Question

Please the solution must be in compact form only not simple form, simple form will be rejected.

Only Compact form accepted only

Question 3: Formulate the LP model for the following problem in compact form The harvest of cane sugar in Australia is highly mechanized. The sugar cane is immediately transported to a sugar house in wagons that run on a network of small rail tracks. The sugar content of a wagon load depends on the field it has been harvested from and on the maturity of the sugar cane. Once harvested, the sugar content decreases rapidly through fermentation and the wagon load will entirely lose its value after a certain time. At this moment, eleven wagons all loaded with the same quantity have arrived at the sugar house. They have been examined to find out the hourly loss and the remaining life span in hours) of every wagon, these data are summarized in the following table. Lot Loss (kg/h) Life span (h) 1 43 8 2 26 8 3 37 2 4 28 8 5 13 4 6 54 8 7 62 8 8 49 8 9 19 6 10 28 8 11 30 8 Every lot may be processed by any of the three, fully equivalent production lines of the sugar house. The processing of a lot takes two hours. It must be finished at the latest at the end of the life span of the wagon load. The manager of the sugar house wishes to determine a production schedule for the currently available lots that minimizes the total loss of sugar.

Answer 1

ANSWER :

Given Data

Proclines = 3 ;

Proctime = 2;

Loss = [43;26;37;28;13;54;62;49;19;28;30] ;

life Span = [ 8;8;2;8;4;8;8;8;8;8;8] ;

$n_{1}$   = length (loss);

$n_{2}$ = Ceil ;

n1 Proclines

Proc = tom('Proc' ,  $n_{1}$ , $n_{2}$ , 'int') ;

% All slots are integers

intcons =

{0 < Proc<1}:

% Production constraint , all lots processed

Prod cons 1 = {Sum (Proc , 2 )==1} ;

% Production constraint , no more than proclines concurrently running

Prod cons 2 = { Sum (Proc , 1) $\leq$ Proclines};

% maximum slot number for a lot

slotcons = { Sum (Proc ,* repmat (1:$n_{2}$, $n_{1}$ ,1) ,2) $\leq$ lifespan / proctime };

% Objective

Objective = Sum (Sum (Repmat (1 : $n_{2}$ , $n_{1}$ , 1 ) * repmat (loss , 1 , $n_{2}$ ) * Proctime ,*Proc ));

Constraints = { intcons , Prodcons 1 , Prodcons 2 , Slotcons };

Options = Struct ;

options . solver = 'cplex ' ;

options . name = ' cane sugar Production' ;

Sol = ezsolve ( Objective , Constraints , $\square$ , options ) ;

f _ k = Subs ( Objective , sol ) ;

Priler = 1 ;

if prilev > 0

lots = Length ( life span ) ;

time slots = ceil ( lots/ proclines (1)) ; % even a fraction means work .

temp = Sol . proc ;

disp [ ' To minimize loss ( ' num 2 str(f_k)' ) use this schema '])

for time = 1 : timeslots ;

disp (['at' num 2 str ((time - 1 )*Proctime (1) +8) !00'.])

idX = find (temp ( : , time )) ;

disp ( [ ' process the lots ' num 2 str (idx')])

end

disp (['at ' num2str((time)*Proctime (1)+8)'00:])

disp ('harvesting completed ')

end.

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