Question

A company prepares two types of packed salads, A and B. During any week, it can prepare at most 650 salads. In week 1 the cost per salad A is S4 and its demand is 400, while the cost per salad B is S3.50 and its demand is 200. In week 2 the cost per salad A is $4.40 and its demand is 300, while the cost per salad B is $4 and its demand is 300. The demands must be met on time. Also, it costs 50e to hold a salad A, and 40e to hold a salad B, in inventory for a week. Formulate an LP to minimize the total cost of meeting the next two weeks' demands

Answer 1

Let be the number of products of A and B produced in the week 1, respectively

Let be the number of products of A and B produced in the week 1, respectively.

Let , be the number of products of A and B produced in the week 1 and kept as inventory for next week respectively.

Objective function:

Minimize: Total cost Z = $4x_{1a}+3.50x_{1b}+4.4x_{2a}+4x_{2b}+0.5I_{1a}+0.4I_{b}$

Subject to:

1. Capacity constraints:

$x_{1a}+x_{1b} \leq 650$

$x_{2a}+x_{2b} \leq 650$

2. Demand Constraints:

$x_{1a} \geq 400$

$x_{1b} \geq 200$

$x_{2a} +I_a \geq 300$

$x_{2b} +I_b \geq 300$

$I_a,I_b, x_{1a}, x_{1b}, x_{2a} \,\,\, and \,\,\, x_{2b} \geq 0$

NOTE: Above model can also be modelled as below:

Objective function:

Minimize: Total cost Z =

Subject to:

1. Capacity constraints:

$x_{1a}+x_{1b} \leq 650$

$x_{2a}+x_{2b} \leq 650$

2. Demand Constraints:

$x_{1a} \geq 400$

$x_{1b} \geq 200$

$x_{2a} +x_{1a}-400 \geq 300$ or $x_{2a} +x_{1a} \geq 700$

$x_{2b} +x_{1b}-200 \geq 300$ or $x_{2b} +x_{1b} \geq 500$

$x_{1a}, x_{1b}, x_{2a} \,\,\, and \,\,\, x_{2b} \geq 0$