Question

A firm produces soft drinks at two factories: R and T. From R and T 4000 bottles should be delivered to shop A, and 3500 to shop B. Factory R can deliver not more than 4500, and factory T not more than 4000 bottles. The cost (in cents) of transportation of one bottle from each factory to each shop is the following: 2.5 from R to A; 2.5 from R to B; 3 from T to A; 2 from T to B. What is the minimum possible transportation cost in cents?

Answer 1

The minimum transportation cost can be found using linear programming by setting up and solving an optimization problem with given supply, demand, and cost parameters.

Step-by-step explanation:

The objective is to calculate the minimum possible transportation cost in cents for delivering soft drinks from two factories to two shops with given cost constraints. We need to determine how many bottles each factory should deliver to each shop to minimize the total transportation costs while not exceeding the production capacities and meeting the demand for each shop.

Let's denote the number of bottles factory R sends to shops A and B as x and y, respectively. Therefore, factory T will deliver (4000 - x) bottles to shop A and (3500 - y) to shop B. The total costs will then be: Cost = 2.5x + 2.5y + 3(4000 - x) + 2(3500 - y).

Simplifying the cost equation, we get: Cost = 12000 + 10000 - 0.5x - y. To minimize the cost, we should maximize the values of x and y while considering the constraints: x + y ≤ 4500 (Factory R's limit) and (4000 - x) + (3500 - y) ≤ 4000 (Factory T's limit).

Applying linear programming, we can use graphical methods or simplex algorithm to solve this optimization problem. By plotting the feasible region determined by the constraints and evaluating the cost at the vertices of this region, we can identify the combination of x and y that gives us the minimum cost.

The minimum cost is the lowest value of the total cost equation in the feasible region. This task involves concepts from linear programming, optimization, and economics.