck is assigned

to a specic center. Each center has a constant G units of fuel to keep the trucks going.

The amount of fuel needed by the trucks at center j to supply x units of toner is a

convex function fuelj(x) where

fuelj(x) =

0:75x if x 60

2x 75 otherwise

Give a linear program that maximizes the prot of the company.

4. For each part, determine if the given mathematical program and the given LP are

equivalent (ie. will give the same optimal value). Justify your conclusion.