Model II: Medical Resource Is Limited
8.3 Numerical Example
8.3.2 Test for Logistic Planning Phase
(1) Parameters setting
Supposing that in a hypothetical influenza outbreaks area, there are 3 pharmaceu- tical plants (S) which supply a critical medicine for the epidemic, 4 distribution centers (DC) that store and distribute the medicine, and 8 local hospitals (H) that are designated to host and treat the infected individuals. Assume that all the transporta- tion costs from the pharmaceutical plants to the DCs and from the DCs to the local hospitals are linear. The coefficient matrix is given by Table8.1.
Particularly, the transportation cost for S1 on day t is a linear function of the shipment quantities from this pharmaceutical plant to the four DCs, namely
u1(t)=2·z11(t)+4·z12(t)+6·z13(t)+8·z14(t).
Table 8.1 Transportation cost between different departments (dollar)
S1 S2 S3 H1 H2 H3 H4 H5 H6 H7 H8
DC1 2 4 3 6 2 6 7 4 2 5 9
DC2 4 3 5 4 9 5 3 8 5 8 2
DC3 6 2 2 5 2 1 9 7 4 3 3
DC4 8 1 3 7 6 7 3 9 2 7 1
For instance, the operation cost for the H3 on day t, is:
h3(t)=2.7×x3(t)+33×α3(t)+40×φ301(t)+33×φk10(t), ∀t =1, . . . ,T. Since the number of patients that can be hosted by H3 on day cannot exceed its capacity 40 and only when it is chosen to operate on dayt, the following constraint must hold
x3(t)≤α3(t)×40
Table8.2describes the operation cost parameters of the local hospitals.
Table 8.3 summarizes the capacity, the operation cost parameters, and initial inventory at each of the four distribution centers.
For example, the daily operation cost for the DC1 is:
g1(t)=1.4×v1(t)+6× 3
i=1
εi1(t), ∀t =1, . . . ,T,
Table 8.2 Hospital parameters Parameter
value k= 1,2, . . . ,8
CapacityXk (person)
Fixed cost h1k (dollar)
Variable cost h2k (dollar)
Switch-on costh3k (dollar)
Switch-off costh4k (dollar)
H1 40 35 2.5 40 35
H2 45 34 2.6 45 34
H3 40 33 2.7 40 33
H4 45 32 2.8 45 32
H5 40 31 2.9 40 31
H6 35 30 3.0 35 30
H7 35 29 3.1 35 29
H8 40 28 3.2 40 28
Table 8.3 DC parameters Parameter
value j= 1,2,3,4
Capacity Vj (dose)
Initial inventory (dose)
Ordering costg1j (dollar)
Carrying costg2j (dollar)
Order size v0j (dose)
Safety stockvsj (dose)
DC1 120 70 6 1.4 40 10
DC2 150 70 6.5 1.3 30 10
DC3 120 80 7 1.2 40 10
DC4 150 80 7.5 1.1 30 10
where
v1(1)=70 andv1(t+1)=v1(t)−8
k=1yk1(t)+3
i=1ε1i(t−Li)+×40,∀t = 1, . . . ,T.
The daily inventory in DC1 must not be less than the safety stock and must not exceed its capacity, or mathematically expressed as:
10≤v1(t)≤120, ∀t =1, . . . ,T.
Table 8.4 gives the lead time, the production capacity, the fixed cost and the variable cost for each of the three pharmaceutical plants.
For example, the production cost for S1 is:
f1(t)=120×ω1(t)+6×d1(t), ∀t=1, . . . ,T,
where the aggregated ordersd1(t)should satisfy the production capacity constraint,
d1(t)= 4
j=1
ε1j(t)×v0j ≤80.
Since any order from a distribution center will trigger the production operation of the pharmaceutical plant S1, we have
ε1j(t)≤ω1(t), ∀j =1, . . . ,4.
(2) Test results
The computation results are summarized in following tables and figures. Initially, Table8.5reports the cumulative number of the daily infected people, the maximum daily infected people, and the computation time, in each of the five cycles.
Following the preventive stocking policy between cycles as discussed in Sect.8.2, Fig.8.3shows the inventory level over time in each of four DCs during the entire 150 days. They all begin with a certain stock for the lead time of the first order as required by the policy and then are maintained at the minimum level (equal to the safety stock) in most of the days in the cycle except for the necessary reorder point. We accredit this low inventory-carrying performance to the merit of the model
Table 8.4 Parameters of pharmaceutical plants Parameter value
i=1,2,3
Lead timeLi (day)
CapacityDi (dose)
Fixed cost f1i (dollar)
Variable costf2i (dollar)
S1 1 80 120 6
S2 2 90 110 7
S3 1 100 115 6.5
Table 8.5 Computation time and daily infected people Decision cycle Cumulative daily
infected people
Maximum daily infected people
Computation time (s)
Cycle 1 2233 142 17.0
Cycle 2 6506 250 47.5
Cycle 3 5151 241 24.1
Cycle 4 1465 91 6.9
Cycle 5 305 21 1.4
0 30 60 90 120 150
0 25 50 75 100
Time
Inventory in DCs
DC1 DC2 DC3 DC4
Fig. 8.3 Inventory level in the DCs
that minimizes the inventory ordering and carrying cost while meeting the hospitals’
need.
Namely, in the first place, the DCs are deposited a certain level of initial inventory, for ensuring supply to the hospitals during the lead time of their first order. In the second place, they are operated to minimize the total logistics cost, therefore the inventory level in the DCs should be as low as possible while coordinated with the medicine order planning and the medicine distribution scheduling. The trajectory in Fig.8.3illustrates an almost just-in-time mechanism in the medicine supply chain of our model.
Figure8.4exhibits the daily production level of the three pharmaceutical plants over time in the solution of our model. It can be seen that the production patterns of all the suppliers are similar where the output levels are high in the second and third cycle, but low in the first, fourth and fifth cycle. This is well expected because in the first cycle when the epidemic just bursts, the infected population size is small and the need for medicine is relatively low. In the second and third cycle when the
0 30 60 90 120 150 0
20 40 60 80 100
Time
The aggregated order
S1 S2 S3
Fig. 8.4 Production level in the pharmaceutical plants (S)
epidemic diffusion is at its maximum scale, the number of infected people reaches the peak and the demand for the medicine is at the highest level, thus so would be the production level. In the fourth and fifty cycle, when the epidemic diffusion is brought under control and the infected population diminishes, the medicine demand decreases and the production slows down.
The solution of our model provides a dynamic assignment of patients at each hospital over the entire 150 days in this numerical example. As a snapshot, Fig.8.5 shows the number of patients assigned to each hospital on the 36th day. On this particular day, Hospital 2, 3, 6 and 8 are operated at capacity, Hospital 4 is used to treat the infected on that day but is not filled up, while Hospital 1, 5 and 7 are not in
1 2 3 4 5 6 7 8
0 10 20 30 40 50
Hospital (the 36th day)
Capacity vs. Assignment
Capacity Assignment
Fig. 8.5 Hospital assignment on 36th day
operation. Considering the cost structure of the hospitals, such assignment explains for avoiding as much as possible the fixed cost, the turning on and turning off costs in the optimal solution. It makes a common sense that it is not desired to have more than one hospital operated under capacity in such a cost structure.