a major coffee supplier has warehouses in seattle and san jose. the coffee supplier receives orders from…

a major coffee supplier has warehouses in seattle and san jose. the coffee supplier receives orders from coffee retailers in salt lake city and reno. the retailer in salt lake city needs 450 pounds of coffee, and the retailer in reno needs 350 pounds of coffee. the seattle warehouse has 750 pounds available, and the warehouse in san jose has 550 pounds available. the cost of shipping from seattle to salt lake city is $2.50 per pound, from seattle to reno $3 per pound, from san jose to salt lake city $4 per pound, and from san jose to reno $2 per pound. find the number of pounds to be shipped from each warehouse to each retailer to minimize the cost.\nship pounds of coffee from seattle to salt lake city, pounds of coffee from seattle to reno, pounds of coffee from san jose to salt lake city, and pounds of coffee from san jose to reno.

a major coffee supplier has warehouses in seattle and san jose. the coffee supplier receives orders from coffee retailers in salt lake city and reno. the retailer in salt lake city needs 450 pounds of coffee, and the retailer in reno needs 350 pounds of coffee. the seattle warehouse has 750 pounds available, and the warehouse in san jose has 550 pounds available. the cost of shipping from seattle to salt lake city is $2.50 per pound, from seattle to reno $3 per pound, from san jose to salt lake city $4 per pound, and from san jose to reno $2 per pound. find the number of pounds to be shipped from each warehouse to each retailer to minimize the cost.\nship pounds of coffee from seattle to salt lake city, pounds of coffee from seattle to reno, pounds of coffee from san jose to salt lake city, and pounds of coffee from san jose to reno.

Answer

Explanation:

Step1: Define variables

Let $x$ be the pounds of coffee shipped from Seattle to Salt Lake City, $y$ be the pounds of coffee shipped from Seattle to Reno. Then the pounds of coffee shipped from San - Jose to Salt Lake City is $450 - x$ and the pounds of coffee shipped from San - Jose to Reno is $350 - y$.

Step2: Set up constraints

We have the following constraints based on the available stock in warehouses: $x + y\leqslant750$ (Seattle warehouse capacity) $(450 - x)+(350 - y)\leqslant550$ which simplifies to $x + y\geqslant250$ Also, $x\geqslant0,y\geqslant0,450 - x\geqslant0,350 - y\geqslant0$ or $0\leqslant x\leqslant450,0\leqslant y\leqslant350$

Step3: Set up the cost function

The cost function $C$ is given by: $C = 2.5x+3y + 4(450 - x)+2(350 - y)$ $C=2.5x + 3y+1800-4x + 700-2y$ $C=-1.5x + y+2500$

Step4: Analyze the cost function within constraints

We want to minimize $C=-1.5x + y+2500$. Since the coefficient of $x$ is negative ($- 1.5$) and the coefficient of $y$ is positive ($1$), to minimize the cost, we want to make $x$ as large as possible within its constraints. The maximum value of $x$ is $450$ (because the retailer in Salt Lake City needs 450 pounds). If $x = 450$, then from the demand of Reno retailer, $y=350$ (because the retailer in Reno needs 350 pounds) The pounds of coffee shipped from San - Jose to Salt Lake City is $450 - 450=0$ The pounds of coffee shipped from San - Jose to Reno is $350 - 350 = 0$

Answer:

450, 350, 0, 0