# (Solved Homework): Suppose you make c_i profit for every x_i you ship. You have n possible items to ship, labeled…

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(Solved Homework): Suppose you make c_i profit for every x_i you ship. You have n possible items to ship, labeled…
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Suppose you make c_i profit for every x_i you ship. You have n possible items to ship, labeled x_1, x_2, x_n. If shipping x_i unit consumes a_ji amount of j resources for every i product respectively, and you only have a total b_m resources to ship it, how would you model this problem?

To solve a complex problem, we identify the problem statement (desired variable) and the boundaries (conditions) existing and build a mathematical equation (model) depending on how they affect the desired variable.

Here, Profit is the desired variable being discussed about that is C.

Generally, we try to attain maximum profits. So, we are trying to attain Cmaximum.

First equation:

For 1 unit of label X1 shipped, profit = C1

For 10 units of label X1 shipped, profit = 10*C1

$\small Cmax = {\color{Red} \max }(X1.Y1+X2.Y2+......+Xn.Yn)$

where Yi is the number of goods labelled Xi being shipped.

Condition:

Here, we have the boundary or limit related to total resources.

For 1 unit of label X1 shipped, resources utilized = Aj1

For 10 units of label X1 shipped, resources utilized = 10*Aj1

Total number of resources utilized = Aj1Y1+Aj2.Y2+….+AjnYn

The total resources available = Bm

So, if we can utilize more resources, we can ship more items there by attaining more profits. So, we should utilize resources to the maximum i.e.; Bm

Therefore, Bm = Aj1Y1+Aj2.Y2+….+AjnYn

The equations deduced are

1. $\small Cmax = {\color{Red} \max }(X1.Y1+X2.Y2+......+Xn.Yn)$ – Containing the decision variable C

2. Bm = (Aj1Y1+Aj2.Y2+….+AjnYn )   – The condition or limiter

So, if we are given the total resources available and the resource utilized per item, we can solve the problem.

It ccan be done in Excel using solver.

C is target cell (maximum)

Bm – constraint

Yi – cells to be changed

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