The Martin-Beck Company operates a plant in St. Louis with an annual capacity of 30,000 units. Product is shipped to regional distribution centers located in Boston, Atlanta, and Houston. Because of an anticipated increase in demand, Martin-Beck plans to increase capacity by constructing a new plant in one or more of the following cities: Detroit, Toledo, Denver, or Kansas. The following is a linear program used to determine which cities Martin-Beck should construct a plant in.

Let

y₁ = 1 if a plant is constructed in Detroit; 0 if not

y₂ = 1 if a plant is constructed in Toledo; 0 if not

y₃ = 1 if a plant is constructed in Denver; 0 if not

y₄ = 1 if a plant is constructed in Kansas City; 0 if not.

The variables representing the amount shipped from each plant site to each distribution center are defined just as for a transportation problem.

x_ij = the units shipped in thousands from plant i to distribution center j

i = 1 (Detroit), 2 (Toledo), 3 (Denver), 4 (Kansas City), 5 (St.Louis) and

j = 1 (Boston), 2 (Atlanta), 3 (Houston)

The complete model for the Martin-Beck distribution system design problem is as follows.

Min 5x₁₁ + 2x₁₂ + 3x₁₃ + 4x₂₁ + 3x₂₂ + 4x₂₃ + 9x₃₁ + 7x₃₂ + 5x₃₃ + 10x₄₁ + 4x₄₂

+ 2x₄₃ + 8x₅₁ + 4x₅₂ + 3x₅₃ + 175y₁ + 300y₂ + 375y₃ + 500y₄

s.t.

x11	+	x12	+	x13					≤ 10y1	Detroit capacity
x21	+	x22	+	x23					≤ 20y2	Toledo capacity
x31	+	x32	+	x33					≤ 30y3	Denver capacity
x41	+	x42	+	x43					≤ 40y4	Kansas City capacity
x51	+	x52	+	x53					≤ 30	St. Louis capacity
x11	+	x21	+	x31	+	x41	+	x51	= 30	Boston demand
x12	+	x22	+	x32	+	x42	+	x52	= 20	Atlanta demand
x13	+	x23	+	x33	+	x43	+	x53	= 20	Houston demand

x_ij ≥ 0 for all i and j; y₁, y₂, y₃, y₄ = 0, 1

 

(a) Modify the original formulation to account for the policy restriction that there must be one plant in either Detroit or Toledo, but not both. new constraint (V V V3 V4) = Place x, in row i column j. value= (b) Modify the original formulation to account for the policy restriction that no more than two plants can be located in Denver, Kansas City, and St. Louis. new constraint (V V V3 V4) = Place Xij value = in row i column j.