1. How much to purchase? How much to produce?
OPERATIONS MANAGEMENT
Exercise classes
Order quantity
How much to purchase?
How much to produce?

2. Classes - Topics Materials flow parameters
SUPPLIERS
PRODUCTION
CUSTOMERS
Materials flow parameters
Quantity parameter (quantity flow)
Time parameter (speed flow)
Choice of product mix (production program optimization) – what and how much to produce and sell?
Time parameter – how long we will produce? (production lead time, methods of lead time reduction)
Quantity parameter – how much to purchase and how much to produce? (economic order quantity, economic production quantity)
Material requirements planning – what, how much and when to produce and purchase?

3. INVENTORY MANAGEMENT (CONTROL) SYSTEMS
Economic Order Quantity Model
DEPENDENT DEMAND INVENTORY CONTROL SYSTEMS
INVENTORY MANAGEMENT (CONTROL) SYSTEMS
FIXED ORDER
QUANTITY SYSTEMS
INDEPENDENT DEMAND INVENTORY CONTROLSYSTEMS
PERIODIC REVIEW SYSTEMS
CONTINUOUS
REVIEW SYSTEMS
Economic Production Quantity Model
Quantity Discounts Model
PERIOD SYSTEMS
MATERIAL REQUIREMENTS PLANNING SYSTEM (MRP)
Back Order Inventory Model

4. Fixed Order Quantity System
Assumptions
Order quantity - fixed
Order cycle, order period - variable
Continuous inventory level control
How much to order ?
In what quantities to place the order?
FIXED ORDER QUANTITY - FOQ
Order quantity is arbitrary defined (e.g.: minimal order = 1 palate = 1000 units)
Fixed Order Quantity - FOQ
ECONOMIC ORDER QUANTITY MODEL - EOQ
Order quantity is calculated using EOQ formule
R. H. WILSON (Wilson’s formule 1915)

5. Economic Order Quantity Model – EOQ
ASSUMPTIONS
Demand is known and constant
Ordering cost and holding cost can be estimated
Lead time (delivery time) is known and constant
Stock replenishment is immediate
Q - order quantity
S - maximal stock
Sav - average stock
R - reorder point
T - order cycle
Cz - stock cycle
LT - lead time (delivery time)
WD - number of working days per year

6. Economic Order Quantity calculation
Annual cost
Total Cost
Holding Cost
Order (Setup) Cost Q*
Cmin
D - annual demand
HC - annual holding cost
H - holding cost per unit per year
SC - total ordering (set up) cost per year
Sc - cost per order (set up cost)
TC – total cost
Cmin – minimal total cost
Q - order quantity
EOQ= Q*- economic order quantity
Economic Order Quantity
Objective :
Total cost minimalization

7. Number of orders per year
EOQ Model - example
DATA
D = 1200 units/year
Sc = 100 $/order
H = 6 $/unit/year
WD = 240 working days /year
Number of orders per year
Annual holding cost
Economic Order Quantity
Total order cost per year
Maximal stock
Total inventory cost
Average stock
Order cycle

8. When to order? - Reorder Point model
Reorder Point (ROP) - Stock level which inform about the necessity of placing an order to replenish inventory
Where:
Q - order quantity
S - maximal stock
SS - safety stock
Sav - average stock
R - reorder point
d - demand rate
LT - lead time (delivery time)

9. EOQ model parameters (1)
Optimization criterion
Total annual ordering OC and holding HC costs minimization
Total annual inventory holding cost
– annual holding cost per unit
Total annual ordering cost
Sc – annual ordering per unit
D – annual demand forecast
Total annual cost

10. EOQ model parameters (2)
Economic order quantity
Maximal inventory
Average inventory
Annual number of orders
Inventory cycle = Order cycle
WD – number of working days per year

11. Options of EOQ model Economic Order Quantity model Basic model
Production Order Quantity Model
EOQ Model with Gradual Replenishment
Back Order Inventory Model EOQ Model with Planned Shortages
Options of EOQ model
Quantity Discount Model
EOQ Model with Quantity Discount 11

12. Production Order Quantity Model – POQ EOQ Model with Gradual Inventory Replenishment
Quantity parameters
Qp - production quantity, production
batch size
S - maximal stock
Sav - average stock
p – production rate
d – inventory consumption rate,
demand rate
Time parameters
T1 - production and consumption
period
T2 - inventory concumption period
IC - inventory cycle
PC – production cycle
MODEL ASSUMPTIONS
EOQ assumptions
Gradual inventory replenishment

13. POQ model parameters (1)
Optimization criterion
Total annual set up and holding costs minimization – min (SC + HC)
Annual inventory holding cost
H – holding cost per unit per year
Annual setup cost (production ordering cost)
Sc – setup cost per order
D – annual (forcasted) demand
Total annual cost

14. POQ model parameters (2)
Economic production quantity
Maximal inventory
Average inventory
Annual setups
Inventory cycle
Production cycle

15. POQ model – example D = 1200 units/year p = 9 units/day
DATA
D = 1200 units/year p = 9 units/day
Sc = 100 $/order d = 5 units/day
= 6 $/unit/year
WD= 240 working days/year
Total annual holding cost
Total annual set up cost
Economic production quantity
Total annual cost
Maximal inventory
Inventory cycle = order cycle
Annual number of setups
Production cycle

16. Back Order Inventory Model (EOQ Model with Planned Shortages)
MODEL ASSUMPTIONS
EOQ assumptions
Stock outs (shortages) are possible - back orders are possible
Quantity parameters
Qb - order quantity
S - maximal stock
SO - maximal stockout
Time parameters
T1 - inventory availability period
T2 - stockout period
IC - inventory cycle

17. Back Order Inventory Model parameters (1)
Optimization criterion
Total annual ordering, holding and stock out costs minimization
Annual holding cost
Annual stock out cost
Cso – stock out cost per unit
Annual ordering cost
Total annual inventory cost

18. Back Order Inventory Model parameters (2)
Economic order quantity with back orders
Maximal stock out
Maximal inventory
Average inventory
Inventory cycle
Inventory availability period
Stockout period
Annual number of orders

19. Back Order Inventory Model – example
DATE
D = 1200 units/year
Sc = 100 $/order
H = 6 $/unit/year
Cso = 13,5 $/unit/year
DN = 240 working days/year
Number of orders per year
Total annual cost (variable)
Inventory cycle = Order cycle
Economic order quantity with back orders
Inventory availability period
Maximal stock out
Stock out period
Maximal inventory

20. Back Order Inventory Model – understanding
Using of Thales theorem (similarity of triangles)
Triangle of stocks is similar to triangle of stock outs
Relation
Example
H = 10 $/unit/year
Cso = 20 $/unit/year
Qb* = 600 units
IC = 30 working days
S = 400 days
SO = 200 days
T1 = 20 days
T2 = 10 days

21. EOQ with Quantity Discounts Model
Quantity Discounts Inventory Model
Model assumptions
EOQ assumptions
Quantity discounts
Discounts
Order quantity
From 1 to Q1
Price P1
From Q1 do Q2
From Q2
P1 > P2 > P3 P2 P3

22. Costs in Quantity Discounts Model
Optimization criterion
Total annual ordering, holding HC and purchasing costs minimization Q
Cost
CP (P1) Q1
CP (P2)
CP (P3)
D · P1
D · P2
D · P3 Q2
Real total cost
Holding costs, ordering costs and purchasing costs depend on the order quantity
Qd* formule depends on holding cost calculation method
Holding cost  fixed value
Holding cost  % of item value (price)

23. 1. Inventory holding cost per unit is fixed
One common economic order quantity Q* for different price
PROCEDURE for calculation of ecomic order quantity when the holding cost does’t depands on price of item – holding cost per unit (H) is fixed Q
Cost Q*
CP (P1)
CP (P2)
CP (P3) OC
HC (P1, P2, P3)
1. Calculate Q* which is common for all prices:
2. Define curve of real total cost TC for Q*
3. If Q* is in real scope of TC curve with the lowest price than Qd* = Q*
If Q* is in real scope of other curve, calculate total cost for Q* and for points where price drop down
Compare costs. Economic order quantity is Q with the lowest TC
Qd* = Q with (TC min)

24. EOQ model with discounts – example 1
DATA
D = 1200 units/year
Sc = 100 $/order
H = 6 $/units/year
= 240 working days/year
Q* for the both prices
Discounts
Order quantity units P1 = 10 $
Order quantity more than 600 units P2 = 9,5 $
Holding cost Ch  fixed value
Real TC curve for Q*  TC (P1)
Annual total cost TC for Q* = 200 units
Annual total cost TC for Q = 600 units
Comparison of the costs
Economic quantity Q* with discount
Number of orders ON = 6 orders
Inventory cycle IC= 40 days

25. 2. Holding cost is calculated as a percentage of the price
PROCEDURE for determining economic order quantity when the holding cost depands on the price Q
Cost
Q1*
TC (P1)
TC (P2)
TC (P3) OC
Q2*
Q3*
HC (P1)
HC (P2)
HC (P3)
Economic order quantities are different because holding cost depands on item price
1. Calculate Q* for different price using the following formula:
i- interest rate of frozen capital
2. Determine the nearest total cost curve with real scope for Q*
3. If Q* is in the real scope of the TC curve for the lowest price, than Qd* = Q*
4. If Q* is placed in the real scope of other curve, than calculate total cost TC for Q* and in the points where the price drop down.
5. Compare the costs. Economic quantity is Q with the minimal TC
Qd* = Q (TC min)

26. Model EOQ model with discounts – example 2
Holding cost H  percentage of the item value (price)
Real TC curve for Q*  TC (P1)
DATA
D = 1200 units/year
Sc = 100 $/order
H = 25% price
DN = 240 working days/year
Annual total cost TC for Q* = 310 units
Annual total cost TC for Q = 600 units
Quantity discounts  as in example 1
Comparison of the costs
Economic order Q* for price P2 or P1
Economic order quantity Q* with discounts
Annual number of orders ON = 2 orders
Inventory cycle IC = 120 days