How Do Restaurants Use Linear Programming for Menu Planning?

1.
Choose the decision variables that apply. In this example, a restaurant needs to produce 250 of its dinner specials per day, one with meat and the other vegetarian. The decision variables are the number of meals and the different menu names (i.e., porterhouse steak and spinach lasagna).

2.
Choose the objective for the restaurant. Normally, the objective is to determine how many of each menu item to prepare that meets the required number of meals yet stays within budget, so this is the objective for the example shown. However, the objective will be the quantity of physical supplies on hand, if there is a shortage of a particular ingredient that several menu items use, such as tomato sauce. Then management can determine how to get the largest number of meals with the quantity of tomato sauce on hand.

3.
Choose the constraints on menu production, which is the day’s monetary budget to produce a specified number of meals. For example, a restaurant has a $1,000 budget for that day’s two dinner specials, and it must prepare 250 meals that cost different amounts to prepare. It cannot spend more than $1,000 and still earn a profit.

4.
Choose the two dinner specials, such as porterhouse steak and spinach lasagna. For this example, the porterhouse steak costs $7 to prepare and the lasagna dinner costs $3. The steak is designated as “S” and the lasagna as “L."

5.
Calculate how many steak dinners the restaurant can prepare for $1,000: S + L = 250 meals. 7S <= $1,000 S <= $1,000 / 7 = 142.85 S = 142 meals for $1,000 (The restaurant cannot serve 85/100 of a meal, so that amount is dropped.)

6.
Calculate how many lasagna dinners the restaurant can prepare for $1,000. 3L <= $1,000 L <= $1,000 / 3 = 333.33 L = 333 meals for $1,000

7.
Calculate the ratio: 142S divided by 333L = 42 percent (drop the decimals). This means that 42 percent of the meals should be steak dinners. Conversely, 58 percent of the dinner specials should be spinach lasagna.

8.
Calculate the number of steak dinners the restaurant can prepare on its budget: 142S times 42 percent = 59 steak dinners (drop the decimals)

9.
Calculate the number of spinach lasagna dinners the restaurant can prepare on its budget: 333L times 58 percent = 193 lasagna dinners.

10.
Verify the quantity of meals: 59 steak dinners plus 193 lasagna dinners = 252 meals. Since the restaurant only has to prepare 250 meals, it is under budget, which means increased profit.

11.
Verify the cost: 59 steak dinners times $7 equals $413. 193 lasagna dinners times $3 equals $579. $413 + $579 = $992, which is under budget.