Case Study

Week 9 Capacitated Facility Location
(application of mixed-integer program)
Class Practical
Case Study – SUNOIL
SunOil is a manufacturer of petrochemical products with worldwide scales. The Vice President of
Supply Chain is considering consolidate plants in few regions: North America, South America, Europe,
Africa and Asia.
The corresponding input data, including:
1) demand by regions,
2) the fixed cost and capacity of plant options (with low and high capacities), and
3) variable production, transport and inventory cost associated with regions, are shown in the
following figure.
Figure 1: Input Data
OK computer
The options of low and high capacities in each region are mutually exclusive. In other words, at the
most only one option in each region can be adopted.
The Vice President wants to:
minimize (Objective Function)
o the plant facility fixed cost, which is independent from the production quantity, plus
o the production, transport and inventory cost, which is dependent on the production
quantity,
by exploring (Decision Variables)
o the allocation of production quantities in supply regions to demand regions, and
o choice of plant facility options (high or low production capacity).
Subject to the following conditions (Constraints):
o demand from regions needs to be met, and
o allocation of production quantities from regions cannot exceed the corresponding
production capacities.
Steps
The decision variables
The constraints are specified as formulas

Both Plants?
Logic Produced Capacity
=Low+High =SUM(C19:G19) =L_Capacity*Low+H_Capacity*High
=Low+High =SUM(C20:G20) =L_Capacity*Low+H_Capacity*High
=Low+High =SUM(C21:G21) =L_Capacity*Low+H_Capacity*High
=Low+High =SUM(C22:G22) =L_Capacity*Low+H_Capacity*High
=Low+High =SUM(C23:G23) =L_Capacity*Low+H_Capacity*High
If Low is open we cannot have
high. In Solver we will set
these vales <=1.
This is the usual transport
model capacity constraint
However, in a capacitated network model
the capacity only is valid if the plant is open.
Here if Low=1 we have Low capacity

Don’t forget the demand constraints (not shown here)
Decision Variables (Values to be found to minimise to the total of “facility cost” and “transport cost” )

N. America S. America Europe Asia Africa
Low High Logic Produced Capacity
N. America
S. America
Europe
Asia
Africa
N. America S. America Europe Asia Africa
0 0 0 0 0
Low Capacity
Plants
(1=Open)
High Capacity
Plants
(1=Open)
Supply Regions Both Plants? Demand Region – Production Allocation (Million Units)

The objective function as a formula

Objective Function
Transport Cost =SUMPRODUCT(UnitCost,ShippingSchedule)
Fixed Costs =SUMPRODUCT(Low,L_FixCost)+SUMPRODUCT(High,H_FixedCosts)
Total Cost =CostProduction+CostFixed

Set up the linear program by calling “Solver” as shown in Figure 2.
Figure 2: Optimisation “Solver”
Don’t forget we still use the linear programming (Simplex method) and non-negative decision
variable constraints

Find out the optimum solution to the problem by clicking “Solve”.
Figure 3: Solution
Questions
1. What is the value of the objective function?
2. Which supply regions are chosen open, with low or high capacity options? Why do the plants in North
America and Europe need to be closed?
3. Is all demand met?
4. Does demand for a plant exceed the corresponding production capacity?
5. Why can we not get a sensitivity report
Save Your Model. Open Solver and choose the Save/Load option. (put the model out of the way/ but
must be on the same sheet … try cell P10)
Now change your model to the transport model with the current configuration. This will allow you to
get a sensitivity report.
Figure 4 Corresponding Transport Model
Decision Variables (Values to be found to minimise to the total of “facility cost” and “transport cost” )

N. America S. America Europe Asia Africa Low High Logic Produced Capacity
N. America
S. America
Europe
Asia
Africa
0 0 0 0 0
12 8 0 0 0
0 0 0 0 0
0 0
4 16 0
0 0
10 0 7
0
0
0
0
0
0
1
0
1
1
0
1
0
1
1
0 0
20 20
0 0
20 20
17 20
N. America S. America Europe Asia Africa
12 8 14 16 7
Low Capacity
Plants
(1=Open)
High Capacity
Plants
(1=Open)
Supply
Regions
Both Plants? Demand Region – Production Allocation (Million Units)

Objective Function

Transport Cost $ 485,100
Fixed Costs $ 1,890,000

 

Total Cost $ 2,375,100

Note: you leave the network in its optimal state (3 high capacity plants operating) and solve the
shipping schedule. It will have the same answer as the integer model, but we run again to sensitivity
report).
Be careful of the shadow prices because these do not include fixed costs. One unit of production at
N America will save $3,600, but even if we were to meet capacity of 10 units it would not justify the
capital cost of $600,000. This is why the integer program settled on the three plants.
o Why is the shadow cost for one more unit delivered to N America $11,700?
o Why do some constraints have have negative shadow prices and others have
positive shadow prices?

Exercise 2
Complete the following example on your own.
Drylce
, Inc., is a manufacturer of air conditioners that has seen its demand grow significantly. The
company
anticipates nationwide demand for the next year to be 180,000 units in the South, 120,000
units in the Midwest, 110,000 units in the East, and 100,000 units in the West. Managers at Dry Ice
are designing the manufacturing network and have selected four potential
sites-New York, Atlanta,
Chicago
, and San Diego. Plants could have a capacity of either 200,000 or 400,000 units. The annual
fixed costs at the four locations are shown in Table below, along with the cost of producing and
shipping an air conditioner to
each of the four markets. Where should Drylce build its factories and
how large
should they be?

Capacity New York Atlanta Chicago San Diego
Fixed Costs
200 000 $6,000,000 $5,500,000 $5,600,000 $6,100,000
400 000 $10,000,000 $9,200,000 $9,300,000 $10,200,000
Variable
Costs
New York Atlanta Chicago San Diego
East $211 $232 $238 $299
South $232 $212 $230 $280
Midwest $240 $230 $215 $270
West $300 $280 $270 $225
Demand
East 110,000
South 180,000
Midwest 120,000
West 100,000

Optimization model:

n = 4: potential sites.
m = 4: number of regional markets.
D
j = annual units needed of regional market j
K
i = maximum possible capacity of potential sites
(Each K
i is assigned value 400000. If actually needed
capacity is less than or equal to 200000, we choose fixed cost accordingly.)
f
i = annualized fixed cost of setting up a potential site
c
ij= cost of producing and shipping an air conditioner from site i to regional market j
y
i = 1 if site i is open, 0 otherwise
x
ij = number of air conditioners from site i to regional market j
1 1 1
n
ij
i 1
m
ij
j 1
Subject to
x 1 (5.1)
x 1 (5.2)
n n m
i i ij ij
i i j
j
i i
Min f y c x
D for j ,…,m
K y for i ,…,n
= = =
= =
+
= =
≤ =
∑ ∑∑
∑ ∑