1. Consider the following LP:
min 3x1 ???? 7x2
s.t. x1 + x2 2
????x1 + 4x2 12
2x1 + x2 12
x1; x2 0:
(a) Sketch the feasible region. (Watch out for signs and subscripts.)
(b) There is exactly one redundant constraint in the above problem. Which one?
(c) Use the Excel solver or Matlab's linprog function to compute the optimal solution.
(d) Use the graphical approach we have seen in class to read o the optimal solution. (Check that it is
the same as what you got with the computer.)
2. A careful investor is trying to determine how to invest $100; 000 in the following ve bonds to maximize
their annual return.
Bond Annual return Maturity Risk Tax free?
1 9.0% Long High No
2 8.5% Short Low Yes
3 9.0% Long Low No
4 8.0% Long High Yes
5 9.0% Short High No
The investor would like to have at least 60% of the money in short-term bonds and no more than 50% in
high-risk ones. At least 25% of the funds should go in tax-free investments, and at least 50% of the total
annual return should be tax free.
(a) Formulate this problem as linear program.
(b) Use the Excel solver or Matlab's linprog function to compute the optimal solution. Show your
spreadsheet or your code. Also report the optimal solution. Which constraints are binding for the
3. Our company is manufacturing toothpicks, at a maximum capacity of one million toothpicks per month.
Currently we have 300,000 unsold toothpicks in our warehouse. The toothpick market is perfectly predictable,
so we can determine that in the next six months the expected cost of manufacturing and the expected
orders we have to fulll is going to be as shown on the table. We must satisfy all the orders.
Month Mfg. cost (per million) Orders
Jan $11000 .8 million
Feb $ 8400 .8 million
Mar $10000 .8 million
Apr $11000 1.2 million
May $12000 1.6 million
Jun $12000 .7 million
Our warehouse capacity is unlimited, but the cost of holding inventory from a month to the next is
estimated to be 4% of the production cost. (For example, to keep an inventory of 1 million unsold
toothpicks at the end of January costs us $440.) A further complication is that owing to a contract with
the trade union, the production can never decrease by more than 10% in a month. (For example, if we
produce 1 million toothpicks in January, we cannot produce less than 900,000 in February.) Last month
(December) the factory was running at maximum capacity.
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MA 325, Module IV Homework|Linear and Integer Optimization April 2, 2018
(a) We need a plan to minimize the total production cost from January through June, meeting all the
constraints given above. Formulate this problem as a linear program. (We can assume that leftover
inventory in June, if any, has no value, but also no associated storage cost.)
(b) Now let's assume that unused warehouse capacity has all sorts of (unspecied) advantages, and we
need a plan to minimize the largest inventory throughout the six months, whatever the cost, meeting
all the constraints given above. Formulate this problem as a linear program.
4. Oak City Logistics operates a single cargo plane that can hold up to 26000 pounds of cargo occupying up
to 18000 cubic feet. They have contracted to transport the following items.
Item type No. of items Weight Volume Cost of subcontract
1 3 4000 1000 $800
2 10 800 1200 $150
3 4 2000 2200 $300
4 5 1500 500 $500
For example, they have agreed to transport 10 items of type 2, each of which weighs 800 pounds and takes
up to 1200 cubic feet of space. The last column refers to the cost per item of subcontracting shipment to
another carrier. For each pound the plane carries, the cost of ying increases by 5 cents.
(a) Which items should they put on the plane, and which should they ship via other carrier, in order to
have the lowest shipping cost? Formulate this problem as an integer program.
(b) Use the Excel solver or Matlab's intlinprog function to compute the optimal solution. (Show your
spreadsheet or your code. Also report the optimal solution.)
5. Howling Cow needs to gure out the production of its ice cream for the Creamery throughout the next
week. We assume that production decisions can be made for each avor independently, and therefore
consider only one avor in this problem. For the avor considered, they anticipate a demand of dt gallons
for each day t (Sunday through Friday; the Creamery is closed on Saturdays.) If they decide to produce
any of the avor on day t, a setup cost of ct is incurred. The production and storage costs (for unsold ice
cream that is saved for the next day) are pt and st dollars per gallon, respectively, on day t. Unsold ice
cream can survive throughout the week, and is thrown out at closing on Friday.
(a) Formulate an mixed-integer linear program to minimize the total cost of production and storage,
while meeting the expected demand every day.
(b) Suppose we allow unsatised demand, at a cost of bt dollars per gallon on day t. Show how to modify
the model to handle this option.
6. Big Bad Airlines (BBA) runs ights between n cities, some of which are hubs for its main competitor,
Three Little Airways (TLA). The distance between city i and city j is given by the number dij . BBA
needs a hub within distance R of each of the n cities, but they cannot choose a city as their hub if it
is already a hub for TLA. BBA wants to determine the smallest number of hubs needed to meet these
requirements. Formulate their problem as an integer program.
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