Question

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)

Answer 1

a. The Company is considering opening atleast one new plant, in order to meet the increased demand in the coming future period.The company will then be able to main its market share and not lose its share to competitors. ALso this will help the company expand its volumes and hence increase turnover and profits.

b. 18 shipment variables.

c. it is important to introduce a Binary Variable, to calculate the investment required in case of opening a new plan in each prospective area and consider the opportunity cost in case the amount is not invested and the loss of opportunity of selling more units in case of increased demand. YA and YB are required in case it is decided to not ship from existing units if the new plants work out more economic in terms of capacity,productivity and efficiency.