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Example 2 Solving a Linear Programming Problem
Find the maximum value of the objective function
Objective function
where x ≥ 0 and y ≥ 0, subject to the constraints

⇒ Constraints

Figure .2.4.

Solution
The region bounded by the constraints is shown in Figure 2.4. By testing the objective function at each vertex, you obtain
⇒ (Maximum value of z)
So, the maximum value of z is 132, and this occurs when x = 15 and y = 12.
Example 3 An Unbounded Region
Find the maximum value of
Objective function
where x ≥ 0 and y ≥ 0 subject to the constraints
x +
Constraints
Solution
The region determined by the constraints is shown below.

Figure 2.5.
For this unbounded region, there is no maximum value of z. To see this, note that the point (x, 0) lies in the region for all values of x ≥ 4. By choosing large values of x, you can obtain values of

that are large. So, there is no maximum value of z.
Example 4 An Application: Optimal Cost
Example 4 in [1]. set up a system of linear equations for the problem below. The liquid portion of a diet is to provide at least 300 calories, 36 units of vitamin A, and 90 units of vitamin C daily. A cup of dietary drink X provides 60 calories, 12 units of vitamin A, and 10 units of vitamin C. A cup of dietary drink Y provides 60 calories, 6 units of vitamin A, and 30 units of vitamin C. Now, assume that dietary drink X costs $0.12 per cup and drink Y costs $0.15 per cup. How many cups of each drink should be consumed each day to minimize the cost and still meet the daily requirements?
Solution
Begin by letting x be the number of cups of dietary drink X and y be the number of cups of dietary drink Y. Moreover, to meet the minimum daily requirements, the inequalities listed below must be satisfied.
Constraints
The cost C is
. Objective function
The graph of the region corresponding to the constraints is shown in Figure 2.6.
To determine the minimum cost, test C at each vertex of the region, as shown below.
At
At
At (Maximum value of C)
At
So, the minimum cost is $0.66 per day, and this occurs when three cups of drink X and two cups of drink Y are consumed each day.

Figure 2.6.
When financial institutions replenish automatic teller machines (ATMs), they need to take into account a large number of variables and constraints to keep the machines stocked appropriately. Demand for cash machines fluctuates with such factors as weather, economic conditions, day of the week, and even road construction. Further complicating the matter in the United States is a penalty for depositing and withdrawing money from the Federal Reserve in the same week. To address this complex problem, a company that specializes in providing financial services technology can create high-end optimization software to set up and solve a linear programming problem with many variables and constraints. The company determines an equation for the objective function to minimize total cash in ATMs, while establishing constraints on travel routes, service vehicles, penalty fees, and so on. The optimal solution generated by the software allows the company to build detailed ATM restocking schedules [1][6].

2.