Hi, This is a formulation of linear programming problem. Please sove it without

Question

Hi,

This is a formulation of linear programming problem. Please sove it without softwares.

An electric company would like to decide on selecting its future investment plan to meet the next 25 years generation plan. This plan could be met exactly by one of the following options: 3. . Option 1: 3 identical nuclear plants each cost 10 million Saudi Riyals (SR); e Option 2: 5 identical nuclear plants each costs 4.5 million SR; . Option 3: 10 identical solar farms each costs 2 million ° Option 4: five solar farms identical to the ones in option 3 and 10 wind farms each costs 0.8ml Option 5: Combination of nuclear plants, solar farms and wind farms as needed to . meet the demand. Formulate a linear programming model to determine the minimum cost plan for meeting the next 25 year generation plan.

Explanation / Answer

Assumption - Option 5 considers any number of resources without any upper limit i.e. there is no upper limit generated from the earlier 4 options.

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From the context, it is clear that there are two types of nuclear plants, one type of solar farm, and one type of wind farm.

Let A, B, S, and W be the number of Nuclear plants (1), Nuclear plants (2), Solar farms, and Wind farms.

Also, note that the first four options give as equivalence between the resources i.e. if demand is D then,

The output of one A = D/3; that of one B = D/5; that of one S = D/10; that of one W = D/20 (as 5S=10W)

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Formulation

Minimize Z = 10A + 4.5B + 2S + 0.8W
Subject to,
(D/3)A + (D/5)B + (D/10)S + (D/20)W = D
cancelling D, we get
A/3 + B/5 + S/10 + W/20 = 1
or, 20A + 12B + 6S + 3W = 60
A, B, S, W >= 0

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Why should we need any software to solve the above!

The objective will be minimized when the decision variable with least cost is given the only positive value to reach the RHS of the constraint. So, the optimal solution will be found when sufficient W will be there to produce RHS =60

i.e. W = 60/3 = 20 and A=B=S=0 should be the optimal solution.

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