TSSFL TECHNOLOGY STACK

Below are typical linear programming problems for undergraduates, are you able to solve them?


Problem 1:

A bookmaker has quoted odds on tennis match between players A and B. The match consists the best two out of three sets. the bookmaker is giving odds of 5 to 2 that player A will win the match and odds of 3 to 2 that player A will win each set. A bettor has which can be distributed by betting on either player A or player B to win the match or any of the sets. All bets are made before the match starts (if there are only two sets all bets on the third set are returned to the bettor).

Find a way of placing bets so that now what happens, the bettor is assured of winning an amount where is as large as possible. Note that might be zero, but in this case it turns out that a bettor can guarantee winning a positive amount. Formulate this as linear programming model.


Problem 2:

The Baruka International Hotel has Tshs. 1 million in available cash. At the beginning of each of the next six months, The Baruka International Hotel will receive revenues and pay bills as shown in the following Table:
It is clear tha the Baruka international Hotel will have a short term cash flow problem until it receives revenues from the Easter holiday sales. To solve this problem, The Baruka International Hotel must borrow money.

At the beginning of July, The Baruka International Hotel may take out a six month loan. Any money borrowed for a six month period must be paid back at the end of December along with interest (early payback does not reduce the interest cost of the loan). The Baruka International Hotel may also meet cash needs through month to month borrowing. Any money borrowed for a one month period incurs an interest cost of per month. Formulate a linear programming model which The Baruka International Hotel can use to determine how to minimize the cost of paying its bills on time.

Problem 3:

A newly graduated Doctor in Tanzania is considering to open a Private Hospital in Dar Es Salaam. She requires different numbers of full time Nurses on different days of the week. The number of full time Nurses required on each day is given in the following table:
Nurses Union rules states that each full time Nurse must work for five consecutive days and then receive two days off. For example, a Nurse who works from Monday to Friday must be off on Saturday and Sunday. If the Doctor wants to meet her daily requirements using only full time Nurses, formulate a linear programming problem that the Doctor can
use to minimize the number of full time Nurses that she must hire.

I will try to formulate the last one regarding nurses and time off.

Assume be the number of nurses working on each day '', where = Monday, Tuesday,, Sunday.

Then the Objective function is:

Minimize .

This is like to say,

Minimize where Monday, Tuesday, etc.

This is the constrained LP problem that depends on the number of days of nurses, if a nurse works on Monday it means he/she has been off Saturday and Sunday, the one whose work starts on Tuesday means he is off Sunday and Monday, etc.

(off on Tuesday and Wednesday this gives all nurses no matter their choice of working day , but they will work on Monday:

(off Wednesday and Thursday)
etc
etc

Then, one can solve this either by MATLAB GUI, LINPROG command in MATLAB or by manual simplex method by introducing variables.

You can always typeset maths formulae by using or \(MathJax\), not easy if you do not know, but I will put a tutorial in future to help others get started.

gaDgeT wrote:You can always typeset maths formulae by using or \(MathJax\), not easy if you do not know, but I will put a tutorial in future to help others get started.

Waoo, i never knew there is this tool in here, that's fantastic. Thank you.