Now I don't need you guys to do my homework for me; however, I am a little stumped
Xara Stores in Canada imports the designer-inspired clothes it sells from suppliers in China and Brazil. Xara estimates that it will have 45 orders in a year, and it must arrange to transport orders (in less-than-full containers) by container ship with shippers in Hong Kong and Buenos Aires. The shippers Xara uses have a travel time of 32 days from Buenos Aires and 14 days from Hong Kong, and Xara wants its orders to have an average travel time of no more than 21 days. About 10% of the annual orders from the shipper in Hong Kong are damaged, and the shipper in Buenos Aires damages about 4% of all orders annually. Xara wants to receive no more than 6 damaged orders each year. Xara does not want to be dependent on suppliers from just one country, so it wants to receive at least 25% of its orders from each country. It costs $3,700$ per order from China and $5,100$ per order to ship from Brazil. Xara wants to know how many orders it should ship from each port in order to minimize shipping costs.
a) Formulate a linear programming model for this problem.
Okay so, I'm not totally clueless here
If I let $X$ be shipments from China, and $Y$ be shipments from Brazil, my cost minimization is
3700X + 5100Y = Min Cost
Also, I assume one of the constraint equations is
.1X + .04Y < 6
At this point I am stumped. My prof told me that one equation will be .75X – .25Y > 0, but I do not understand how I can graph that line
Any help is appreciated
Best Answer
Draw the line 0.75X-0.25Y = 0. The solution must be above and to the right of (or on) the line, as you have a $\gt $. (But I've no idea where your prof gets this constraint from. It might be an error).
The second line tells you to assume that X + Y = 45.
The third line tells you that a constraint is that $\frac{14X+32Y}{45} \le 21$.
The next constraint is that 0.1X+0.04Y $\le$6, so your assumption was almost correct.
Next 2 constraints: $\frac{i}{45} \ge 0.25$, i = X,Y
In actual fact 3700X + 5100Y = Cost, rather than 'Min Cost'. So the way you write the problem formulation is like this:
Minimise 3700X + 5100Y
subject to
X + Y = 45
$\frac{14X+32Y}{45} \le 21$
0.1X+0.04Y $\le$6
$\frac{i}{45} \ge 0.25$, i = X,Y
You can find the solution using Excel's Solver, but if you have to do it analytically, you can graph it or use the branch and bound method. There may be other methods you can use too (possibly quicker than branch and bound, but these are specialist so don't work for all lps, whereas branch and bound does, so you have to learn which method is best for a given problem, or settle for a possibly slower method if you're not sure). To graph it, draw the lines, dealing with $\ge$ and $\le$ as I described to find the feasible region. For constaints with $\le$, you change the $\le$ to = to draw the line and then you know the solution must lie below and to the left of (or on) the line. Once you've determined the feasible region, you know (because someone proved it) that the solution must lie at a vertex i.e. where 2 or more of the constraint lines meet, so you find the points where the lines cross using simultaneous equations and then plug the values into the objective function (cost) to see which minimises it.
I've no idea why it's mentioned that the containers must be less than full: seems irrelevant. Also, technically the shipments are from Argentina rather than Brazil, although the orders may originate in Brazil.