Chapter 4, Problem 45AP

(a)

Summary Introduction

To determine: The optimal rotational cycle time.

Introduction:Economic order quantity refers to the approach which is used to evaluate flow and volume of order needed to fulfill consumers demand while reducing the cost per order.Such tool is used by the organization to handle andmanage operations and logistic operations.

Explanation of Solution

Optimal cycle time of the fiber store is as follows.

Apply variables of each type of items and determine optimal cycle time by the following formula.

  $\begin{array}{l}\text{T*= }\sqrt{\frac{\text{2×}{\displaystyle {\sum }_{\text{j=1}}^{\text{n}}{\text{K}}_{\text{j}}}}{{\displaystyle {\sum }_{\text{j=1}}^{\text{n}}{\text{H'}}_{\text{j}}{\text{×λ}}_{\text{j}}}}}\\ \text{Where,}\\ {\text{K}}_j=\text{ Ordering Cost }\\ {\text{H'}}_j=\text{ Modified holding cost }\\ {\lambda }_j=\text{ Total demand per year }\\ \text{Determine the variables for each 8 items given in the table by the following method}\text{.}\\ \text{Modified holding rate, H'=H(1-}\frac{\lambda }{\text{P}}\text{)}\\ \text{Where, }\\ \text{H= Holding cost }\\ \lambda \text{= Total demand per year }\\ \text{P= Production units per years }\\ \text{On putting the above values in the formula }\\ \text{H'= (}\text{.03+}\text{.18)}\times \text{.03(1-}\frac{25,900}{4,500\times 250}\text{)}\\ \text{= \$0}\text{.000615}\\ \end{array}$

Similarly, the modified holding cost for each of the item should be calculated by the following method.

Item	Annual Sales	$\text{Productivity per day }\times \text{250}$	Cost	$\begin{array}{l}\text{Holding Cost }\\ \text{H=0}\text{.18+0}\text{.03}\times \text{cost}\end{array}$	$\begin{array}{l}\text{Modified Holding Cost }\\ \text{H'=H(1-}\frac{\text{D}}{\text{Production rate}}\text{)}\end{array}$	$\begin{array}{l}\text{Ordering cost }\\ \text{K=20}\times \text{setup time}\end{array}$	$\text{H}\times \text{D}$
A	25,900	1125000	.003	.00063	.000615	120	15.9285
B	42,000	1375000	.002	.00042	.000407	80	17.094
C	14,400	825000	.008	.00168	.00165	160	23.76
D	46,000	800000	.002	.00042	.000395	80	18.17
E	12,500	450000	.001	.0021	.002041	60	25.5125
F	75,000	975000	.005	.00105	.000969	120	72.675
G	30,000	725000	.004	.00084	.000805	20	24.15
H	18,900	300000	.007	.00147	.001377	60	26.0253
SUM					0.008259	700	223.315

Put the value of each item to determine the optimal cycle time.

  $\begin{array}{l}\text{T*= }\sqrt{\frac{2\times 700}{223.315}}\\ =2.50\end{array}$

Hence, the optimal rotation cycle time is 2.50 per year

(b)

Summary Introduction

To determine:TheSize of the lots.

Introduction: Economic order quantity sometimes EOQ refers to the technique used by the organizations to determine the volume and frequency or order needed to fulfill the customer demand while minimizing the cost of the item.

Explanation of Solution

Determine the optimal quantity of each item by the following formula.

  $\begin{array}{l}{\text{Q}}_j={\lambda }_j\times \text{ T*}\\ \text{where, }\\ {\lambda }_j=\text{ total demand per year }\\ \text{T*= optimal cycle time }\\ {\text{Q}}_j=\text{ optimal quantity of each item }\\ \text{Determine the value for the first itme A by the following method}\text{.}\\ {\text{Q}}_A=\text{ 25,900}\times \text{2}\text{.50}\\ \text{= 64750}\end{array}$

Same procedure will be followed to calculate optimal quantity for each item

Item	Demand	Time	Quantity (Q)
A	25,900	2.5	64750
B	42,000	2.5	105000
C	14,400	2.5	36000
D	46,000	2.5	115000
E	12,500	2.5	31250
F	75,000	2.5	187500
G	30,000	2.5	75000
H	18,900	2.5	47250

(C)

Summary Introduction

To determine:Average annual cost of holding and setup cost at the optimal solution.

Introduction: Economic order quantity sometimes EOQ refers to the technique used by the organizations to determine the volume and frequency or order needed to fulfill the customer demand while minimizing the cost of the item.

Explanation of Solution

Procedure to find average annual cost of holding and setup cost is discussed below.

Determine the annual holding and setup cost by the following formula.

  $\begin{array}{l}{\text{G (Q}}_j\text{)=}\frac{{\text{K}}_j\times {\lambda }_j}{{\text{Q}}_j}\times \frac{\text{H'}\times {\text{Q}}_j}{2}\\ \end{array}$

Determine the values to calculate annual holding cost for the item A by the following method

  $\begin{array}{l}{\text{G (Q}}_j\text{)= }\frac{120\times 25,900}{64750}+\frac{.000615\times 64,750}{2}\\ =\text{ 48+19}\text{.91}\\ \text{=67}\text{.91}\\ \end{array}$

Same procedure will be followed to calculate optimal quantity for each item

Item	Setup cost (K*D/Q)	Holding Cost (H*Q/2)	Total
A	48	19.91	67.91
B	32	21.37	53.37
C	64	29.70	93.70
D	32	22.71	54.71
E	24	31.89	55.89
F	48	90.84	138.84
G	8	30.19	38.19
H	24	32.53	56.53
SUM			559.14

Hence, annual cost of holding and setup is $559.14

(d)

Summary Introduction

To determine: Contractual obligation that G might have with the fabrics that would stop them from implementing the policy mention in part (a) and (b).

Introduction:Economic order quantity sometimes EOQ refers to the technique used by the organizations to determine the volume and frequency or order needed to fulfill the customer demand while minimizing the cost of the item.

Explanation of Solution

With the responsibility or obligation, the time required to supply 3 shipments will be 4 times in a year instead of 2.5. Hence, the size of the optimal quantity will vary as the cycle time being changed to 4 times in a year.

Hence, to make shipment thrice in a year it is very much important for the firm to change its cycle time and optimal quantity of every item to remain focused in their commitment.