Solving of this problem will become simple with Linear Programming (LP). LP is a mathematical technique for optimization of linear objective function. In 1939, Leonid Kantorovich, a Russian mathematician, founds this technique. Then, George B. Datzig, John von Neumann, Leonid Khaciyan, and Narendra Karmarkar develop this technique. Clearance sale shirt and t-shirt example below will show that LP can be used to solve this problem. For example, a department store opens New Year clearance sale for shirt and t-shirt. One shirt prices $25 and $15 for a t-shirt. A shop looks this opportunity to make a $400. In addition, his inventory only can keep 100 clothes, whereas he has kept 53 shirts and 29 t-shirts. Then, he considers that he can sell shirt at least twice than t-shirt in one week. How much shirt and t-shirt should he buy in this clearance sale?

Consider that X_{1} is for shirt and X_{2} is for t-shirt. The objective of this problem is maximize profit from (40-25) X1 + (35-15) X_{2} = 15 X_{1} + 20 X_{2}. There are three constraint of this problem. First, sales demand constraint is X_{1} – 2X_{2} >= 0. Second, budget constraint is 25X_{1} +15 X_{2} <= 400. Third, capacity constraints is X_{1} + X_{2} <= (100-(53+29)) ~ 18. So the mathematical equation is:

Max Profit: z = 15X_{1} + 20 X_{2}

Constraint: X_{1 }– 2X_{2} >= 0

25X_{1} +15 X_{2} <= 400

X_{1} + X_{2} <= 18

There are many ways to solve this problem such as LP Graphic and Excel Solver. The graphic and the excel solver output will be showed below.

From graphics, there are three alternatives of optimum solutions. First, in point (18, 0) will make profit $270. Second, in point (13, 5) will make profit $295. Third, in point (12.3, 6.15) will make profit $307.5, but this answer is invisible because shirt and t-shirt cannot be decimal. Hence, the nearest options are (12, 6) with profit $300. Furthermore, He should buy 12 shirts and 6 t-shirts. It is very suitable with the answer from Excel Solver table below.

The owner of shop also can analyze the changing in the parameters of this LP problem. This analysis is called sensitivity analysis. In this case, Excel Solver is also used to define the sensitivity analysis.

The table above shows that the final answer will not be change if:

1. The profit of Shirt changes in range from 0 to 20 infinity

2. The profit of T-Shirt changes in range from 15 to infinity

3. The constraint of budget change in range from 290 to infinity

4. The constraint of capacity change in range from 0 to 18.461538462

5. The constraint of demand change in range from -36 to 3

Hence, have fun in shopping.

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