Question

Use this mini-case to answer the following question(s). (Total marks = 100)
The Delux Nut Company produces a deluxe mix of almonds, cashews, peanuts and walnuts. The deluxe mix must contain at least 10% of each kind of nut. At least half of the mix must consist of almonds and cashews and it can contain at most 20% peanuts.
The company also produces a companion mix of cashews, walnuts and raisins. This mix must contain at least 10% cashews, at least 30% walnuts, and between 20 and 40% raisins. Raisins are available in unlimited supply at a purchasing cost of R3,60 per kilogram. The quantities of nuts available and the purchasing costs in rand per kilogram are as follows:

Nut	Purchasing costs in rand	Kilogram available
Almonds Cashews Peanuts Walnuts	12,00 10,60 2,80 8,40	400 200 600 300

Both mixes are sold in 100 g packages. The company must produce at least 4 000 packages of the deluxe mix and at least 5 000 packages of the companion mix.
(a) Formulate an LP model for this problem.
(b) Solve the LP using SOLVER (NB: Please use solver for me and not LINGO)
(c) Write down the optimal purchasing plan and the associated purchasing costs.

Use only the initial printout of the optimal solution to answer the following questions. (This means that you may not change the relevant parameters in the model and do re-runs.) Explain how you arrive at your answers.
(d) The suppliers inform the company that the availability of almonds has been reduced to 200 kg and that of peanuts to 100 kg. How will this influence the optimal plan and purchasing costs? Give reasons for your answer.
(e) An extra 50 kg of walnuts can be bought at R11,00 per kg. Would you make use of this offer? Give reasons for your answer.
(f) Assume that the optimal plan is obtained from the original available quantities of nuts. The company now decides that it must produce at least 4 500 packages of the deluxe mix. Will the total purchasing costs be influenced? Give reasons for your answer.
(g) The supplier of peanuts announces a 40% increase in price. What will the effect of this be? Give reasons for your answer.

Answer 1

Given that

The Delux Nut Company produces a deluxe mix of almonds, cashews, peanuts and walnuts. The deluxe mix must contain at least 10% of each kind of nut. At least half of the mix must consist of almonds and cashews and it can contain at most 20% peanuts.

a.

In each pack of deluxe mix, let the amount of almond be xa, cashew be xc and so on in grams

In each pack of companion mix, let the amount of almond be ya, cashew be yc and so on in grams

So, ya=0, yp=0 and xr=0.

Let packages of deluxe mix be d and companion mix be c

Min Total cost : Cost of almond + Cost of Cashews + .. + Cost of Raisins

Cost z = 0.001*{1200*(d*xa) + 1060*(d*xc+c*yc)+280*(d*xp) + 840*(d*xw+c*yw) + 360*(yr)}, 0.001 factor is to convert grams to kilograms

Constraints :

Deluxe mix:

xa,xc,xp,xw>=10, at least 10% of each

xa+xc>=50, al least half of almonds and cashews

xp<=20, at most 20% of peanuts

xa+xc+xp+xw=100

Companion mix:

yc>=10, at least 10% of cashews

yw>=30, at least 30% of walnuts

20<=yr<=40, between 20 and 40% of raisins

yc+yw+yr=100

Amount of packs:

d>=4000, at least 4000 packs of deluxe

c>=5000, at least 5000 packs of companion mix

Amount of nuts:

d*xa<=400000

d*xc+c*yc<=200000

d*xp<=600000

d*xw+c*yw<=300000

b.

Solving the LP we get,

c.

The optimal cost and purchasing plan is marked in green.

d.

If almonds available is 200 kg and peanuts is 100kg then optimal plan is: