Multi-Item EOQ Model with Both Demand-Dependent Unit Cost and Varying Leading Time via Geometric Programming

The objective of this paper is to derive the analytical solution of the EOQ model of multiple items with both demand-dependent unit cost and leading time using geometric programming approach. The varying purchase and leading time crashing costs are considered to be continuous functions of demand rate and leading time, respectively. The researchers deduce the optimal order quantity, the demand rate and the leading time as decision variables then the optimal total cost is obtained.


Introduction
The problem of the EOQ model with demand-dependent unit cost had been treated by some researchers.Cheng [1] studied an EOQ model with demand-dependent unit cost of single-item.The problem of inventory models involving lead time as a decision variable have been succinctly described by Ben-Daya and Abdul Raouf [2].Abou-El-Ata and Kotb [3] developed a crisp inventory model under two restrictions.Also, Teng and Yang [4] examined deterministic inventory lot-size models with time-varying demand and cost under generalized holding costs.Other related studies were written by Jung and Klein [5], Das et al. [6] and Mandal et al. [7].Recently, Kotb and Fergany [8] discussed multi-item EOQ model with varying holding cost: a geometric programming approach.
The aim of this paper is to derive the optimal solution of EOQ inventory model and minimize the total cost function based on the values of demand rate, order quantity and leading time using geometric programming technique.In the final a numerical example is solved to illustrate the model.

Notations and Assumptions
To construct the model of this problem, we define the following variables:  The following basic assumptions about the model are made: 1) Demand rate is uniform over time.
r 2) Time horizon is finite.
is inversely related to the demand rate.Where is called the price elasticity.5) Lead time crashing cost is related to the lead time by a function of the form   , 1,2,3, , , where ,   are real constants selected to provide the best fit of the estimated cost func-tion.6) Our objective is to minimize the annual relevant total cost.

Mathematical Formulation
The annual relevant total cost (sum of production, order, inventory carrying and lead time crashing costs) which, according to the basic assumptions of the EOQ model, is: To solve this primal objective function which is a convex programming problem, we can write it in the form: Applying Duffin et al. [9] results of geometric programming technique to (3), the enlarged predual function could be written in the form: Since the dual variable vector j r , is arbitrary and can be chosen according to convenience subject to: We choose jr such that the exponents of r r are zero, thus making the right hand side of (4) independent of the decision variables.To do this we require: These are called the orthogonality conditions which together with (5) are sufficient to determine the values of , 1 2, 3, 4, 5, 5) and ( 6) for jr W , we get: , and the partial derivatives were taken relative to .Setting it to equal zero and simplifying, we get: where: It is clear that   0 0 f  and which means that there exists a root The trial and error method can be used to find this root.However, we shall 0,1 W   first verify the root 5 calculated from (8) to maximize .This is confirmed by the second derivative to with respect to , which is always negative.
Thus, the root calculated from (8) maximize the dual function .Hence, the optimal solution is , where 5 is the solution of ( 8) and are evaluated by substituting value of in (7).To find the optimal values r r r , we apply Duffin et al. [9] of geometric programming as indicated below: and 2 By solving these relations, the optimal demand rate is given by: The optimal order quantity is: The optimal lead time is: By substituting the values of in (3), we deduce the minimum total cost as: and As a special case, we assume 0 , 0  .This is the classical EOQ inventory model.

Conclusions
his paper is devoted to study multi-item inventory tal cost is found at T model that consider the order quantity, the demand rate and the leading time as three decision variables.These decision variables , and , 1,2, 3, ,  are evaluated and the is deduced.The classical system is derived as special case and a numerical example is solved.
The smallest value of the minimum to minimum annual total cost min TC the smallest values of b and  .

rD=
Annual demand rate (decision variable).

=
purchase (production) cost.hr = Unit holding (inventory carrying) cost per item per unit time.Safety stock.n = Number of different items carried in inventory.r L Q = Leading rate time (decision variable).r= Production (order) quantity batch (decision variable).= Average annual total cost.For the r th item.
, we get the dual function .To find 5r W which maximize   5r g W , the logarithm of both side of

6 
 unit/year and K = 2.For some different values of  and b, we use equation (8) to determine 5 , whose value is to be determined to obtain * r W * jr W , j = 1, 2, 3, 4, r = 1, 2, 3 from (6).It follows that the optimal values of the production batch quantity Q demand rate * r D ad time * r L and minimum annual total cost are given in Tables 2-7.

Figure 1 .
Figure 1.The optimal order quantity against b (for all  ).

Figure 2 .
Figure 2. The optimal demand rate against b (for all  ).

Figure 3 .
Figure 3.The minimum total cost against b (for all  ).

Figure 4 .
Figure 4.The minimum total cost against  (for all b).

Table 6 . The optimal solution of and as a function of b (
* r