Utopia Company operates a plant in Avalon with a monthly capacity of 35,000 units and a plant in Bridgetown with a monthly capacity of 40,000 units. Product is shipped to regional distribution centers located in Riverdale, Seaside and Teatown.

The long-range planning group at Utopia reevaluated their forecasts and determined that due to an anticipated increase in demand in the next 5 years, the company should consider increasing capacity by constructing a new plant in one or more of the following cities: Cedar, Daphne, Elm or Florence.

Let Xij = the units shipped from plant i to distribution center j, with i = A, B, C, D, E or F and j = 1, 2, 3. For example, XD1 designates the # of units shipped from Plant D to Distribution Center 1.

(a) Why is the company considering opening at least one new plant?

(b) How many shipment variables will the LP formulation have? (That is, how many Xij variables are

included in the formulation?)

(c) Why do we need to introduce binary variables YC, YD, YE and YF? Do we need a binary

variables YA and YB? Why?

(d) Formulate a linear programming model for minimizing total costs (shipping costs+fixed costs);

you should have a total of 9 constraints. You can use steps (i)-(iv) below as a guide for writing the LP

Use suitable variables to express the objective function (minimize shipping costs + fixed costs).

Write the demand constraints for Distribution Centers 1, 2 and 3.

Write the supply constraints for plants A and B.

Use suitable variables to write the supply constraints for plants C, D, E and F.

(e) Write a constraint to express a requirement that the company will open only one new plant

(f) If the company will open only one plant, which plant do you think it will be C, D, E or F? Why?

(g) Write a constraint to express a requirement that the company will either open both plants C and D or neither of them (that is, if plant C opens then plant D has to open as well and if plant D is open then Plant C has to open too)