Production and Operation Managements Inventory Control Subject to Known Demand Prof.JIANG Zhibin Dr.GENG Na Department of Industrial Engineering Management Shanghai Jiao Tong University
Production and Operation Managements Prof. JIANG Zhibin Dr. GENG Na Department of Industrial Engineering & Management Shanghai Jiao Tong University Inventory Control Subject to Known Demand
Contents .Introduction .Types of Inventories .Motivation for Holding Inventories; .Characteristics of Inventory System; Relevant Costs; .The EOQ Model: EOQ Model with Finite Production Rate Quantity Discount Models .Resource-constrained multiple product system .EOQ models for production planning .Power-of-two policies
Contents •Introduction •Types of Inventories •Motivation for Holding Inventories; •Characteristics of Inventory System; •Relevant Costs; •The EOQ Model; •EOQ Model with Finite Production Rate •Quantity Discount Models •Resource-constrained multiple product system •EOQ models for production planning •Power-of-two policies
Resource-constrained multiple product system When the EOQ model is used in companies stocking many different items,although we can obtain the optimal order quantities separately for each of different items,there could exist constraints (such as stock space,financial budget)that would make the resulting solution infeasible
Resource-constrained multiple product system When the EOQ model is used in companies stocking many different items, although we can obtain the optimal order quantities separately for each of different items, there could exist constraints (such as stock space, financial budget) that would make the resulting solution infeasible
Resource-constrained multiple product system Example 4.5:three items are produced in a small fabrication shop =0.25 The shop never have more than $30,000 invested in the inventory of these items at one time What lot sizes should the shop be producing so that they do not exceed the budget? 1 2 3 Demand rate 1,850 1,150 800 Variable cost 50 350 85 Setup cost 100 150 50
Resource-constrained multiple product system Example 4.5: three items are produced in a small fabrication shop. I=0.25 The shop never have more than $30,000 invested in the inventory of these items at one time What lot sizes should the shop be producing so that they do not exceed the budget? 12 3 Demand rate 1,850 1,150 800 Variable cost 50 350 85 Setup cost 100 150 50
Resource-constrained multiple product system First check whether the budget is feasible when using EOQ values of these three items E00= 2*100*1,850 =172, 0.25*50 E00,= 2*150*1,150 =63, 0.25*350 E00= 2*50*800 =61. 0.25*85 If the EOQ value is used,the maximum investment in inventory would be72*50+63*350+61*85=$35835>$30000 We therefore need to reduce these lot sizes.How to do?
Resource-constrained multiple product system First check whether the budget is feasible when using EOQ values of these three items If the EOQ value is used, the maximum investment in inventory would be 72*50+63*350+61*85=$35835>$30000 We therefore need to reduce these lot sizes. How to do? 1 2 3 2*100*1,850 172, 0.25*50 2*150*1,150 63, 0.25*350 2*50*800 61. 0.25*85 EOQ EOQ EOQ
Resource-constrained multiple product system We multiply each EOQ value by the ratio 30,000/35,835 0.8372.In order to guarantee that we do not exceed the $30,000 budget,we round each value down,we get the optimal values as follows: Q1=172*0.8372≈144 Q2=63*0.8372≈52 Q3=61*0.8372≈51， Lastly,we can increase the lot sizes of some items if there is remaining budget
We multiply each EOQ value by the ratio 30,000/35,835 = 0.8372. In order to guarantee that we do not exceed the $30,000 budget, we round each value down, we get the optimal values as follows: Q1=172*0.8372 ≈144 Q 2=63*0.8372 ≈52 Q 3=61*0.8372 ≈51, Lastly, we can increase the lot sizes of some items if there is remaining budget. Resource-constrained multiple product system
Resource-constrained multiple product system Suppose that n items have unit costs of cc2....c respectively,and the total budget available for them is C. Then the budget constraint can be written C101+c202+...,cnOn<=C Let EOC,= 2K, h In general,budget-constrained problems can be solved as follow:
Suppose that n items have unit costs of c1, c 2,…, c n, respectively, and the total budget available for them is C. Then the budget constraint can be written c1Q1 + c 2 Q2 +…, c n Q n <=C In general, budget-constrained problems can be solved as follow: 2 Let i i i i K EOQ h Resource-constrained multiple product system
Resource-constrained multiple product system Case 1:IfC.then the optimal solution isE Case 2:If cEoQ>C,then if c/h=c2/==c/h i=l we can prove that the optimal solution is O=m*EO: C where m= 2cio0 Note that c/h=c(Ic)=11,,the condition is equivalent to the requirement that the same interest rate be used,which is reasonable in most circumstances
1 Case 1: If , then the optimal solution is ; n ii i i i c EOQ C Q EOQ Note that ( ) 1 , the condition is equivalent to the requirement that the same interest rate be used, which is reasonable in most circumstances. i i i ii i c h c Ic I Resource-constrained multiple product system 1 12 2 1 * 1 Case 2: If , then if , we can prove that the optimal solution is * ; where . n i i nn i i i n i i i c EOQ C c h c h c h Q m EOQ C m c EOQ
Contents .Introduction .Types of Inventories .Motivation for Holding Inventories; .Characteristics of Inventory System; Relevant Costs; .The EOQ Model: EOQ Model with Finite Production Rate Quantity Discount Models .Resource-constrained multiple product system .EOQ models for production planning Power-of-two policies
Contents •Introduction •Types of Inventories •Motivation for Holding Inventories; •Characteristics of Inventory System; •Relevant Costs; •The EOQ Model; •EOQ Model with Finite Production Rate •Quantity Discount Models •Resource-constrained multiple product system •EOQ models for production planning •Power-of-two policies
EOQ models for production planning We consider an extension of the EOQ model with a finite production rate,discussed in sec.4.6,to the problem of production n products on a single machine 国 The goal is to determine the optimal procedure for producing n products on the machine to minimize the cost of holding and setup,and to guarantee that no stock-outs occur during the production cycle. Assumption∑2，/P≤1 is required,.which is stronger than i1 A<P for each i.Why??
EOQ models for production planning We consider an extension of the EOQ model with a finite production rate, discussed in sec. 4.6, to the problem of production n products on a single machine The goal is to determine the optimal procedure for producing n products on the machine to minimize the cost of holding and setup, and to guarantee that no stock-outs occur during the production cycle. 1 Assumption 1 is required, which is stronger than for each . Why?? n i i i i i P P i