Question

pen 15 22 11896 | Use this mini-case to answer the following question(s). 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 contain at least 10% cashews, at least 30% walnuts, and between 20 and 40 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: (Total marks 100) raisins. This mix must % raisins. Raisins Nut Almonds Cashews Peanuts Walnuts Purchasing costs in rand 12,00 10,60 2.80 8.40 Kilogram available 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 LINGO or SOLVER (c) Write 20 marks) (20 marks down the optimal purchasing plan and the associated purchasing

Answer 1

A, B, C

The boxes in green, are decision variables
The boxes in blue are constrains
The boxes in orange is the objective function

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The formulation:

	Decision variable	g in one pack
	Production in packs	Almonds	Cashews	Peanuts	Walnuts	Raisins	Total
Deluxe Mix	4000	12.4999975552772	37.5	20.0000012223595	30.0000012223633		=SUM(C3:G3)	=	100
	>=	=C3/$H$3	=D3/$H$3	=E3/$H$3	=F3/$H$3
	4000	>=	>=	>=	>=
		0.1	0.1	0.1	0.1
	Almond + Cashews	=SUM(C3:D3)/H3		<=
		>=		0.2
		0.5
Companion mix	5000		10		29.9999995653828	60.0000004346173	=SUM(C11:G11)	=	100
	>=		=D11/$H$11		=F11/$H$11	=G11/$H$11
	5000		>=		>=	>=
			0.1		0.3	0.2
						<=
						0.4
	Total	=$B$3*C3+$B$11*C11	=$B$3*D3+$B$11*D11	=$B$3*E3+$B$11*E11	=$B$3*F3+$B$11*F11	=$B$3*G3+$B$11*G11
	Availability constraint (g)	<=	<=	<=	<=
		400000	200000	600000	300000
Cost per KG		12	10.6	2.8	8.4	3.6
	Total cost	=SUMPRODUCT(C17:G17,C20:G20)
	Objective function	Minimize

The solution:

	Decision variable	g in one pack
	Production in packs	Almonds	Cashews	Peanuts	Walnuts	Raisins	Total
Deluxe Mix	4000	12.50	37.50	20.00	30.00		100	=	100
	>=	12.50%	37.50%	20.00%	30.00%
	4000	>=	>=	>=	>=
		10%	10%	10%	10%
	Almond + Cashews	50.00%		<=
		>=		20%
		50%
Companion mix	5000		10		30	60	100	=	100
	>=		10.00%		30.00%	60.00%
	5000		>=		>=	>=
			10%		30%	20%
						<=
						40%
	Total	50000	200000	80000	270000	300000
	Availability constraint (g)	<=	<=	<=	<=
		400000	200000	600000	300000
Cost per KG		12	10.6	2.8	8.4	3.6
	Total cost	6292000
	Objective function	Minimize

D.

It does not change the solution, as the required for Almond is 50 Kg and for peanuts is 80 Kg.

So, the cost remains the same and so does the mix.

E.

It is of no use, as only 270 Kg is used, hence there is already 30 kg unused, hence 50 kg extra is of no use. Optimal solution persists.

F.

So we change the constraint from 4000 to 4500.

So the solution changes: As below

	Decision variable	g in one pack
	Production in packs	Almonds	Cashews	Peanuts	Walnuts	Raisins	Total
Deluxe Mix	4500	16.67	33.33	20.00	30.00		100	=	100
	>=	16.67%	33.33%	20.00%	30.00%
	4500	>=	>=	>=	>=
		10%	10%	10%	10%
	Almond + Cashews	50.00%		<=
		>=		20%
		50%
Companion mix	5000		10		30	60	100	=	100
	>=		10.00%		30.00%	60.00%
	5000		>=		>=	>=
			10%		30%	20%
						<=
						40%
	Total	75000.0	200000	90000.01	285000	300000
	Availability constraint (g)	<=	<=	<=	<=
		400000	200000	600000	300000
Cost per KG		12	10.6	2.8	8.4	3.6
	Total cost	6746000
	Objective function	Minimize