J g carlsson u of mn isye lecture 10 inventory models

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Unformatted text preview: f MN ISyE Lecture 10: Inventory Models November 7, 2013 7 / 55 The basic EOQ model We will be ordering Q units in “cycles” of length Q/a, and our objective is to determine Q The purchasing and setup cost per cycle is K + cQ The holding cost per cycle is the area under one of the triangles: ˆ Q/a hQ2 h · (Q − at) dt = 2a 0 so the total cost per cycle is K + cQ + J. G. Carlsson, U of MN ISyE hQ2 2a Lecture 10: Inventory Models November 7, 2013 8 / 55 The basic EOQ model The total cost per cycle is hQ2 2a and therefore, the total cost per unit time is K + cQ + 2 K + cQ + hQ /2a aK hQ = + ac + Q/a Q 2 We differentiate this with respect to Q to find the optimal Q∗ : d dQ ( aK hQ + ac + Q 2 ) aK h = − 2 + = 0 =⇒ Q∗ = 2 Q √ 2aK , h the EOQ formula J. G. Carlsson, U of MN ISyE Lecture 10: Inventory Models November 7, 2013 9 / 55 Observations The optimal order quantity Q∗ is √ ∗ Q= which costs us aK hQ + ac + Q 2 √ Q= 2aK h = √ 2aKh + ac 2aK/h per unit time Note that Q∗ becomes larger as a and K increase and smaller as h increases Also note that the optimal cost doesn’t change if we change K → β K and h → h/β because √ √ 2a(β K)(h/β ) + ac = 2aKh + ac Also, the optimal cost doesn’t change if we change a → β a and h → h/β and c → c/β because √ √ 2(β a)K(h/β ) + (β a)(c/β ) = 2aKh + ac J. G. Carlsson, U of MN ISyE Lecture 10: Inventory Models November 7, 2013 10 / 55 Planned shortages A shortage or stockout occurs when we cannot meet demand currently because the inventory is depleted Under certain circumstances, it may be desirable to permit a limited planned shortage – this requires that customers be willing to accept a delay in fulfilling their orders The EOQ model with planned shortages addresses this kind of situation as follows [1] : When a shortage occurs, the affected customers will wait for the product to become available again. Their backorders are filled immediately when the order quantity arrives to replenish inventory. We incorporate a shortage cost p per unit short per unit of time short: $ unit × hour We are also interested in the inventory level just after a batch of Q units is added to inventory, which we’ll call S p: [1] F.S. Hillier and G.J. Lieberman. Introduction to Operations Research. Introduction to Operations Research. McGraw-Hill Higher Education, 2010. J. G. Carlsson, U of MN ISyE Lecture 10: Inventory Models November 7, 2013 11 / 55 Planned shortages The costs per cycle (of length Q) are given as follows: The purchasing and setup cost per cycle, as before, is K + cQ Inventory level is positive for time S/a, so the holding cost is the area under one ´ S/a 2 of the “positive triangles”: 0 h · (S − at) dt = hSa 2 The shortage cost is the area above one of the “negative triangles”: ´ Q/a −2 − S/a p · (S − at) dt = p(Q2a S) J. G. Carlsson, U of MN ISyE Lecture 10: Inventory Models November 7, 2013 12 / 55 Planned shortages The total cost per cycle is then K + cQ + hS2 p(Q − S)2 + 2a 2a The total cost per unit time is aK hS2 p(Q − S)2 K + cQ + hS /2a + p(Q−S) /2a = + ac + + Q/a Q 2Q 2Q 2 2 This has two variables, and we can differentiat...
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