Scheduling and lot sizing models for the single-vendor multi-buyer problem under consignment stock partnership

Abstract

We consider a centralized supply chain composed of a single vendor serving multiple buyers and operating under consignment stock arrangement. Solving the general problem is hard as it requires finding optimal delivery schedule to the buyers and optimal production lot sizes. We first provide a nonlinear mixed integer programming formulation for the general scheduling and lot sizing problem. We show that the problem is NP-hard in general. We reformulate the problem under the assumption of ‘zero-switch rule’. We also provide a simple sequence independent lower bound to the solution of the general model. We then propose a heuristic procedure to generate a near-optimal delivery schedule. We assess the cost performance of that heuristic by conducting sensitivity analysis on the key model parameters. The results show that the proposed heuristic promises substantial supply-chain cost savings that increase as the number of buyers increases.

Appendix

A.1. Mathematical model

Notation

Input parameters:

In what follows, the index i refers to the buyer and indices j and k designate deliveries.

n::

number of buyers

m::

number of shipments sent to all buyers during a cycle

A _bi ::

the cost of placing an order by the ith buyer

h _bi ::

ith buyer's cost of holding one unit in stock for one unit of time

m _i ::

number of shipments sent to ith buyer during a cycle

d _i ::

demand rate from buyer i

A _v ::

Vendor's setup cost

h _v ::

Vendor's cost of holding one unit in stock for one unit of time

D::

total of buyers demand rate=∑_i=1ⁿd _i

P::

vendor's production rate

Decision variables:

T :

=replenishment cycle length

t _j :

=time to produce the quantity shipped in the jth delivery

u _j :

=time interval between the jth and (j+1)th replenishments, interpreted cyclically. As can be seen in Figure A1, we have

Figure A1

Inventory variation with time in the vendor's facility.

u _j :

=

where t_m+1 is the duration of the production downtime period at the vendor facility.

Figure A1 depicts the inventory variation over time at the vendor's facility for the case of two buyers. Note that the first, third, and fourth shipments are sent to the first buyer and the second delivery is made for the second buyer.

x _ij :

y _ijk :

=1 when the next delivery to the ith buyer after the jth replenishment is made at the kth shipment, and is equal zero otherwise.

Y _ijk :

T _ij :

=time interval until an order is shipped to the ith buyer after the jth delivery

T _ij :

q _ij :

=quantity received at the ith buyer facility during the jth replenishment. Note that when the jth replenishment is not sent to the ith buyer, q _ij is equal to zero

q _ij :

=P t _j x _ij

q _i1 :

=P(u _m −t_m+1) x_i1

q _i1 :

=P(u _m −t_m+1) x_i1

q _ij :

=Pu_j−1x _ij for j=2,3,…,m

I _ij ^b :

=inventory at buyer i facility just before the receipt of the jth order

I _ij ^a :

=inventory at buyer i facility just after the receipt of the jth order

I _ij ^a :

I _ij ^b :

Model formulation

Using Equations (A2) and (A3), one can easily show that

which reduces the inventory level decision variables to the I_i1^b's only.

Note that using (A4) for j=m, we have

or

which ensures that the inventory schedule is cyclic. In other words, the last equation states that the inventory at the beginning of the cycle for the ith buyer is equal to his/her inventory at the time of the last replenishment minus the demand over the time interval between the mth delivery and the first delivery of the next cycle.

We need also to restrict I _ij ^b to be non-negative to avoid any shortages. Therefore, I _ij ^b⩾0.

Using the above defined variables, the buyers’ average holding costs per cycle are

The term between brackets of the above expression is the area of the trapezoid between the jth delivery and the next delivery to the buyer receiving the jth order (see Figure A2).

Figure A2

Sketch of a buyer i's inventory level.

The total system wide cost per cycle is

where the first term is the vendor setup cost, the second term is the sum of buyers’ ordering costs, the third term is the buyers’ average inventory holding costs per cycle, and the last term is vendor's average inventory cost per cycle.

Therefore, the average total cost per unit of time for the centralized supply chain is then

Based on the above, the mathematical model for the centralized single-vendor n buyers supply chain under CS partnership agreement can be stated as follows:

CSP:

Min ATC

s.t.

The first set of constraints (A7) states that buyer i receives exactly m _i orders. The second set of constraints (A8) ensures that each delivery is sent to only one buyer. Constraints (A9) require that there has to be exactly one delivery to be sent to the ith buyer just after the jth replenishment. Constraints (A10) enforce x _ik to be one if y _ijk is equal to one. Constraints (A11) define the variables Y _ijk as function of y _ijl . Constraints (A12), together with constraints (A9), guarantee that no deliveries is sent to buyer i between the jth delivery and the first delivery to buyer i after the jth delivery. Constraints (A13) and (A14) define T _ij and the cycle length, respectively, as explained earlier. The remaining constraints (A15-A20) are related to the inventory levels and replenishment sizes.