A furniture company produces chairs and tables. The company aims to maximize its profit, which

is $$20$ per chair and $$30$ per table. Each chair requires $2$ hours of labor and $1$ unit of wood. Each

table requires $4$ hours of labor and $3$ units of wood. The company has $120$ labor hours available per

week. There are $90$ units of wood available per week. Calculate the shadow price for each of the

constraints.

A company manufactures two types of gadgets, A and B$.$ The profit from each type A gadget is $$40,$

and from each type B gadget is $$50.$ The production process is constrained by two factors: machine

hours and raw material availability. Each type A gadget requires $3$ machine hours and $4$ units of raw

material. Each type B gadget requires $5$ machine hours and $2$ units of raw material. There are $180$

machine hours available per week. There are $160$ units of raw material available per week.

Formulate the linear programming problem and calculate the shadow price for each of the

constraints.

A logistics company needs to transport goods from its warehouse to two different regions using

trucks and trains. The objective is to minimize transportation costs. The cost of transporting a unit

of goods by truck is $$30$ and by train is $$20.$ The constraints include: The maximum capacity for

trucks is $100$ units per week. The maximum capacity for trains is $150$ units per week. The total

demand in both regions is $200$ units per week. Formulate the linear programming problem and

calculate the shadow price for each constraint.