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

SCHEDULING CLASS (MANUFACTURING ENGINEERING) !!!!! 4. 4. Suppose you make ci profit for every xi you ship. You have n possible items Xn. If shipping x, unit consumes aj amount of j resources for every i product have a total bm r to ship, labeled x1, x2, esources to ship it, how would you model this problem?

SCHEDULING CLASS (MANUFACTURING ENGINEERING) !!!!!

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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?

Expert Answer

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 = A_j1

For 10 units of label X1 shipped, resources utilized = 10*A_j1

Total number of resources utilized = A_j1Y₁+A_j2.Y₂+….+A_jnY_n

The total resources available = B_m

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.; B_m

Therefore, B_m = A_j1Y₁+A_j2.Y₂+….+A_jnY_n

The equations deduced are

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

2. B_m = (A_j1Y₁+A_j2.Y₂+….+A_jnY_n ) – 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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