II - University of Oklahoma



Supply Chain Design

This problem considers several production plants, warehouses and distribution centers. The location and number of Warehouses and Distribution centers is to be determined. A multiperiod model is assumed.

Manufacturing Warehouses Distribution Customer

Plants Centers Zones

(Fixed Location) (Location to be selected) (Location to be selected) (Fixed Location)

Subscripts:

p: Products i:Prod. Plants j:Warehouses k:Dist. Centers

m:Customer Zones t:Periods

• Mass balance: the steady state is supposed through the Supply Chain

[pic] (1)

[pic] (2)

[pic] (3)

• Capacity constraints: there are maximum and minimum capacity constraints for the plants and warehouses of the supply chain

[pic] (4)

((pj: Production capacity factor of p at i ; Capacity per unit produced)

[pic] (5)

((it: Binary variable that states that Manufacturing Plant i is already build in period t)

[pic] (6)

((pj: Handling capacity factor of p at j ; Capacity per unit handled)

[pic] (7)

([pic]: Binary variable that states that Warehouse j is already build in period t)

[pic] (8)

((pk: Handling capacity factor of p at k ; Capacity per unit handled)

[pic] (9)

([pic]: Binary variable that states that Distribution Center k is already build in period t)

• Logical Constraints

[pic] (10)

[pic] (11)

[pic] (12)

Only one construction is allowed. The rest of the constraints define [pic]

[pic] (13)

[pic] (14)

[pic] (15)

[pic] (16)

[pic] (17)

[pic] (18)

Objective Function

The objective function is to maximize the NPV:

[pic] (Capital investment has negative cash flow) (19)

[pic](If period 1 is reserved for construction only.) (20)

[pic] (21)

[pic] (22)

[pic] (23)

Manufacturing Plants + Warehouses + Distribution

[pic] (24)

[pic] (25)

[pic] (26)

[pic] (27)

[pic] (27)

[pic] (28)

[pic] (29)

[pic] (30)

Manufacturing Plants + Warehouses + Distribution Centers

[pic] (31)

[pic] (32)

[pic] (33)

[pic] (34)

[pic] (34)

We assume straight line depreciation

[pic] (35)

New Capital cost Constraints:

For simplicity we will assume that the investment cost will have two different cost coefficients for different values of the capacity for the manufacturing plants only, according to the next figure:

[pic] [pic]

[pic] [pic]

Thus, the following constraints

[pic] (24)

should be substituted by the following:

[pic] (24)

and the following constraints should be added.

[pic] (36)

[pic] (37)

[pic] (38)

So for example, if [pic] , then

[pic] (24)

However, in the above formula , there is a bilinear term [pic], which needs to be linearized. Thus, we introduce new variables ([pic],[pic]) in (24) as follows:

[pic] (24)

where

[pic]

[pic]

which can in turn be substituted by:

[pic] (39)

[pic] (40)

[pic] (41)

[pic] (42)

[pic] (43)

[pic] (39)

[pic] (40)

[pic] (41)

[pic] (42)

[pic] (43)

Indeed, when [pic], we have:

[pic]

[pic]

[pic]

[pic]

and the last two will force [pic] while the first two are satisfied. Alternatively, when [pic], we have

[pic]

[pic]

[pic]

[pic]

and the first two will force [pic].

Another way of doing this is writing:

[pic] (24)

[pic] (39)

[pic] (40)

[pic] (41)

where[pic] are the capital investment corresponding to capacities of 0(zero), [pic].

Thus, when [pic] and [pic], we have [pic]=0, because of equation (40) and therefore:

[pic]

Conversely, when [pic] and [pic], we have [pic]=0, because of equation (39) and therefore:

[pic]

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Xpijt

Zpkmt

Ypjkt

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