Operations Research Lecture 14
Operations Research Inventory Control
Inventory Physical stock of items held in any business for the production or sales
Type of Inventory Finished Goods ! Finished Product Raw Material In-Process Finished Goods !
Inventory Costs Shortage Cost Rs C1 /item/unit time Item Cost, Rs C1/item Ordering Cost, Rs C2 /order Holding Cost Rs C1 /item/unit time Shortage Cost Rs C1 /item/unit time
Inventory Models (EOQ Models) The inventory control model can be broadly classified into two categories: Deterministic inventory problems Probabilistic inventory problems
Model 1 Purchasing Model with no Shortages
Assumptions Demand (D) is at a constant rate. Replacement of items is instant The C1, C2, and C3 are constant. No shortage cost, C4 = 0.
Cont ! Quantity Time
D is the demand per year
Model 2 Purchasing Model with Shortages
Example
The demand for an item is 18000 units/year The demand for an item is 18000 units/year. The cost of one purchase is Rs 400. The holding cost is Rs 1.2 per unit per year. The item cost is Rs 1 per item. The shortage cost is Rs 5 per unit per year. Determine: (a) The optimum order quantity. (b) The time between orders. (c) The number of orders per year. (d) The optimum shortages. (e) The maximum inventory. (f) The time of items being held. (g) The optimum annual cost.
= 1500 units / month C1 = Rs 1.0 / item C3 = Rs 1.2 / year/ item Data D = 18000 units / year = 1500 units / month C1 = Rs 1.0 / item C3 = Rs 1.2 / year/ item C2 = Rs 400 /order C4 = Rs 5.0 / year / item
Optimum Order Quantity
Time Between Orders
Number of Orders per year Number of orders per year = 12 / 2.57 = 4.66
Optimum Shortages
Maximum Inventory
The Time when Items being held
Order cost = Rs 400 per order Optimum Annual Cost Annual cost = Item cost + Ordering cost + Holding cost + Shortage cost Item cost = Rs 3857 per order Order cost = Rs 400 per order
Holding Cost
Shortage Cost
Total Cost per order
Model 3 Manufacturing Model with no Shortages
Demand rate (D) is constant Assumptions: Demand rate (D) is constant All cost coefficients (C1, C2, C3) are constants There is no shortage cost, or C4 = 0 The replacement rate is finite and greater than the demand rate. This is also called replenishment rate or manufacturing rate, denoted by R
Demand rate (D) is constant Assumptions: Demand rate (D) is constant All cost coefficients (C1, C2, C3) are constants There is no shortage cost, or C4 = 0
Cont ! The replacement rate is finite and greater than the demand rate. This is also called replenishment rate or manufacturing rate, denoted by R
Slope = D Slope = (R-D) Q Im t t2 t1 t = t1 + t2
Q T Q Im s t t2 t1 t = t1 + t2
The total cost of inventory per period is the sum of three components: The total cost of inventory per period is the sum of three components: item cost, order cost, and items holding cost.
Let Im be the maximum inventory, t1 be the time of manufacture and t2 be the time during which there is no supply. In this model, all items required for a cycle are not stored at the beginning as in Wilson’s Model. The items are manufactured at a higher rate than the demand so that the difference (R–D) is the existing inventory till the items are exhausted.
Item cost/period = C1Q Order cost/period = C2 Item holding cost/period =
Differentiating C with respect to Q and setting equal to zero for minimum C, we get,
Example
The demand for an item in a company is 18000 units/year and the company can produce at the rate of 3000 per month. The cost of one set up is Rs. 500 and the holding cost of 1 unit per month is 15 paisa. Determine: (a) The optimum manufacturing quantity. (b) The maximum inventory. (c) The time between orders. (d) The number of orders/year. (e) The time of manufacture. (f) The optimum annual cost if the cost of the item per unit is Rs. 2. Assume no shortages.
C2 = Rs 500 per order. C3 = Rs 0.15 / item per month Data C1 = Rs 2 per item. C2 = Rs 500 per order. C3 = Rs 0.15 / item per month D = 18000 / year = 1500/ month R = 3000/month
Operations Research Lecture 14