42 manufacturing model with no shortages in this

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4.4.2 Manufacturing model with no shortages In this model the following assumptions are made: (1) Demand is at a constant rate ( D ). (2) All cost coefficients ( C 1 , C 2 , C 3 ) are constants. (3) There is no shortage cost, or C 4 = 0. (4) The replacement rate is finite and greater than the demand rate. This is also called replenishment rate or manufacturing rate, denoted by R .
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6 Schematically, this model is illustrated in fig 1 Slope= (R-D) Slope = D Q I m t 1 t 2 Time t Fig. 1 The total cost of inventory per period is the sum of three components: item cost, order cost and items holding cost. Let I m be the maximum inventory, t 1 be the time of manufacture and t 2 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 = C 1 Q Order cost/period = C 2 Item holding cost/period = 1 2 3 ( ) 2 m C I t t × × + 3 2 m t C I = × 1 ( ) m I t R D = - But 1 t Q R = Therefore ( )( ) m I Q R R D = - Substituting the value of I m , we get the total cost of inventory per period.
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7 2 3 1 2 ( )( ) C C Q Q R t C C R D ′ = + - × Total cost of inventory per unit time C C t = 3 1 2 2 ( )( ) C Q t Q R t C C R D t = + + - × But t = Q/D Substituting the value of t we get 3 1 2 ( ) 2 ( ) C C D C D Q Q R R D C = + - + Differentiating C with respect to Q and setting equal to zero for minimum C , we get, 2 3 2 ( ) 0 0 2 dC C D C R D dQ R Q - = - + = Solving equation (43), we get 2 3 2 * C D R Q C R D = - This gives the economic order quantity and is a balance between holding and set up costs. 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.
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