OPERATIONS MANAGEMENT Exercise classes Order quantity How much

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

Classes - Topics SUPPLIERS PRODUCTION CUSTOMERS Quantity parameter (quantity flow) Materials flow parameters Time parameter (speed flow) 1. Choice of product mix (production program optimization) – what and how much to produce and sell? 2. Time parameter – how long we will produce? (production lead time, methods of lead time reduction) 3. Quantity parameter – how much to purchase and how much to produce? (economic order quantity, economic production quantity) 4. Material requirements planning – what, how much and when to produce and purchase? 2

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

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

Economic Order Quantity Model – EOQ ASSUMPTIONS u Demand is known and constant u Ordering cost and holding cost can be estimated u Lead time (delivery time) is known and constant u 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

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

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

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)

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 D – annual ordering per unit – annual demand forecast Total annual cost 9

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 10

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

Production Order Quantity Model – POQ EOQ Model with Gradual Inventory Replenishment MODEL ASSUMPTIONS u u EOQ assumptions 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 T 1 - production and consumption period T 2 - inventory concumption period IC - inventory cycle PC – production cycle 12

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 D – setup cost per order – annual (forcasted) demand Total annual cost 13

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

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

Back Order Inventory Model (EOQ Model with Planned Shortages) MODEL ASSUMPTIONS u u EOQ assumptions Stock outs (shortages) are possible - back orders are possible Quantity parameters Qb S SO - order quantity - maximal stockout Time parameters T 1 - inventory availability period T 2 IC - stockout period - inventory cycle 16

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 17

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

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 Economic order quantity with back orders Maximal stock out Maximal inventory Number of orders per year Total annual cost (variable) Inventory cycle = Order cycle Inventory availability period Stock out period 19

Back Order Inventory Model – understanding Using of Thales theorem (similarity of triangles) u Relation Triangle of stocks is similar to triangle of stock outs Example H Cso Qb* IC = 10 $/unit/year = 20 $/unit/year = 600 units = 30 working days SO days T 1 days = 400 = 20 20

EOQ with Quantity Discounts Model Quantity Discounts Inventory Model assumptions u u EOQ assumptions Quantity discounts Discounts Order quantity Price From 1 to Q 1 P 1 From Q 1 do Q 2 P 2 From Q 2 P 3 P 1 > P 2 > P 3 21

Costs in Quantity Discounts Model Optimization criterion Total annual ordering, holding HC and purchasing costs minimization Cost Holding costs, ordering costs and purchasing costs depend on the order quantity Real total cost CP (P 1) CP (P 2) CP (P 3) Qd* formule depends on holding cost calculation method D · P 1 D · P 2 D · P 3 Q 1 Q 2 u Holding cost fixed value u Holding cost % of item value (price) Q 22

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 1. Calculate Q* which is common for all prices: Cost CP (P 1) CP (P 2) CP (P 3) 2. Define curve of real total cost TC for Q* HC (P 1, P 2, P 3) OC Q* Q 3. If Q* is in real scope of TC curve with the lowest price than Qd* = Q* 4. If Q* is in real scope of other curve, calculate total cost for Q* and for points where price drop down 5. Compare costs. Economic order quantity is Q with the lowest TC Qd* = Q with (TC min) 23

EOQ model with discounts – example 1 Holding cost Ch fixed value DATA D = 1200 units/year Sc = 100 $/order H = 6 $/units/year = 240 working days/year Discounts Real TC curve for Q* TC (P 1) Annual total cost TC for Q* = 200 units Annual total cost TC for Q = 600 units Comparison of the costs Order quantity 1 - 599 units P 1 = 10 $ Order quantity more than 600 units P 2 = 9, 5 $ Economic quantity Q* with discount Q* for the both prices Number of orders ON = 6 orders Inventory cycle IC= 40 days 24

2. Holding cost is calculated as a percentage of the price Economic order quantities are different because holding cost depands on item price Cost TC (P 1) PROCEDURE for determining economic order quantity when the holding cost depands on the price 1. Calculate Q* for different price using the following formula: TC (P 2) i- interest rate of frozen capital TC (P 3) HC (P 1) 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* HC (P 2) HC (P 3) OC Q 1* Q 2* Q 3* 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) 25

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