PROBLEM 2: LP Applications in Business

Swedes are crazy about strawberries, and Judit Jordgubbar is no exception. In fact, she owns an orchard in the Skåne region, and each year she has to allocate her harvest of strawberries to various markets. The three possible markets are packaged strawberries for retail sales, production of pastries, and production of smoothies. In order to decide what to do with each strawberry, Judit uses a sorting machine that grades each strawberry on a scale of 1 to 3 according to its size, weight, and colour, where a grade of 1 is the worst one possible. Based on the sorted harvest, Judit must, therefore, determine how best to allocate her strawberries to the three markets.

This year’s harvest has rendered 100,000 kilograms (kg) of grade-1 strawberries, 170,000 kg of grade-2 strawberries, and 150,000 kg of grade-3 strawberries, which will be allocated to packaging for retail sales, pastry production, or smoothie production. Prior to the allocation, Judit must examine each market’s required average grade, the revenues from sales, and the costs of production, which are different depending on the grade of the strawberry and the target market.

The following table summarises this information (selling price, cost of production, and the average grade) for each market:

Judit is already committed to delivering 75,000 kg of strawberries for retail packaging, 50,000 kg of strawberries for pastries, and 50,000 kg of strawberries for smoothies. The average grade is calculated as a function of the component grades and quantities. Thus, if Judit sends a shipment of 1,000 kg of grade-1 strawberries, 2,000 kg of grade-2 strawberries, and 2,000 kg of grade-3 strawberries, then the average grade of the shipment will be:

You are asked to formulate a linear programming (LP) model that will help Judit to allocate her harvest among the different markets in order to maximise her profit.

Question 1:

Define clearly the decision variables for this model.

Question 2:

Formulate the objective function for this model. What does it represent?

Question 3:

There are four types of constraints for this LP: availability of each grade of strawberries, demand for each market, quality of average grade for each market, and non-negativity constraints. Write all of these constraints.