lot sizing problem in case when demand varies over different time periods. . Key WordsâLot sizing, total cost, ordering cost, holding cost, Wagner and Whitin, ...
Dynamic lot sizing in Inventory Management using spreadsheet 1. Mrs. N.R. Rajhans 2.
Abstract-- This paper presents a model for calculating the parameters of an inventory replenishment scheduling policy with dynamic demand considerations. The optimal policy parameters specify how many times and in what quantities to replenish the stock at a place such as warehouse or company. The problem is represented using a dynamic demand during one year or twelve periods of one month per period. Although the paper is motivated by a third-party manufacturing industry application, the underlying model is applicable in the general context of coordinating inventory ordering decisions. The problem is challenging due to the dynamic demand situation in which variation in demand causes varying inventory levels for different periods which in turn causes holding costs for different periods to differ leading to variation in total cost. The main aim is to optimize inventory policy so as to minimize total inventory costs incurred. The paper presents several methods for optimizing inventory costs by deciding specific ordering time and quantity. The best method is selected among those which has minimum total cost (ordering cost +holding cost) with the help of spreadsheet document of Microsoft excel. 2006 (Brahimi et al, )first presented a comprehensive review of lot sizing problem in case when demand varies over different time periods. . Key Words—Lot sizing, total cost, ordering cost, holding cost, Wagner and Whitin, dynamic demand pattern.
1.
INTRODUCTION
Order management has been defined (Cox and Blackstone) as the directing, monitoring, planning and controlling of the processes related to customer orders, manufacturing orders. In every ordering process a policy decision or set of rules and procedure is established to determine lot size and ordering time for any particular order. Generally lot-sizing involves the determination of the size of the order or quantity of order and the timing of such any decision taken to satisfy the requirements of demand over a fixed future period. In most of manufacturing and process organizations inventory management is of vital importance as large part of capital is invested in inventory materials. Capital invested in inventory generates no returns and usually called as idle capital and increase in holding costs. Large inventory causes reduction in return on investment (ROI) for a company
and also increases the risk of damage and obsolescence. Also very less inventory leads to shortage costs and increased ordering costs leading dissatisfaction leading to long term losses to the company. Thus it is of extreme importance to minimize total inventory costs and also to improve customer satisfaction. As there are different kinds of products it is necessary to develop a decision support system which will yield a correct quantity and timing of orders. Dissatisfied with the “square root formula” to find the economic lot size under the assumption of steady-state 1958 (constant) demand, Wagner and Whitin developed an elegant forward algorithm based on dynamic programming principles to make optimal lot size 2007 1998 decisions. Dai and Qi , and Rosenblatt et al. examined this problem where the demand of the product was constant and developed the well known economic order quantity method. The main aim behind this paper is to put forth a method for inventory lot sizing and ordering timing which reduces overall cost. Here a demand pattern of a medium scale manufacturing organization is considered during a span of one year divided into twelve periods. Thus we have a demand pattern for each month of a particular period. Considering this demand pattern lot sizing and ordering timing is done by using various methods. The final decision is to select the best method which has the lowest overall cost. The data is representative and can be generalized to different components and industries. 2. Model Description Here a representative demand data of a manufacturing organization over a period of one year is considered which is divided in twelve parts of one month each. The data is as shown in table 1. Here the demand data is dynamic. The lot size and ordering activity must be such that the total cost that is the sum of holding costs and ordering costs must be lowest. The various methods for deciding lot size and ordering timing are given below with the cost associated with each of them.
Table 1 (Demand Pattern)
Period
1
2
3
4
5
6
7
8
9
10
11
12
Demand
10
62
12
94
90
129
88
52
124
60
238
41
Totals
Table 2(Lot for Lot) Period
1
2
3
4
5
6
7
8
9
10
11
12
Beginning inventory
0
0
0
0
0
0
0
0
0
0
0
0
Demand
10
62
12
94
90
129
88
52
124
60
238
41
Ordering (setup) cost/order
54
54
54
54
54
54
54
54
54
54
54
54
Carrying cost/unit/period
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
Order quantity (lotsizes)
10
62
12
94
90
129
88
52
124
60
238
41
Ending inventory
0
0
0
0
0
0
0
0
0
0
0
0
Ordering cost
54.00
54.00
54.00
54.00
54.00
54.00
54.00
54.00
54.00
54.00
54.00
54.00
648.0
Carrying cost
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
Total period cost
54.00
54.00
54.00
54.00
54.00
54.00
54.00
54.00
54.00
54.00
54.00
54.00
648.0
1000
1000
1. Lot for Lot : 2. Least total cost Lot-for-lot (LFL) ordering is the simplest approach of all. An order is scheduled for each period in which a demand occurs. Here in case of lot for lot items are purchased in the exact quantities required for each period. In case of lot for lot ordering there is hardly any inventory which is carried over to next year. This results in virtually zero inventory holding costs. But on the other hand the ordering activity has to be increased to cope up with the demand resulting in higher ordering costs. Thus it is suitable to use lot for lot size in case of large holding costs and low ordering costs. The table shows the inventory position at opening and closing time of each month. Here the total costs are calculated using excel spread sheet and represented in the table 2.
This method is almost akin to EOQ method. Here it is attempted to find the least total cost for a lot which will occur when setup cost and inventory carrying costs are nearly the same. In least total cost technique different lots are considered and setup costs and carrying costs for each is calculated. The lot for which setup cost and inventory carrying cost are almost same that lot is selected. The ordering schedule and total cost is depicted in table 3.
Table 3(Least total cost)
LEAST TOTAL COST Period
1
2
3
4
5
6
7
8
9
10
11
12
Totals
Beginning inventory
0
74
12
0
90
0
88
0
124
0
238
0
Demand
10
62
12
94
90
129
88
52
124
60
238
41
Ordering (setup) cost/order
54
54
54
54
54
54
54
54
54
54
54
54
Carrying cost/unit/period
0.4
0.4
0.4
0.4
0.4
0.40
0.4
0.4
0.4
0.4
0.4
0.4
Order quantity (lotsizes)
84
0
0
184
0
217
0
176
0
298
0
41
Ending inventory
74
12
0
90
0
88
0
124
0
238
0
0
Ordering cost
54
0.00
0.00
54
0.00
54
0.00
54.00
0.00
54
0.00
54
324
Carrying cost
29.60
4.80
0.00
36.00
0.00
35.20
0.00
49.60
0.00
95.20
0.00
0.00
250.40
Total period cost
83.6
4.8
0.0
90.0
0.0
89.2
0.0
103.6
0.0
149.20
0.00
54.00
574.40
1000
1000
Table 4(Least Unit Cost)
LEAST UNIT COST Period
1
2
3
4
5
6
7
8
9
10
11
12
Beginning inventory
0
74
12
0
154
0
88
0
124
0
0
0
Demand
10
62
12
130
154
129
88
52
124
160
238
41
Ordering (setup) cost/order
54.0
54.00
54.00
54.00
54.00
54.00
54.00
54.00
54.00
54.00
54.00
54.00
Carrying cost/unit/period
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
Order quantity (lotsizes)
84
0
0
284
0
217
0
176
0
160
238
41
Ending inventory
74
12
0
154
0
88
0
124
0
0
0
0
Ordering cost
54.00
0.00
0.00
54.00
0.00
54.00
0.00
54.00
0.00
54.00
54.00
54.00
378.0
Carrying cost
29.60
4.80
0.00
61.60
0.00
35.20
0.00
49.60
0.00
0.00
0.00
0.00
180.8
Total period cost
83.60
4.80
0.00
115.60
0.00
89.20
0.00
103.60
0.00
54.00
54.00
54.00
558.8
3. Least Unit cost In case of least unit cost, ordering cost and carrying costs are added together. The sum so obtained is divided by total number of units to get least unit cost. Here demand for different consecutive period is considered. Here by summing cumulative demands different lot sizes are obtained. The total of ordering and holding cost is calculated for each lot and divided by total number of units to get least total cost. The ordering schedule and total cost is depicted in table 4.
Totals
1200
1200
4. Wagner and Whitin algorithm The Wagner-Whitin algorithm (Evans, J. R) gives an optimum solution to the deterministic dynamic order size problem over a finite horizon. The Wagner-Whitin algorithm is a dynamic programming method which can be used to determine the policy in case of minimum cost.
Table 5(Wagner-Whitin lot-size)
Wagner-Whitin lotsizes -- optimal lotsizes Period
1
2
3
4
5
6
7
8
9
10
11
12
Beginning inventory
0
74
12
0
0
129
0
52
0
0
0
41
Demand
10
62
12
130
154
129
88
52
124
160
238
41
54
54
54
54
54
54
54
54
54
54
54
54
Carrying cost/unit/period
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
Order quantity (lotsizes)
84
0
0
130
283
0
140
0
124
160
279
0
Ending inventory
74
12
0
0
129
0
52
0
0
0
41
0
Ordering cost
54.00
0.00
0.00
54.00
54.00
0.00
54.00
0.00
54.00
54.00
54.00
0.00
378.00
Carrying cost
29.60
4.80
0.00
0.00
51.60
0.00
20.80
0.00
0.00
0.00
16.40
0.00
123.20
Total period cost
83.60
4.80
0.00
54.00
105.60
0.00
74.80
0.00
54.00
54.00
70.40
0.00
501.20
Ordering cost/order
Totals
1200
(setup)
Wagner and Whitin Algorithm
1. Define Zce to be the total variable cost in periods c through e of placing an order in period c which satisfies requirements in periods c through e : Zce = C + h P
( Qce – Qci )for 1≤c≤e≤N
2. Define fe to be the minimum possible cost in periods , given that the inventory level at the end of period e is zero. fe = Min ( Zce + fc-1 ) for c = 1,2…..e. The value of fN is the cost of the optimal order schedule.
3. To translate the optimum solution (fN) obtained by the algorithum to order quantities FN = Z w N + f w – 1 F w – 1 = Z v w-1 + f v – 1 upto f u–1 = Z 1
u-1
+ f0
1200
3. CONCLUSION Here in the problem under consideration the demand pattern with different lot sizing methods is analyzed. The ordering cost and inventory holding cost is considered same for all methods of lot sizing. Here costs of inventory by different lot sizing techniques is calculated by using Microsoft excel spreadsheet. The total cost is lowest for Wagner-Whitin algorithm as in the table below. As the Wagner-Whitin algorithm gives the lowest total cost this method of lot sizing is best suited for minimizing total inventory costs. Here the cost was also calculated for different demand patterns which results the same. Table 6 depicts the total cost of inventory by different lot-sizing methods. Total costs for each lotsizing methods are represented by a chart in figure 1. Table 6
Method of lotsizing 1. Lot for Lot 2. Least total cost 3. Least unit cost 4. WagnerWhitin
Total Cost 648 574 558 501
800
Total Cost
600 400 200 0
Figure 1
4. REFERENCES 1.
Wagner, H. M. and T. M. Whitin (1958), “Dynamic Version of the Economic Lot Size Model,” Management Science vol 50 No.12
2.
Whitin, T. M. 1957. The Theory of Inventory Management, 2nd edition. Princeton University Press, Princeton, NJ.
3.
Quantitative Techniques by N. D. Vora
4.
Operations Management by Joseph G. Monks 2nd edition
5.
Performance analysis of conjoined supply chains Benita M. Beoman. International journal of production research(2001)vol 39,No.14,pp.3195-3218 Evans, J. R. (1985), “An Efficient Implementation of the Wagner-Whitin Algorithm for Dynamic Lot-Sizing,” Journal of Operations Management, 5, 2, 229-235.
6.
