Network Flow Model for Problem II of One Denomination

We first describe the construction of the minimum cost flow (MCF) network model and then illustrate it with an example. In the construction, we start with one DI containing one node U for the unpackaged coin inventory and one node P for the packaged coin inventory plus duplicate nodes for the unpackaged and packaged coin

inventory in different ordering and receiving periods. A source node O is included for providing the deposits from the DI’s branches and packaged coins supplied by the FRS. In the end, a sink node S is also added for receiving withdrawals of the DI’s branches and unpackaged coins send from the DI to the FRS for packaging. Now, we complete the whole network for a DI’s coin ordering, packaging, holding, and supply. Notation needed for Problem II is in Table 3.7. An example for one DI is shown in Figure 3.11 considering a 3-period rolling horizon.

Figure 3.11: An Example of Problem II for One DI.

3 D 1 X 3 D 3 D 2 D 1

U

U

2

U

3 1 u X 2 u X 1 u I 2 u I 2 D 2 D ∞ hu ∞ hu

O

1 D 3 u X 1 u Y ₂ Y 1 D 1 D

O

_S

0 X u Y ₂ u Y 3 W 1

P

2

P

P

3 0 p X 1 p I 2 p I 3 W 3 W 2 W 1

P

2

P

P

3 1 p X 2 p X p I (0, , )∞ p h ∞ hp 2 W 2 W 1 W ( , ,0)1 W 1 W Period 3 Period 2 Period 1 ( , ,0)W W

We now describe each arc with its lower bound, upper bound, and unit cost for the flow on that arc. We assume that there is no beginning and ending period coin inventory, neither unpackaged nor packaged, left in the DI. In other words, I_u0 = I_p0 = I_uT = I_pT = 0. We also assume that at the beginning of period 1 (end of period 0) and at the end of period T, neither the FRS nor the 3PLP receives unpackaged coins from the DI or sent packaged coins to the DI. Thus, X0

u = XuT = Xp0 = XpT =

Y0

Table 3.7: Parameters and Variables for Formulating the MCF Model of Problem II.

Parameters

T Number of periods in the planning horizon.

Dt The deposits of coin (in bags) during period t at the DI, t = 1, 2, . . . , T . Wt _{The withdrawals of coin (in bags) during period t at the DI, t =}

1, 2, . . . , T , k = 1, 2, . . . , K.

hu The per bag per period holding cost for the unpackaged coin inventory.

hp The per bag per period holding cost for the packaged coin inventory.

c The per bag transportation and handling cost incurred for the DI for making deposits (or withdrawals) of coins at the FRS.

f The per bag fee paid by the DI (in addition to the face value) to the FRS for purchasing packaged coins.

g The per bag packaging fee paid by the DI to the 3PLP.

Variables

Xt

u The amount of unpackaged coins (in bags) shipped from the DI to the

FRS at the end of period t, t = 0, 1, 2, . . . , T . Xt

p The amount of packaged coins (in bags) shipped from the FRS to the DI

at the end of period t, t = 0, 1, 2, . . . , T .

Y_ut The amount of unpackaged coins (in bags) shipped from the DI to the 3PLP for packaging at the end of period t, t = 0, 1, 2, . . . , T .

It

u The unpackaged coin inventory (in bags) at the end of period t at the DI,

t = 0, 1, . . . , T . For each period t, I_ut is the unpackaged coin inventory level right after the DI ships out Xt

u to the FRS and Yut to the 3PLP.

It

p The packaged coin inventory (in bags) at the end of period t at the DI,

t = 0, 1, . . . , T . For each period t, I_pt is the packaged coin inventory level right after the DI receives Xt

p and Yut bags of packaged coins from the

FRS and from the 3PLP.

packaged coins as safety stock. Here, we assume that this safety stock is zero, which will not affect the analysis.

• An arc from source node O to node Ut_{, where U}t _{represents the unpackaged}

coin inventory at the end of period t. The lower bound, upper bound and unit cost of this arc of deposits from the DI’s branches are Dt_{, D}t_{, and 0.}

• An arc from source node O to node Pt_{, where P}t _{represents the packaged coin}

arc. For 0 < t < T , the lower bound, upper bound and unit cost of this arc are 0, ∞, and c + f.

• An arc from node Ut−1 _{to node P}t _{represents a flow from the DI’s unpackaged}

coin inventory to its own packaging process. For t = 0 and t = T , there is no flow on this arc. For 0 < t < T , the lower bound, upper bound and unit cost of this arc are 0, ∞, and g. Note that in the reality, g < (c + f).

• An arc from node Ut−1 _{to node U}t _{represents a flow of unpackaged coin inven-}

tory carried from on period to the next. The lower bound, upper bound and unit cost of this arc are 0, ∞, and hu. Note that since Iu0 = IuT = 0, they are

not included in the network.

• An arc from node Ut _{to the sink node S represents a flow of unpackaged coin}

inventory sent to the FRS for packaging at the end of period t. For t = 0 and t = T , there is no flow on this arc. For 0 < t < T , the lower bound, upper bound and unit cost of this arc are 0, ∞, and c.

• An arc from node Pt−1 _{to node P}t _{represents a flow of packaged coin inventory}

carried from on period to the next. The lower bound, upper bound and unit cost of this arc are 0, ∞, and hp. Note that since Ip0 = IpT = 0, they are not

included in the network.

• An arc from node Pt _{to the sink node S represents a flow of packaged coin}

withdrawal from the DI’s branches. The lower bound, upper bound and unit cost of this arc are Wt_{, W}t_{, and 0.}

The objective of the MCF model is to send ∑_tWt _{from source node O to sink}

MIP formulation for Problem II

The objective function minimizes the total cost for one specified DI, which includes coin packaging cost (either paid to the FRS or paid to the 3PLP), circulating coin transportation and handling cost between the DI and the FRS, and the packaged and unpackaged coin inventory holding cost. The last two terms compute the packaged and unpackaged coin inventory holding cost. Note that when considering the average packaged coin inventory during the period j, the beginning packaged coin inventory level is I_pt−1 and the instantaneous packaged coin inventory level right before the end of period j is It−1

p − Wt. Thus, the average packaged coin inventory level during

the period j is equal to the half of the sum of the beginning inventory and the instantaneous ending inventory. Similarly, for the average unpackaged coin inventory during the period j, the beginning unpackaged coin inventory level is It−1

u and the

instantaneous unpackaged coin inventory level right before the end of period j is I_ut−1 + Dt. Thus, the average unpackaged coin inventory level during period j is equal to the half of the sum of these two.

Problem II: Min T_∑−1 t=1 f · X_pt + T_∑−1 t=1 g· Y_ut+ c· ( T_∑−1 t=1 X_pt+ T_∑−1 t=1 X_ut) + hu· T ∑ t=1 (I_ut−1+ D t 2 ) + hp· T ∑ t=1 (I_pt−1−W t 2 ) Subject to:

Constraints (3.31)-(3.32) are the coin flow balance equations.

I_pt = I_pt−1− Wt+ X_pt+ Y_ut, ∀t (3.31) I_ut = I_ut−1+ Dt− X_ut − Y_ut, ∀t (3.32) Constraints (3.33) enforce the initial inventory level and the inventory level in the end of the planning horizon to be zero.

I_p0 = I_u0 = I_pT = I_uT = 0 (3.33) Constraints (3.34) are non-negativity constrains.

All variables are nonnegative & ingeter, ∀t (3.34)

Lemma 5 Solving the minimum cost flow problem constructed above is equivalent

to optimizing DIs’ coin ordering, packaging, holding and supply process.

Proof: Since each feasible integer flow corresponds to a feasible sourcing for DIs’

coin receiving, transportation, and inventory management process and vice versa, the optimal solution to the minimum cost flow problem constructed above is equivalent to the optimal process for DIs’ for Problem II. This result holds in general.

Property 4 Since the flows in MCF are integers with integer lower and upper ca-

pacity limits, each feasible integer flow corresponds to a feasible shipment of coins for Problem II and vice versa.

In document Analysis of Operations Management Problems in Currency and Food Supply Chains (Page 100-105)