A student is trying to decide on the lowest cost diet that meets the nutrition requirements, with two choices: steak and peanut butter. Steak costs 7 dollars a pound and peanut butter costs 5 dollars a pound. Steak has 2 units of protein per pound and peanut butter has 1 unit of protein per pound. A proper diet needs at least 5 units of protein per day. Unfortunately, steak also has 8 units of cholesterol per pound and peanut butter has 4 units of cholesterol per pound. High quantity of cholesterol is bad and the daily limit for cholesterol is 10 units. How many pounds of steak and peanut butter should the student consume daily?
Write this LPP in standard form, then convert into canonical form. Why do we not get a initial simplex tableau just using the original decision variables and the slack variables?
Let $x_1$ = pounds of steak
Let $x_2$ = pounds of PB
In standard form --
Minimize: $$z=7x_1+5x_2$$
Subject to:
$$2x_1+1x_2 \geq 5$$ $$8x_1+4x_2 \leq 10$$ $$x_1, x_2 \geq 0$$
In canonical form --
Minimize: $$z=7x_1+5x_2$$
Subject to:
$$2x_1+1x_2 + s_1 = 5$$ $$8x_1+4x_2 + s_2 = 10$$
$$x_1, x_2 \geq 0$$
Thus, my main question is why do we not get a initial simplex tableau just using the original decision variables and the slack variables?
Well the way I think about it is, that when you draw this graphically, there really is no region of feasibility. Perhaps we need to convert this into a maximization problem before setting up the initial simplex tableau. But what is the actual, mathematical reason behind why we cannot get an initial simplex tableau? How would I explain this to someone?