This exercise was also contributed by Dr. Rick Wilson of Oklahoma State University. The following simple scenario mimics the “Black Book” described in a Business Week article (http://www.businessweek.com/articles /2013-01-31/coke-engineers-its-orange-juice-with-an algorithm, accessed February 2013) about Coca-Cola’s production of orange juice. Create an appropriate LP model for this scenario. For the next production period, there are five different batches of raw orange juice that can be blended together to make orange juice products, SunnyQ, GlowMorn, and OrenthalJames. In creating the optimal blend of the three products from the five different batches, an LP model should seek to maximize the net of the sales price per gallon of the products less the assessed per-gallon cost of the raw juice. The five raw batches of orange juice are described here. Brix is a measure of sweetness, pulp, available stock, and cost—all self-explanatory:

Batch 1—Pineapple Orange A, brix = 16, pulp = 1.2, 250 gallons, $2.01/gallon

Batch 2—Pineapple Orange B, brix = 17, pulp = 0.9, 200 gallons, $2.32/gallon

Batch 3—Mid Sweet, brix = 20, pulp = 0.8, 175 gallons, $3.14/gallon

Batch 4—Valencia, brix = 18, pulp = 2.1, 300 gallons, $2.41/gallon

Batch 5—Temple Orange, brix = 14, pulp = 1.6, 265 gallons, $2.55/gallon

Note that to make sure that the raw juice doesn’t get too “old” over time, one production requirement is that at least 50% of each batch’s available stock must be used in blending the three orange juice products (obviously, more than what is available cannot be used). From a product perspective, there must be at least 100 gallons of SunnyQ blended and at least 125 gallons each of GlowMorn and OrenthalJames. Likewise, the projected future demand for the products indicates that in this period, there should be a maximum of 400 gallons of SunnyQ, a maximum of 375 gallons of GlowMorn, and a maximum of 300 gallons of OrenthalJames produced. Also, when blending the products from the five batches, an individual batch can provide no more than 40% of the total amount of a given product. This is to be enforced individually on each product. Attributes of the three products include sales price, the maximum average brix of the final mixed product, the minimum average brix of the final mixed product, and the maximum average pulp content. In the three “average” requirements, this implies the weighted average of all juice mixed together for that product must meet that specification.

SunnyQ—Sales = $3.92/gallon, Max Brix = 19, Min Brix = 18.5, Max Pulp = 1.6

GlowMorn—Sales = $4.13/gallon, Max Brix = 17, Min Brix = 16.75, Max Pulp = 1.8

OrenthalJames—Sales = $3.77/gallon, Max Brix = 17.75, Min Brix = 17.55, Max Pulp = 1.1