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

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?

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

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)

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

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

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

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

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

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 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

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

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

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

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

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

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

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

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

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

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)

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)

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 1 - 599 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

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)

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