IE504 Facilities Design
Spring 2000 Exam II
May 1, 2000
Closed Books and Notes
Max Score: 100 points
1. Does the introduction of barriers for a Rectilinear metric location problem always
increase the number of all cells (regions) that are formed? Explain your answer,
supporting your choice.
(6 points)
2. Explain the difference between a barrier and a forbidden region. Give two
examples of barriers in a plant and two examples of forbidden regions. (8 points)
3. Explain why the algorithm discussed in lecture for the single facility Euclidean
metric minimax problem will eventually terminate.
(6 Points)
4. In the case of the dedicated storage, let T denote the length of the planning
horizon in time periods, Q denote the storage capacity, d t denote the storage
space required in period t, c0 denote the cost per unit storage capacity, c1, t denote
the cost per unit stored in owned space during period t, and c2, t denote the cost per
unit stored in leased space during period t.
(i) Write an expression for the fixed cost of storing items in the planning
horizon, explain its validity.
(5 Points)
(ii) Write an expression for the cost of using owned space in period t. Explain
its validity.
(5 Points)
(iii) Write an expression for the cost of using leased space in period t. Explain
its validity.
(5 Points)
(iv) Write an expression for the total cost of storing items in the planning
horizon. Explain its validity.
(5 Points)
5. Consider the problem of assigning products to storage/retrieval locations.
Let xj, k = 1 if product j is assigned to storage/retrieval location k, and 0 otherwise.
Also assume that product j needs S j storage locations. Furthermore, let S denote
the total number of available storage locations, with s =  S j .
j
(i) Write a constraint that ensures that each storage location gets assigned one
product. Explain its validity.
(5 Points)
(ii) Write a constraint that ensures that each product gets assigned its requisite
number of storage locations. Explain its validity.
(5 Points)
6. Explain how a randomized storage location policy is implemented. Also explain
why its space requirement is typically less than that for a dedicated storage
location policy. Finally, explain why the total distance for handling product is
typically higher than for a dedicated storage location policy.
(15 Points)
7. The Objective function for the single-facility minisum Euclidean distance location
problem is given by
m
f ( x, y )   wi [(x-ai)2 + (y-bi)2]1/2 , where (x,y) are the coordinates of the new
i 1
facility, (ai ,bi) are the coordinates of existing facility i, for i = 1,....,m, and wi is
the weight of existing facility i.
(a) Show that f ( x, y ) / x and f ( x, y ) / y are undefined when (x, y) = (ai ,bi)
for some i = 1,....,m.
(5 Points)
(b) Explain how to "get around" the difficulty in (a).
(5 Points)
(c) Explain why we need to perform an iterative procedure to find the optimal
values of x and y.
(5 Points)
8. Consider the following problem instance of a single facility Rectilinear metric
location problem, in which the objective function is the weighted distance
between the new facility and the existing facilities
y
Existing facility
5
3
4
4
3
1
Travel
Directions
2
1
x
2
4
6
10
(a) Explain why the location (1,4) cannot be an optimal location for the facility,
regardless of the weights of the existing facilities.
(5 Points)
(b) let the weights of the existing facilities be w1 = 0.4 , w2 = 0.2 , w3 = 0.3 ,
w4 = 0.1. Let the new facility be located at a point with coordinates (x, y).
Write the objective function as a function of x and y. Explain your answer.
(5 Points)
(c) For the problem description in (b), write the objective function as a linear
function of x and y when 4 < x < 6 and 1 < y < 3. (Note that cx + dy + e is a
linear function of x and y, where c, d, and e are constants.) Explain your
answer.
(10 Points)