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Blending Constraints

Blending problems typically introduce a unique kind of constraint that is based upon the notion of a weighted average. Consider the following data for an automaker who is managing the MPG average for the fleet of vehicles it produces.

Car Type / Mileage
Subcompact / 45
Compact / 35
Midsize / 28

(1)The automaker produces equal amounts, or it has a 1-1-1 (Subcompact-Compact-Midsize) mixture (in cars). What is the average MPG?

But this can also be written as

(2)I have a 2-1-1 mixture. What is the average MPG?

(3)I have a 3-1-2 mixture. What is the average MPG?

(4)Now suppose that my mixture is x-y-z. How do I compute my average MPG?

(5)Imagine that I want my average MPG to be at least 38 for my x-y-z mixture. How do I write such a constraint?

From (4) we have

This implies

Including this constraint in a linear program will make the average MPG at least 38 in the solution of the model.

Another type of blending constraint arises in the following way. Assume than we wish to constrain our mixture such that no more than 40% of the fleet is made up of subcompacts. Our constraint can be written as

We then multiply through by the term in order to create a linear constraint:

which implies the linear constraint

.

Consider how we might use these types of constraints in the following problem.

Example 1: Accelerators Unlimited (AU) is a supplier of alternators used in Toyota cars. The company recently received a $750,000 order for three types of alternators it produces. Each alternator produced must first pass through two processes called wiring and blocking. AU does not have the capacity to fill the order by its due date because the company has 10,000 hours of wiring capacity and 5,000 hours of blocking capacity available to devote to this order. Nevertheless, AU can subcontract a portion of this order to one of its competitors as long as the total number of alternators subcontracted does not exceed 2500. In addition, management wants to make sure that Alternator 3 represents at least 25% of the total in house production of alternators.

The following table summarizes the requirements for the three types of alternators.

Alternator 1 / Alternator 2 / Alternator 3
Number ordered / 3000 / 2000 / 900
Hours or wiring per unit / 2.0 / 1.5 / 3.0
Hours of blocking per unit / 1.0 / 2.0 / 1.0
Cost to produce / $50 / $83 / $130
Cost to buy / $61 / $97 / $145

Create a linear program which determines the number to produce and buy for each alternator type allowing AU to fulfill the order at least cost.

Example 2: Popeye Doyle, Inc. (PDI) blends olive oils from California, Spain, and Italy into a mixture that it sells nationally (known as the French Connection (FC), even though it contains no French oil!). Each type of olive oil is characterized by its aroma and flavor. The table below shows the aroma and flavor rating for each oil provided by PDI’s olive oil taster, as well as the cost (per fluid ounce).

Californian / Spanish / Italian
Aroma / 70 / 85 / 95
Flavor / 55 / 62 / 65
Cost / 0.50 / 1.50 / 2.00

PDI wishes to sell a mixture with an aroma rating of at least 87 and a flavor rating of at least 60. PDI has 23000, 10000, and 25000 fluid ounces, respectively, of the Californian, Spanish, and Italian oils. PDI wishes to fill an order of 40000 fluid ounces at these specifications at the least cost.

Example 3: (Test yourself) Now assume that in addition to FC, PDI also makes a mixture called Superman (MOS) which has an aroma rating of 75 and a flavor rating of 58. If the FC sells at $2.30 per fluid ounce, and MOS sells at $1.80 per fluid ounce, how should PDI blend the ingredients to maximize the profitability of the blends? (Note: you should ignore the order of 40000 fluid ounces in solving this problem.)

Homework: Develop constraints for Example 1 which represent the following restrictions. (Note: You do not have to resolve the problem unless indicated.)

H1: Management wants to make sure that Alternators 2 and 3 represents at least 40% of the total in house production of alternators.

H2: A union contract restricts the average time that an alternator spends in wiring to no more than two hours.

H3: No more than 40% of the alternators sold can be outsourced. How much does this policy cost AU?