Brighter Ideas produces two types of residential light bulbs: incandescent and h

Question

Brighter Ideas produces two types of residential light bulbs: incandescent and halogen. Brighter Ideas has four manufacturing sites for its light bulbs and two principal markets, but they do not know how to ship their product to best satisfy demand over the coming week. Brighter Ideas four manufacturing centers are Phoenix, Portland, Madison, and Rochester with a weekly production potential of 2,000, 13,000, 6,000, and 15,000 respectively for the incandescent bulbs and 19,000, 9,000, 9,000, and 8,000 respectively for the halogen bulbs. The two markets are Atlanta and Baltimore and have a weekly demand of 17,000 and 19,000 respectively for incandescent bulbs and 34,000 and 11,000 respectively for halogen bulbs. Due to contract requirements, Brighter Ideas must satisfy this demand.

Brighter Ideas has contracted a shipping company to ship the bulbs and has contracted the following shipping costs (given in $/bulb):

Incandescent Bulbs

Demand

Atlanta

Baltimore

Manufacture Site

Phoenix

2.81

0.66

Portland

3.29

1.85

Madison

3.9

4.15

Rochester

2.18

1.89

Halogen Bulbs

Demand

Atlanta

Baltimore

Manufacture Site

Phoenix

2.26

2.5

Portland

1.37

3.87

Madison

1.66

1.74

Rochester

2.4

3.68

Using this information, complete the following tasks/questions:

Mathematically formulate (i.e. type out or write down) a linear optimization model to minimize Brighter Ideas’ transportation costs for the week including all decision variables, the objective function, and all constraints.

Incandescent Bulbs

Demand

Atlanta

Baltimore

Manufacture Site

Phoenix

2.81

0.66

Portland

3.29

1.85

Madison

3.9

4.15

Rochester

2.18

1.89

Halogen Bulbs

Demand

Atlanta

Baltimore

Manufacture Site

Phoenix

2.26

2.5

Portland

1.37

3.87

Madison

1.66

1.74

Rochester

2.4

3.68

Explanation / Answer

Mathematical formulation of Linear optimization model is as follows:

Decision variables: Let Nij be the number of incandescent bulbs to be shipped from plant i to market j and, Hij be the number of Halogen bulbs to be shipped from plant i to market j

Objective: Min 2.81N11+0.66N12+3.29N21+1.85N22+3.9N31+4.15N32+2.18N41+1.89N42+2.26H11+2.5H12+1.37H21+3.87H22+1.66H31+1.74H32+2.4H41+3.68H42

Constraints:

N11+N12 <= 2000

N21+N22 <= 13000

N31+N32 <= 6000

N41+N42 <= 15000

N11+N21+N31+N41 = 17000

N12+N22+N32+N42 = 19000

H11+H12 <= 19000

H21+H22 <= 9000

H31+H32 <= 9000

H41+H42 <= 8000

H11+H21+H31+H41 = 34000

HN12+H22+H32+H42 = 11000

Nij, Hij >= 0

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