Steps for Solving Quantity Discount
Quantity Discount Models Object is to Minimize total inventory costs; includes material
Inventory Control costs
Material costs relevant in total cost: (Production Planning & Control)
) + Q/2(C )
- – TC = DC + D/Q(C o h
where
- D = unit annual demand
- C = unit cost
C = each order cost
- IE 2353
o Pratya Poeri Suryadhini
- C = carrying cost per unit per year
h
IC must be used in place of C in decision-making h
Quantity Discount Models Steps for Solving Quantity Discount
Text example:
1. Compute EOQ for each discount price:
Quantity Discount Schedule
2DC
- * o
Q
IC 2. If EOQ < discount minimum level, make Q = minimum.
Table 6.1
3. For each EOQ, compute total cost:
Material cost:
- Total material cost is affected by the Discount (%)
TC = DC + D/Q(C ) + Q/2(C ) o h
- Unit cost if first $5.00, then $4.80, and finally $4.75 4. Choose the lowest cost quantity from all levels.
- – A Review Quantity Discount Steps
Total Cost Curves for each of the 3 discount plans
Figure 6.7 1. Calculate Q for each discount.2. Adjust Q upward if quantity is too low for discount.
3. Compute total cost for each discount.
4. Select Q with the lowest total cost.
Quantity Discount Example Quantity Discount Example
The Smith company purchases 8000 units of a product each year. The supplier offers the units for sale at $10.00 per unit for orders up to 500 units andThe EOQ at $9.00 is invalid, since it is not available for quantities less than 500 units. The at $9.00 per unit for orders of 500 units or more. EOQ at $10.00 is valid. Therefore, the total cost of the valid EOQ is compared with the total cost at the larger price-break quantity:
What is the economic order quantity if the order cost is $30.00 per order and the holding cost is 30% of per unit cost per year?
Comparing the total costs of the single price-break quantity and the valid EOQ , the minimum cost order quantity is 500 units. Economic Production Quantity Economic Production Quantity
The assumptions that the entire orders is received into inventory at one time (instantaneously is often not true. The EPQ assumes continuous gradual additions to stock (finite replenishment rate) over the production period.
The EPQ formula is obtained:
p = production rate d = demand rate Economic Production Quantity EPQ Example
Optimum length of production run
The demand for an item is 20,000 units per year, and there are 250 working days per year. The production rate is 100 units per day, and the lead time is 4 days.
Production reorder point in units
The units production cost is $50, the holding cost is $10 per unit per year and the setup cost is $20 per run. What are the economic production quantity,
Total annual cost = production cost + setup cost + holding cost the number of runs per year, the reorder point, and the minimum total annual cost?
EPQ Example The Use of Safety Stock
Stock-outs occur when there are uncertainties with:
- Demand - Lead time
Safety stock is extra stock on hand to avoid stock-outs
- ROP = d*L + SS
- d = average daily demand
- L = average lead time, time for an order to be delivered
- SS
= safety stock ROP is adjusted to implement safety stock policy:
The Use of Safety Stock In ve n to ry o n Ha n d
- Given probability of demand, find total cost for each safety stock alternative
In ve n to ry o n Ha n d
Stockout Time
- Set service level; use normal distribution
Stockout is avoided Time
Safety Stock Fig. 6.8. The Use of Safety Stock
The Use of Safety Stock and ROP Known stock-out costs:
Unknown stock-out costs:
- ABCO example:
- ABCO example:
Calculations for a given ROP, N:
EMV(40) = 0.20*$50 + 0.20*$0 + 0.30*$2,400 + 0.20*$4,800 + 0.10*$7,200
ABCO example: Last step for ROP = 40 is to calculate the EMV: EMV = P(D) * Cost of Being Short/Over
Known Stock-out Costs continued
D(50) = (50-40)*$40*6 = $2,400 D(60) = (60-40)*$40*6 = $4,800
D(30) = (40-30)*$5 = $50 D(40) = $0
h
2. Being Over: D(O) = (D-O)* C
ss
1. Being Short: D(S) = (N-S)* C
ABCO example:
Calculation the EMV (expected monetary value) for each ROP alternative Known Stock-out Costs continued
Table 6.3
Known Stock-out Costs continued
= $40/ unit (stock-out cost) D/Q = 6 times per year
ss
= $5 C
h
ROP = 50 (d*L) C
Table 6.2 Initial calculations:
Known Stock-out Costs
- D/Q,
- where S = demand during lead time >where O = demand under ROP Calculations for an ROP of
- Being Over
- Being Short
= $2,410
Unknown Stock-out Costs Unknown Stock-out Costs continued
Hinsdale Company example:
1. Lead time demand ~N(350, 10) When stock-out costs are not quantifiable or not
- where = 350, = 10
applicable:
2. Desired Policy:
P(Stock-out) = 5% • Use a service level to determine safety stock level. Therefore, service level = 95% • Service Stock: the % of time an item is out of stock. Visualization of Desired Inventory Policy:
- Service Level = 1 – P(Stock-out), Or • P(Stock-out) = 1 – Service Level
Figure 6.9
Unknown Stockout Costs continued Unknown Stock-out Costs continued
Hinsdale Company example: Hinsdale Company example: Find Z using a Normal table, like in Appendix A:
Z = 1.65 for a 5% right tail Rewrite equation:
Figure 6.10
SS SS Z = 1.65 = = X = + Safety Stock (SS)
Solving for SS yields 16.5, or 17, units. SS = X – = Z
Therefore, ROP = 350 + 17 = 367
X- SS
Z = = 23 24
Service Level versus Carrying Costs ABC Analysis The following curve depicts the tradeoff
between carrying costs and service level for ABC analysis divides on-hand inventory into three classifications
on the basis of dollar (TL) volume. the previous example such dramatic tradeoffs exist for all similar problemsIt is also known as Pareto analysis. (which is named after principles dictated by Pareto). The idea is to focus resources on the critical few and not on the trivial many. (Annual Dollar Volume of an Item) = (Its Annual Demand) x (Its Cost per unit)
Figure 6.11
ABC Analysis ABC Analysis
Class A items are those on which the annual Class B items are those on which dollar volume is high. annual dollar volume is medium.
They represent 70-80% of total They represent 15-25% of total inventory costs, but they account dollar value, and they account for for only 15% of total inventory 30% of total inventory items on the items. average.
ABC Analysis ABC Analysis
Class C items are low dollar volume items. They represent only the 5% of total dollar volume, but they include as many as 50-60% of total inventory items.
Summary of ABC Analysis ABC Analysis
- Group A Items - Critical
Some of the Inventory Management Policies that may be based on ABC analysis include:
- Group B Items - Important • Group C Items - Not That Important a) Class A items should have tighter inventory control.
Are Complex Quantitative Inventory Dollar Inventory Control
b) Class A items may be stored in a more secure area.
Group Usage (%) Items (%) Techniques Used?
Yes A
70
10 c) Forecasting Class A items may warrant more care.
In some cases B
20
20 No C
10 32
70 ABC Inventory Policies ABC Inventory Analysis l a
Greater expenditure on supplier development u n ge 100 for A items than for B items or C items n sa
90 A
A U Items
80
of r t la
Tighter physical control on A items than on B
70
n e ol
60
items or on C items D rc
B Items
50
e P C Items
40
30 Greater expenditure on forecasting A items 20 1 2 3 4 5 6 7 8 9 10
than on B items or on C items
10 Percent of Inventory Items
33 34 ABC Inventory Example ABC Inventory Example
Rank by percentage of usage
Item Item Annual Dollar Usage Total Annual Persentage Usage Cumulative Percentage Annual Annual Dollar Total Annual
Classification Item Unit cost Usage Usage Persentage Usage
7 3400
46.7
46.7 A 1 2500
34.3
81 A
1 0.05 50000 2500
34.3 9 450
6.2
87.2 B
2 0.11 2000 220
3.0 5 336
4.6
91.8 B
3 0.16 400
64
0.9 2 220
3.0
94.8 B
4 0.08 700
56
0.8 6 195
2.7
97.5 B
3
64
0.9
98.4 C
5 0.07 4800 336
4.6
4
56
0.8
99.2 C
6 0.15 1300 195
2.7
10
48
0.6
99.8 C
7 0.20 17000 3400
46.7
8
12 0.2 100 C
8 0.04 300
12
0.2 Rank by Classification
9 0.09 5000 450
6.2 Item Classification Items Percentage Percentage Value
10 0.12 400
48
0.6 A 7, 1
20
81 B 9, 5, 2, 6
40
16.5 C 3, 4, 10, 8
40
2.5