## Thursday, May 7, 2009

### LINEAR PROGRAMMING AND SENSITIVITY ANALYSIS FOR SHOPPING

One problem when people usually meet in shopping is what people should buy in “clearance sale” with aim to sale again. Usually people confuse to buy between/among two types of products or more. The objective of this problem usually is maximize profit, but this problem will be more difficult because people should decide among different constraints such as sales demand, budget, and capacity. How do we solve this problem?

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 X1 is for shirt and X2 is for t-shirt. The objective of this problem is maximize profit from (40-25) X1 + (35-15) X2 = 15 X1 + 20 X2. There are three constraint of this problem. First, sales demand constraint is X1 – 2X2 >= 0. Second, budget constraint is 25X1 +15 X2 <= 400. Third, capacity constraints is X1 + X2 <= (100-(53+29)) ~ 18. So the mathematical equation is:

Max Profit: z = 15X1 + 20 X2

Constraint: X1 – 2X2 >= 0

25X1 +15 X2 <= 400

X1 + X2 <= 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.