When several commodities are kept in stock, their inventory policies are intertwined because of common constraints and joint costs, as we now discuss in two separate cases. In the first case, a limit is placed on the total investment in inventories, or on the warehouse space. In the second case, commodities share joint ordering costs. For the sake of simplicity, both analysis will be performed under the EOQ model hypotheses.
4.7.1 Models with capacity constraints
Letnbe the number of commodities in stock andqj, j =1, . . . , n, the amount of commodityjordered at each replenishment. The inventory management problem can be formulated as follows.
Minimize
à(q1, . . . , qn) (4.24)
subject to
g(q1, . . . , qn)⩽b, (4.25)
q1, . . . , qn⩾0, (4.26)
where the objective function (4.24) is the total average cost per time unit. Under the EOQ hypothesis, the objective functionà(q1, . . . , qn)can be written as
à(q1, . . . , qn)= n j=1
àj(qj), where, on the basis of Equation (4.12),
àj(qj)=kjdj/qj+cjdj+12hjqj, j =1, . . . , n,
and quantitieskj, dj, cj, hj, j =1, . . . , n, are the fixed reorder cost, the demand rate, the value, and the holding cost of itemj, respectively. As is customary,hj = pjcj, j=1, . . . , n, wherepj is the interest rate of commodityj.
Equation (4.25) is a side constraint (referred to as acapacity constraint) represent- ing both a budget constraint or a warehouse constraint. It can usually be considered
SOLVING INVENTORY MANAGEMENT PROBLEMS 137 as linear,
n j=1
ajqj ⩽b, (4.27)
whereaj, j = 1, . . . , n, and b are constants. As a result, problem (4.24)–(4.26) has to be solved through iterative methods for NLP problems, such as theconjugate gradient method. Alternatively, the following simple heuristic can be used if the capacity constraint is linear and the interest rates are identical for all the commodities (pj =p, j =1, . . . , n).
Step 1. Using Equation (4.13), compute the EOQ order sizesqj, j =1, . . . , n:
qj =
2kjdj pcj
, j =1, . . . , n. (4.28)
If the capacity constraint (4.27) is satisfied,STOP, the optimal order size for each productj, j =1, . . . , n, has been determined.
Step 2. Increase the interest ratepof aδquantity to be determined. Then, the order sizes become
qj(δ)=
2kjdj
(p+δ)cj, j =1, . . . , n. (4.29) Determine the valueδ∗satisfying the relation,
n j=1
ajqj(δ∗)=b.
Hence,
δ∗= 1
b n j=1
aj
2kjdj
cj 2
−p. (4.30)
Insertδ∗in Equations (4.29) in order to determine the order sizesq¯j, j=1, . . . , n.
New Frontier distributes knapsacks and suitcases in most US states. Its most suc- cessful models are the Preppie knapsack and the Yuppie suitcase. The Preppie knap- sack has a yearly demand of 150 000 units, a value of $30 and a yearly holding cost equal to 20% of its value. The Yuppie suitcase has a yearly demand of 100 000 units, a value of $45 and a yearly holding cost equal to 20% of its value. In both cases, placing an order costs $250. The company’s management requires the average capital invested in inventories does not exceed $75 000. This condition can be expressed by the following constraint,
30q1/2+45q2/2⩽75 000,
138 SOLVING INVENTORY MANAGEMENT PROBLEMS where it is assumed, as a precaution, that the average inventory level is the sum of the average inventory levels of the two items. The EOQ order sizes, given by Equation (4.28),
q1 =
2×250×150 000
0.2×30 =3535.53 units, q2 =
2×250×100 000
0.2×45 =2357.02 units,
do not satisfy the budget constraint. Applying the conjugated gradient method starting from the initial values(q1, q2)=(1,1), the following solution is obtained after 300 iterations,
¯
q1=2500 units,
¯
q2=1666.66 units,
whose total cost is $9 045 000. Applying the heuristic procedure, by using Equa- tion (4.30), the same solution is obtained. In effect,
δ∗= 1
75 000 30
2
2×250×150 000
30 +45
2
2×250×100 000 45
2
−0.2
=0.2, hence,
¯ q1=
2×250×150 000
(0.2+0.2)30 =2500 units,
¯ q2=
2×250×100 000
(0.2+0.2)45 =1666.66 units.
4.7.2 Models with joint costs
For the sake of simplicity, we assume in this section that only two commodities are kept in the inventory. Letk1 andk2 be the fixed costs for reordering the two commodities at different moments in time, and letk1−2be the fixed cost for ordering both commodities at the same time (k1−2< k1+k2). In addition, letT1andT2be the time lapses between consecutive replenishments of commodities 1 and 2, respectively (see Figure 4.9). Then,
q1=d1T1, (4.31)
q2=d2T2. (4.32)
SOLVING INVENTORY MANAGEMENT PROBLEMS 139
t q1
q2 I(t)
T1 T2
Figure 4.9 Inventory level as a function of time in case of synchronized orders.
The periodicity of a joint replenishment policy is T =max{T1, T2}. In each periodT, the orders issued for the two items are
N1=T /T1, N2=T /T2.
N1andN2are positive integer numbers, one of them being equal to 1 (in the situation depicted in Figure 4.9,N1=3 andN2=1). During each periodT, two items are ordered simultaneously exactly once. Moreover,Nj−1 single orders are placed for each itemj, j =1,2. Hence, the total average cost per time unit is
à(T , N1, N2)
= k1−2+(N1−1)k1+(N2−1)k2
T +c1d1+c2d2+h1d1T
2N1 +h2d2T 2N2
. (4.33) By solving the equation,
∂
∂Tà(T , N1, N2)
T=T∗=0, the valueT∗(N1, N2)that minimizesà(T , N1, N2)is obtained,
T∗(N1, N2)=
2N1N2[k1−2+(N1−1)k1+(N2−1)k2] h1d1N2+h2d2N1
, (4.34)
as a function ofN1andN2.
140 SOLVING INVENTORY MANAGEMENT PROBLEMS Shamrock Microelectronics Ltd is an Irish company which assembles printed cir- cuit boards (PCBs) for a number of major companies in the appliance sector. TheY23 PCB has an annual demand of 3000 units, a value of€30 and a holding cost equal to 20% of its value. TheY24 PCB has an annual request of 5000 units, a value of€40 and a holding cost equal to 25% of its value. The cost of issuing a joint order is€300 while ordering a single item costs€250. If no joint orders are placed, the order sizes are, according to Equation (4.13),
q1∗=
2×250×3000
0.2×30 =500 units, q2∗=
2×250×5000
0.25×40 =500 units.
From Equations (4.31) and (4.32):
T1∗=500/3000=1/6, T2∗=500/5000=1/10.
This means that Shamrock would issue 1/T1∗=6 orders per year ofY23 PCB and 1/T2∗=10 orders per year of theY24 PCB. Since
à1(q1∗)= 250ì3000
500 +30×3000+0.2×30×500 2
=93 000 euros per year, à2(q2∗)= 250ì5000
500 +40×5000+0.25×40×500 2
=205 000 euros per year,
the average annual cost is€298 000 per year. If a joint order is placed andN1=1, N2=2, the periodicity of joint orders is, according to Equation (4.34),
T∗=
2×1×2×(300+250)
0.2×30×3000×2+0.25×40×5000×1 =0.16.
Shamrock would issue 1/T∗=6.25 joint orders per year. The annual average cost, computed through Equation (4.33), is equal to
à(T∗,1,2)=300+250
0.16 +30×3000+40×5000+0.2×30×3000×0.16 2
+0.25×40×5000×0.16 2×2
=296 877.5 euros per year.
SOLVING INVENTORY MANAGEMENT PROBLEMS 141