Questions: A greeting card manufacturer has 400 boxes of a particular card in warehouse I and 270 boxes of the same card in warehouse II. A greeting card shop in San Jose orders 325 boxes of the card, and another shop in Memphis orders 250 boxes. The shipping costs per box to these shops from the two warehouses are shown in the table. How many boxes should be shipped to each city from each warehouse to minimize shipping costs? What is the minimum cost? (Hint: Use x, 325-x, y, and 250-y as the variables.) San Jose Memphis Warehouse I 0.25 0.22 Warehouse II 0.24 0.19 From warehouse I, ship box(es) to San Jose and box(es) to Memphis, and from warehouse II, ship box(es) to San Jose and box(es) to Memphis, for a minimum cost of .

Solution Steps

Let:

• \( x \) = number of boxes shipped from Warehouse I to San Jose.
• \( 325 - x \) = number of boxes shipped from Warehouse II to San Jose.
• \( y \) = number of boxes shipped from Warehouse I to Memphis.
• \( 250 - y \) = number of boxes shipped from Warehouse II to Memphis.

The constraints are based on the available boxes in each warehouse:

1. From Warehouse I: \( x + y \leq 400 \).
2. From Warehouse II: \( (325 - x) + (250 - y) \leq 270 \), which simplifies to \( x + y \geq 305 \).
3. Non-negativity constraints: \( x \geq 0 \), \( y \geq 0 \), \( 325 - x \geq 0 \), and \( 250 - y \geq 0 \).

The total shipping cost \( C \) is: \[ C = 0.25x + 0.22y + 0.24(325 - x) + 0.19(250 - y). \] Simplify the cost function: \[ C = 0.25x + 0.22y + 78 - 0.24x + 47.5 - 0.19y, \] \[ C = (0.25x - 0.24x) + (0.22y - 0.19y) + 125.5, \] \[ C = 0.01x + 0.03y + 125.5. \]

The feasible region is determined by the constraints:

1. \( x + y \leq 400 \),
2. \( x + y \geq 305 \),
3. \( x \geq 0 \), \( y \geq 0 \),
4. \( x \leq 325 \), \( y \leq 250 \).

The feasible region is a polygon bounded by the lines \( x + y = 305 \) and \( x + y = 400 \), with \( x \) and \( y \) non-negative.

To minimize \( C = 0.01x + 0.03y + 125.5 \), evaluate the cost at the vertices of the feasible region:

1. At \( x = 0 \), \( y = 305 \): \[ C = 0.01(0) + 0.03(305) + 125.5 = 9.15 + 125.5 = 134.65. \]
2. At \( x = 305 \), \( y = 0 \): \[ C = 0.01(305) + 0.03(0) + 125.5 = 3.05 + 125.5 = 128.55. \]
3. At \( x = 325 \), \( y = 75 \): \[ C = 0.01(325) + 0.03(75) + 125.5 = 3.25 + 2.25 + 125.5 = 131. \]
4. At \( x = 150 \), \( y = 250 \): \[ C = 0.01(150) + 0.03(250) + 125.5 = 1.5 + 7.5 + 125.5 = 134.5. \]

The minimum cost occurs at \( x = 305 \), \( y = 0 \), with \( C = 128.55 \).

From Warehouse I:

• Ship \( x = 305 \) boxes to San Jose.
• Ship \( y = 0 \) boxes to Memphis.

From Warehouse II:

• Ship \( 325 - x = 20 \) boxes to San Jose.
• Ship \( 250 - y = 250 \) boxes to Memphis.

From Warehouse I, ship \( 305 \) box(es) to San Jose and \( 0 \) box(es) to Memphis, and from Warehouse II, ship \( 20 \) box(es) to San Jose and \( 250 \) box(es) to Memphis, for a minimum cost of \$ \( 128.55 \).

Final Answer