MCF Model

We now formulate a minimum cost flow (MCF) problem on a network to optimize sourcing for all regional vaults. Additional notation used to formulate the MCF model is in Table 2.3. See Appendix A.5 for the detailed IP formulation of the MCF model. The unit cost on arc (m, n) connecting any two nodes representing vaults m and n is cmn = min{2_Qbt_tdtmn+ ct,_Q1_a(fa+ 2badamn) + ca}, which chooses the less

expensive transportation mode between IU and IM.

Table 2.3: Additional Notation Used to Formulate Problem 2 in the MCF Model: Parameters.

S_j− The amount of maximum monthly surplus (deposits minus withdrawals) in

million dollars per period of net negative regional vault j.

S_k+ The amount of maximum monthly deficit (withdrawals minus deposits) in

million dollars per period of net positive regional vault k.

Qa Capacity of one airplane, by value of currency.

Qt Capacity of one truck, by value of currency.

We first describe the construction of the network model and then illustrate it with an example. We start with one node for each big or regional vault, then add a source node O and a sink node S, plus duplicate nodes for the big vaults to represent their receipt of excess cash from net negative regional vaults. Because big vaults are assumed to have infinite capacities, they could supply all regional vaults, if needed. A net negative regional vault can supply a net positive regional vault either by using its cash inventory or by being the transition node between either a big vault or another net negative regional vault and the net positive regional vault. We now describe each arc with its lower bound, upper bound, and unit cost for the flow on that arc. An example is shown in Figure 2.7.

• An arc from source node O to node Bi, where Bi represents a big vault. The

• An arc from source node O to node Nj, where Nj represents a net negative

regional vault. This arc represents the surplus at the net negative regional vault Nj. The lower bound, upper bound and unit cost of this arc are Sj−, Sj−,

and 0.

• An arc from node Bi to node Pk, where Pk represents a net positive regional

vault. The lower bound, upper bound and unit cost of this arc are 0, S_k+, and cik = min{2_Qbt_tdtik + ct,_Q1_a(fa + 2badaik) + ca}, which is the unit cost for

transporting cash from big vault i to net positive regional vault k.

• An arc from node Nj to node Pk. The lower bound, upper bound and unit cost

of this arc are 0, S_k+, and cjk = min{2_Qbt_tdtjk+ ct,_Q1_a(fa+ 2badajk) + ca}, which

is the unit cost for transporting cash from net negative regional vault j to net positive regional vault k.

• An arc from node Bi to node Nj. The lower bound, upper bound and unit cost

of this arc are 0, min{J,∑_kS_k+}, and cij = min{2_Qbt_tdijt +ct,_Q1_a(fa+2badaij)+ca},

which is the unit cost for transporting cash from big vault i to net negative regional vault j. Recall that J is the capacity limit for regional vaults.

• An arc from node Nj to node Nl. The lower bound, upper bound and unit cost

of this arc are 0, min{J,∑_kS_k+}, and cjl= min{2_Qbt_tdjlt +ct,_Q1_a(fa+2badajl)+ca},

which is the unit cost for transporting cash from net negative vault j to net negative regional vault l.

• An arc from node Nj to node Bi′. The lower bound, upper bound and unit cost

of this arc are 0, S_j−, and cji = min{2_Qbt_tdtji + ct,_Q1_a(fa+ 2badaji) + ca}, which

is the unit cost for transporting excess cash from net negative vault j to big vault i.

• An arc from node Pk to sink node S. The lower bound, upper bound and unit

cost of this arc are S_k+, S_k+, and 0. These bounds enforce Pk’s deficit.

• An arc from node B′

i to sink node S. The lower bound, upper bound and unit

cost of this arc are 0, ∑_kS_k−, and 0.

Figure 2.7: Example for the MCF Model [lmn: Lower Bound; umn: Upper Bound;

cmn: Unit Cost].

B

B

P

O

B

P

S

N

S

Bˊ

N

Bˊ

Bˊ

N

Bˊ

Example: There are 2 big vaults, 2 net negative regional vaults, and 2 net positive

regional vaults. In this example, we let S₁+, S₂+, S₁−, and S₂− be 15, 20, 5 and 10, respectively (Figure 2.7).

For clarity, we assume that the cost structure makes it more economical for net negative regional vault 2 to be supplied by big vault 1 and for net negative regional vault 1 to be supplied by big vault 2. Additionally, net negative regional vaults 1 and 2 can supply each other. Consider the following five different types of feasible flows:

• Type 1: Big vault 1 or 2 is the only source for supplying net positive regional vaults. The net negative vaults return all of their cash to a big vault:

– 20 units of flow move with sequence: O → B1 → P2 → S. – 5 units of flow move with sequence: O → N1 → B2′ → S. – 10 units of flow move with sequence: O → N2 → B1′ → S. • Type 2: All units of flow S−

j from the net negative regional vaults go to

the same net positive regional vault. The other net positive regional vault is supplied by a big vault:

– 5 units of flow move with sequence: O → N1 → P1 → S. – 10 units of flow move with sequence: O → N2 → P1 → S. – 20 units of flow move with sequence: O → B1 → P2 → S.

• Type 3: One net positive regional vault is supplied by a big vault and a net negative regional vault while another net negative vault sends its surplus back to a big vault. The other net positive regional vault is supplied by a big vault:

– 5 units of flow move with sequence: O → B1 → P1 → S. – 10 units of flow move with sequence: O → N2 → P1 → S. – 20 units of flow move with sequence: O → B2 → P2 → S. – 5 units of flow move with sequence: O → N1 → B2′ → S.

• Type 4: Net negative regional vaults are supplied by big vaults, which then supply net positive regional vaults. This flow may happen when there is no road between big vaults and net positive regional vaults and the distance and demand do not justify IM transportation. This flow also indicates that the surplus at net negative regional vaults is not enough to satisfy the demand from net positive regional vaults:

– 10 units of flow move with sequence: O → N2 → P1 → S. – 15 units of flow move with sequence: O → B2 → N1 → P2 → S. – 5 units of flow move with sequence: O → N1 → P2 → S.

• Type 5: One net negative regional vault supplies the other, which then supplies a net positive regional vault. This flow path happens because there is no road between N1 and P1, N2 has only 10 units of surplus, and no big vault supplies P1.

– 5 units of flow move with sequence: O → N1 → N2 → P1 → S. – 10 units of flow move with sequence: O → N2 → P1 → S. – 20 units of flow move with sequence: O → B2 → P2 → S.

Clearly, solving the minimum cost flow problem constructed above is equivalent to optimizing the sourcing for Problem 2. Since all deposits and withdrawals in MCF are scaled as integers and all flows in the network have integer capacity, each feasible integer flow corresponds to a feasible sourcing for the currency network and vice versa.