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.

Answer 1

Decision variables:

Let A_ij denotes the number of Incadescent lamp and B_ij denotes the number of halogen lamps manufactured at locations i , i = 1 to 4 and supplied to destiantion j, j = 1 and 2 where i = 1,2,3 and 4 denotes the locations Phoenix, Portland, Madison and Rochester respectively and j = 1 and 2 denotes the destinations Atlanta and Baltimore respectively. The total cost function C is given as

C = 2.81 A₁₁ + 0.66 A₁₂ + 3.29 A₂₁ + 1.85 A₂₂ + 3.9 A₃₁ + 4.15 A₃₂ + 2.18 A₄₁ + 1.89 A₄₂ +

    2.26 B₁₁ + 2.5 B₁₂ + 1.37B₂₁ + 3.87 B₂₂ +1.66 B₃₁ + 1.74 B₃₂ + 2.4 B₄₁ + 3.68 B₄₂

The objective is to minimize the value of C. So the objective function is

Minimize C = 2.81 A₁₁ + 0.66 A₁₂ + 3.29 A₂₁ + 1.85 A₂₂ + 3.9 A₃₁ + 4.15 A₃₂ + 2.18 A₄₁ + 1.89 A₄₂ + 2.26 B₁₁ + 2.5 B₁₂ + 1.37B₂₁ + 3.87 B₂₂ +1.66 B₃₁ + 1.74 B₃₂ + 2.4 B₄₁ + 3.68 B₄₂

Constraints: Production capacity

A₁₁ + A₁₂ + A₁₃ + A₁₄ 2,000 (Phoenix Incadescent potential)

A₂₁ + A₂₂ + A₂₃ + A₂₄ 13,000 (Portland Incadescent potential)

A₃₁ + A₃₂ + A₃₃ + A₃₄ 6,000 (Madison Incadescent potential)

A₄₁ + A₄₂ + A₄₃ + A₄₄ 15,000 (Rochester Incadescent potential)

B₁₁ + B₁₂ + B₁₃ + B₁₄ 19,000 (Phoenix halogen potential)

B₂₁ + B₂₂ + B₂₃ + B₂₄ 9,000 (Portland halogen potential)

B₃₁ + B₃₂ + B₃₃ + B₃₄ 9,000 (Madison halogen potential)

B₄₁ + B₄₂ + B₄₃ + B₄₄ 8,000 (Rochester halogen potential)

Constraints: Demand satisfaction

A₁₁ + A₂₁ + A₃₁ + A₄₁ 17,000 (Atlanta weekly requirement for incadescent)

B₁₁ + B₂₁ + B₃₁ + B₄₁ 34,000 (Atlanta weekly requirement for halogen)

A₁₂ + A₂₂ + A₃₂ + A₄₂ 19,000 (Baltimore weekly requirement for incadescent)

B₁₂ + B₂₂ + B₃₂ + B₄₂ 11,000 (Baltimore weekly requirement for halogen)

Non-negativity and integer constraints:

A_ij , B_ij 0 and integers for i = 1, 2, 3 and 4 and j = 1 and 2

The linear optimization model to minimize Brighter Ideas' transportation costs for the week is summarised as below.

Minimize C = 2.81 A₁₁ + 0.66 A₁₂ + 3.29 A₂₁ + 1.85 A₂₂ + 3.9 A₃₁ + 4.15 A₃₂ + 2.18 A₄₁ + 1.89 A₄₂ + 2.26 B₁₁ + 2.5 B₁₂ + 1.37B₂₁ + 3.87 B₂₂ +1.66 B₃₁ + 1.74 B₃₂ + 2.4 B₄₁ + 3.68 B₄₂

Subject to the constraints

A₁₁ + A₁₂ + A₁₃ + A₁₄ 2,000

A₂₁ + A₂₂ + A₂₃ + A₂₄ 13,000

A₃₁ + A₃₂ + A₃₃ + A₃₄ 6,000

A₄₁ + A₄₂ + A₄₃ + A₄₄ 15,000

B₁₁ + B₁₂ + B₁₃ + B₁₄ 19,000

B₂₁ + B₂₂ + B₂₃ + B₂₄ 9,000

B₃₁ + B₃₂ + B₃₃ + B₃₄ 9,000

B₄₁ + B₄₂ + B₄₃ + B₄₄ 8,000

A₁₁ + A₂₁ + A₃₁ + A₄₁ 17,000

B₁₁ + B₂₁ + B₃₁ + B₄₁ 34,000

A₁₂ + A₂₂ + A₃₂ + A₄₂ 19,000

B₁₂ + B₂₂ + B₃₂ + B₄₂ 11,000

A_ij , B_ij 0 and integers for i = 1, 2, 3 and 4 and j = 1 and 2