Simplex Tableau (word problem)

Question

A Gamestop sells the following consoles: PS4's, XboxOne's, and Nintendo Switches. For each PS4 it stocks, it needs to have 6 games on hand, for each XboxOne it needs to have 5 games, and for each Switch it needs 10 games. They want to have at most 2100 games on hand. The store wants to stock at most 300 consoles in total. The revenue from each console is $300 per PS4, $400 per XboxOne, and $600 per Switch.

 

List all the constraints for the linear programming problem.

 

Introduce slack variables and create the simplex tableau.

 

Using the tableau and your calculator, nd the amount of each console that the store should stock to maximize revenue.

 

 

Answer 1

Variables:

1. x₁ = number of PS4s
2. x₂ = number of XboxOnes
3. x₃ = number of Nintendo Switches

Objective function (maximize revenue): Z = 300x₁ + 400x₂ + 600x₃

Constraints:

1. Games constraint: 6x₁ + 5x₂ + 10x₃ ≤ 2100
2. Console stock constraint: x₁ + x₂ + x₃ ≤ 300
3. Non-negativity: x₁ ≥ 0, x₂ ≥ 0, x₃ ≥ 0

Introducing slack variables s₁ and s₂ to convert inequalities to equalities:

1. 6x₁ + 5x₂ + 10x₃ + s₁ = 2100
2. x₁ + x₂ + x₃ + s₂ = 300
3. Z − 300x₁ − 400x₂ − 600x₃ = 0

The initial simplex tableau and optimal solution are shown below:

V

visualize

V

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Summary of the solution:

To maximize revenue, the store should stock 0 PS4s, 180 XboxOnes, and 120 Nintendo Switches, yielding a maximum revenue of $144,000.

The simplex method reached the optimum in just 2 iterations. The key insight is that the Switch generates the most revenue per unit ($600) but is also the most game-intensive (10 games each), so the optimal mix balances the game constraint against the console limit — settling on more XboxOnes (which are cheaper in game slots at 5 each) alongside as many Switches as the games constraint allows