Planning Phase

Model II: Medical Resource Is Limited

8.2 The Mathematical Model

8.2.2 Planning Phase

We definex_k(t)to be the number of beds assigned for epidemic patients in hospital kat timet, which must be under the designated capacityX_k, i.e.,

xk(t)≤ Xk, ∀k=1,2, . . . ,K, t =1,2, . . . ,T. (8.8) Since no bed can be used to host any epidemic patient in a designated hospital unless it is chosen to operate for that day, one has the following relationship

xk(t)≤αk(t)Xk. (8.9)

Various costs could occur at a designated hospital during the course of the epi- demic control and treatment. First, there will be a fixed costh1k that may account for administrative manpower overhead, utilities, and other resources allocated. The resources include patient rooms and facilities allocated exclusively for treating the epidemic patients and thus not available for any other use. Such cost does not change daily during the time when the hospital is in operation of epidemic treatment.

Secondly, there is a variable cost that is represented by doctor-hours, nurse-hours, medicine cost (including medicine expense and medicine inventory cost) meals etc., which is in proportion to the number of patients hospitalized. Leth_2k indicate the rate of the variable cost in hospitalkduring epidemic operation, then the variable cost for daytish_2kxk(t). Thirdly, when a designated hospital is opened for treat- ing epidemic patients, there is a switch-on cost, including such costs for storing up needed medicines, preparing of equipment, separating the ward, etc. Likewise, when a hospital from operational in one period changes to non-operational in the follow- ing period, there will be a switch-off cost related to such processes as sterilization.

Denoteh3kthe switch-on cost andh4kthe switch-off cost for hospitalk. We define another set of auxiliary 0–1 variablesφk⁰¹(t), φk¹⁰(t), ∀t=1,2, . . . ,T.

φ⁰¹k (t)

=1,if hospitalkis non-operational on dayt−1 and operational on dayt

=0,otherwise ;

φ¹⁰_k (t)

=1,if hospitalkis operational on dayt−1 and non-operational on dayt

=0,otherwise ;

The following relationships between the 0–1 variables αk(t),∀t = 1,2, . . . ,T and the auxiliary 0–1 variables φk⁰¹(t), φk¹⁰(t), ∀t = 1,2, . . . ,T must hold in accordance with the definitions:

φk⁰¹(t)≤αk(t); φ⁰¹k (t)≤1−αk(t−1); φ⁰¹k (t)≥αk(t)−αk(t−1); (8.10) φk¹⁰(t)≤αk(t−1); φk¹⁰(t)≤1−αk(t); φ¹⁰k (t)≥αk(t−1)−αk(t); (8.11) As a matter of fact, the values of the 0–1 variablesαk(t−1)andαk(t)together will uniquely determine the values ofφk⁰¹(t), φk¹⁰(t)by the inequalities of Eqs. (8.10) and (8.11), and thus introducing the auxiliary variables φk⁰¹(t), φk¹⁰(t)does not

actually increasing the computational complexity of our model. The aggregated cost at hospitalkon dayt,h_k(t),is given by

hk(t)=h_2kxk(t)+h_1kαk(t)+φk⁰¹(t)h_3k+φk¹⁰(t)h_4k,

∀t=1, . . . ,T, k=1, . . . ,K. (8.12) Inventory cost at the hospital level is not explicitly addressed in our model but reflected as a part of the variable cost h2k. The foremost reason is that, although there is inventory cost for carrying the medicine at the hospital level, there is no separate decision making at this level. The hospitals are not allowed to carry exces- sive inventory during this special time because the medicine resource is critically important and tightly controlled to prevent spoilage and waste. Holding an excessive amount of medicine at one hospital may cause shortage at another hospital, which is detrimental to the overall effort in treating the patients and controlling the epidemic.

On the other hand, no hospital wants to run any risk of medicine supply shortage in exchange of carrying cost advantage. This assumption is made in contrast to most inventory models dealing with consumer commodities. In practice, hospitals are sup- plied a just-right amount of medicine according to the number of their inpatients for their usage in a very short period of time (one to three days). Our model includes the medicine inventory cost in the variable cost, which is in proportion to the number of the inpatients, but does not model the ordering point and/or ordering quantity as decision variables. Secondly, the timet in our model, although referred to a day in a modeling term, can actually be two or three days in application as a decision time within which some inventory holding cost is insignificant but the guarantee of supply is paramount.

(2) Logistics part of distribution center tier

The district distribution centers (DCs) are used to store the medicine and distribute them to the local hospitals based on their demand. Theoretically, we assume any distribution center can send the needed amount of medicine to any hospital on any day. Let y_k^j(t) be the amount of medicine shipped from distribution center j to hospitalkon dayt,vj(t)be the inventory of distribution center j on dayt,V_j be the storage capacity at DC j andv^sj be the safety stock at DC j. Then, DC j must satisfy the upper bound and the lower bound on each day, i.e.,

v^sj ≤vj(t)≤V_j, ∀t =1, . . . ,T, j=1, . . . ,J. (8.13) Suppose that each distribution center has a fixed order size and it can order from any of the medicine suppliers. Letv⁰j be the order size of DCjand letLⁱbe the lead time of pharmaceutical planti, and define the 0–1 variable:

εⁱj(t)

=1,if DC jorders from supplieri on dayt

=0,otherwise ,

then the orders received by DC jon daytisI

i=1εⁱj(t−Lⁱ)⁺v⁰j and it should hold the following relationship between the ending inventories of two consecutive days at DC j:

vj(t+1)=vj(t)− K k=1

y_k^j(t)+ I

i=1

εⁱj(t−Lⁱ)⁺v⁰j,

∀t =1, . . . ,T, j=1, . . . ,J. (8.14) Suppose that the inventory carrying cost rate (per day) isg2jand the ordering cost isg1jat DC j. Then the total daily operation cost incurred at DC jis:

gj(t)=g2jvj(t)+g1j

I i=1

εⁱj(t), ∀t =1, . . . ,T. (8.15)

Meanwhile, we can relate the number of patients in hospitalk on dayt,x_k(t), with the medicine supplies that are distributed from the DCs to the hospital on day t,y_k^j(t), which are given as below:

mx_k(t)= J

j=1

y_k^j(t), (8.16)

wheremis the average daily dose for a patient across the area in this epidemic spread.

(3) Logistics part of supplier tier

We now consider the pharmaceutical plants at the top tier of this medical supply chain network. Since the plants are not necessarily located in the area and they need a production plan to acquire the raw materials and reset the facilities to make this medicine, we suppose each pharmaceutical plant has an independent and possibly different lead time for the order of the distribution centers, namelyLⁱ, ∀i =1, . . . ,I. Also, each supplier has a production capacity,D_i, ∀i=1, . . . ,I. The operation cost for supplieri, ∀i =1, . . . ,I, is comprised of two parts, as in most manufacturing firms, a constant fixed cost f1i that is incurred whenever the production of this medicine takes place and is regardless of the output level, and a variable cost that is in proportion to the production output, with the unit cost being denoted by f2i. The aggregated order that supplierireceives on daytis:

di(t)= J

j=1

εⁱj(t)v⁰j, (8.17)

which forms the dynamic demand for its production plan. We can get the following constraint pertaining to production capacity and the aggregated orders:

di(t)= J

j=1

εⁱj(t)v⁰j ≤ Di. (8.18)

The production cost for supplieri, ∀i =1, . . . ,I can be formulated as:

fi(t)= f_1iωi(t)+ f_2idi(t), ∀t =1, . . . ,T. (8.19) whereωi(t)is a 0–1 variable for planti pertaining to its operation schedule on day t and is defined as:

ωi(t)

=1,if supplieriruns its operation on timet

=0,otherwise .

Since any order from a distribution center will trigger the production operation of the plant who receives the order, the following relationship between the 0–1 variables of DC order schedule and the supplier operation schedule must be satisfied:

εⁱj(t)≤ωi(t), ∀j =1, . . . ,J, i=1, . . . ,I. (8.20) Namely,ωi(t)cannot be zero unless allεⁱj(t)’s are zero. On the other hand, since the production cost Eq. (8.19) is eventually to be minimized,ωi(t)will be zero in the optimal solution if allεⁱj(t)’s are zero, although not directly implied by Eq. (8.20).

Note that Eqs. (8.19) and (8.20) imply a pull system in which the production system will be turned on by any order received from a distribution center, which is not unreasonable in an urgent rescue campaign such as an epidemic spread. One sees that in Eq. (8.19), f_i(t)is zero if and only if allεⁱj(t),∀j =1, . . . ,Jare zero, i.e., no order is received on dayt, and that even a small size ordered by one DC will trigger the production system to bear the fixed cost. Such a system may result in a higher supplier cost. However, the problem is modeled as a centralized control system to minimize the overall social cost, and therefore, all the costs regardless at which tier they occur are equally accounted for in the minimization. Consequently, the 0–1 decision variables,εⁱj(t),∀j=1, . . . ,J, defined for the distribution centers tier will eventually take into account of the production cost minimization too. That is, the central agent shall consider coordinating the ordering times of various distribution centers so that they will pool together a larger size of aggregated orders to trigger the production system of a supplier. This can be observed in the numerical example.

(4) Transportation costs and objective function

We now turn to look at the shipments both between the supplier tier and the DC tier, and between the DC tier and the location hospital tier. By the definition ofεⁱj(t)and the lead timeLⁱ, the shipment from supplierito arrive at DC jon dayt is:

zⁱ_j(t)=εⁱj(t−Lⁱ)⁺v⁰j, (8.21)

which is zero if DC j didn’t place an order on dayt −Lⁱ. Let our model employ a general function for the transportation cost so that it can take a special form in a particular application. Note that a line haul cost is the major part of this transportation cost because the suppliers in our model may be located outside of the area and thus geographically far away from the distribution centers that are within the epidemic spread area. Therefore, some types of consolidation in truck load can take effect in reducing the transportation cost. For instance, shipments from the same supplier to several DC’s in the area can be consolidated in transportation and result in significant saving, or two close-by suppliers can also consolidate their shipments. Let uⁱ(t) denote the general transportation cost for supplieri on dayt, which is a general function ofzⁱ(t)=(zⁱ₁(t), . . . ,zⁱ_J(t))∈ R₊^J, namely

uⁱ(t)=uⁱ(zⁱ(t)). (8.22)

Likewise, let w^j(t)denote the general transportation cost for DC j on day t, which is a function ofy^j(t)=(y₁^j(t), . . . ,y_K^j(t))∈ R₊^K, namely

w^j(t)=w^j(y^j(t)). (8.23)

The system optimization problem is to minimize the total cost of operations occur- ring at the tier of local hospitals, district distribution centers, the pharmaceutical plants as well as the transportation costs for shipping the medicine from the suppli- ers to the distribution centers and shipping the medicine from the distribution centers to the local hospitals. Letdenote the total cost, i.e.,

(x,y, α, ε, ω)= T

t=1

⎛

⎝^K

k=1

h_k(t)+ J

j=1

(gj(t)+w^j(t))+ I

i=1

(f_i(t)+uⁱ(t))

⎞

⎠.

(8.24) Given the forecasted number of infected people in the next cycle I(t),t = 1, . . . ,T provided by the forecast phase of the model, we can now define a fea- sible region:

=

(x,y, α, ε, ω): K

k=1

x_k(t)=I(t),

and (9−11), (13−14), (16−18), (20−21) hold} (8.25) Then, the optimal planning for hospital resource allocation, medicine distribution and transport for the next cycle can be obtained by solving the following optimization problem:

Min_{(x,y,δ,ε,ω)∈}(x,y, α, ε, ω). (8.26)

Since all the constraints in this optimization problem are linear, and the cost functions defined in the model are linear, so the system minimization problem (8.26) is a mixed 0–1 linear integer programming problem. Such a problem can be solved by some available optimization software such as CPLEX for an optimal solution.