Linear-programming, object function

linear programminglinear-transformationsproblem solvingprogrammingsolution-verification

I have this programming problem with the notation: production units i = 1,.., I there is known from data. But cost, capacity and demand are all assumed to be parameters. We denote the production cost of unit 𝑖 with 𝑐𝑖>0 (in USD/MWh) and the capacity is π‘π‘šπ‘Žπ‘₯𝑖>0 (in mW). The demand is denoted as d> 0 (in MWh). Decision variables, on the other hand, are unknown (endogenous). The decisions consist of how much it takes produced on each production unit. The production level for unit i is denoted as pi β‰₯ 0.

I have to formulate the object function as a function of the parameters 𝑐𝑖 , i = 1,… , I and the decision variables 𝑝𝑖 , i = 1,. . ., I.
2)Formulate the conditions that for each unit ensure that the production level does not exceed the capacity.

Can someone help me with this problem? I maybe think that the object function is
π‘šπ‘Žπ‘₯βˆ‘π‘–=0π‘˜π‘π‘–π‘π‘–
where 𝑐𝑖 is from data and 𝑝𝑖 is unknown? Or what? I'm totally new to lineary programming

I have this programming problem with the notation: production units 𝑖=1,.., I there is known from data.

But cost, capacity and demand are all assumed to be parameters. We denote the production cost of unit 𝑖 with 𝑐𝑖>0 (in USD/MWh) and the capacity is 𝑝 max 𝑖>0 ( in mW ). The demand is denoted as 𝑑>0 (in MWh). Decision variables, on the other hand, are unknown (endogenous).

The decisions consist of how much it takes produced on each production unit. The production level for unit 𝑖 is denoted as 𝑝𝑖 β‰₯ 0.

  1. I have to formulate the object function as a function of the parameters 𝑐𝑖 :

𝑖=1,…,𝐼

And the decision variables 𝑝𝑖 :

𝑖=1,…,𝐼

  1. Formulate the conditions that for each unit ensure that the production level does not exceed the capacity.

Can someone help me with this problem? I maybe think that the object function is:
maxβˆ‘π‘–=0π‘˜π‘π‘–π‘π‘–
where 𝑐𝑖 is from data and 𝑝𝑖 is unknown? Or what? I'm totally new to lineary programming

Best Answer

YouΒ΄re almost right with the first question. Usually the goal is to minimize the cost. Thus we have

$$\textrm{min} \ \ \sum_{i=1}^I c_i\cdot p_i$$

The upper bound for the index $i$ is $I$ and it starts at $1$. Next you have the capacity constraints:

$$p_i\leq p_i^{\textrm{max}} \ \ \forall \ i\in\{1,2,...,I\}$$

Finally the non-negativity constraints: $p_i\geq 0 \ \ \forall \ i\in\{1,2,...,I\}$

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