The EZ-Zees Bed Company make three sizes of the \"Sweet Dreams\" brand of bed th

Question

The EZ-Zees Bed Company make three sizes of the "Sweet Dreams" brand of bed that they sell (annual demand of 500, 700, and 1100 units for King, Queen, and Double sizes, respecively). They produce this brand on a special machine that requires a number of setup activities every time they want to switch from a different brand to the Sweet Dreams brand, and they also require setup activities to switch from one size to another within the brand. The cost to setup to produce the brand is estimated to be $124; once this setup has been performed, the cost to set up for producing King beds is $52, for Queen it is $48, and for Double it is $35. Inventory holding costs are estimated to be 25% of item value. The company's book value cost for the beds is $325, $275, $200 for King, Queen, and Double beds, respectively.

Using a joint lot-sizing model, the optimal number of production cycles for the "Sweet Dreams" brand will be ______ times per year (to two decimals, think about why we don't need to round to an integer). Each time they produce ______ the Sweet Dreams brand, they should produce  Queen Beds (provide an integer for this value - think about why it needs to be rounded).

Explanation / Answer

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There are 2 models in an inventory control - Buy and Make;

Our question deals about the Make model;

Economic Batch Quantity = EBQ;

Initial Set up Cost = O1 = $124;

Carrying cost = Co = Inventory holding cost = 25% of item value = 0.025 * unit price of each bed = the fixed set up cost for the group of items = the fixed set up cost for all the 3 bed sizes from king, queen to double;

sqrt = square root;

nr = Numerator; dr = denominator;

Co = fixed set up cost in $ for group of items = all 3 bed sizes

Coi = marginal set up cost for item i

Di = demand in units per annum for item i

Dr = Annual demand in $ for group of 3 items

dri = annual demand for item i in the group

Ki = production in units per annum of item i

K1 = Number of units of King size beds made

K2 = Number of units of Queen size beds made

K3 = Number of units of double size beds made

pi = purchase price per unit of item i in the group of items made

I = Inventory carrying cost in % of the unit cost = 25% = 0.025

Qr = Economic batch quantity in Dollars for the group of 3 bed items

qri = Economic batch quantity in Dollars for the item 1 in the group of 3 bed items made

qr1 = Economic batch quantity in Dollars for the item 1 king size bed in the group of 3 bed items made

qr2 = Economic batch quantity in Dollars for the item 2 queen size bed in the group of 3 bed items made

qr3 = Economic batch quantity in Dollars for the item 3 double size bed in the group of 3 bed items made

Qi = EBQ in units of the item 1

Q1 = EBQ in units of the king size bed

Q2 = EBQ in units of the queen size bed

Q3 = EBQ in units of the double size bed

N = number of batch intervals per year

Size

Demand per annum = Di

Custom set up cost = Coi in $

Cost on book

Dr = Di * book cost

King

D1 = 500

Co1 = 52

p1 = 325

500*325 = 162500

Queen

D2 = 700

Co2 = 48

p2 = 275

700*275 = 192500

Double

D3 = 1100

Co3 = 35

200

1100*200 = 220000

Dr = 162500 + 192500 + 220000 = $575,000

m = 3 = for 3 bed sizes

EBQ = Qr = sqrt(nmr/dmr)

where

nmr = 2 * (Co + Sigma 1 to m of Coi) * Dr

dmr = I * (1 - (Sigma 1 to m of Di/Sigma 1 to m of Ki)

nmr = 2 * (124 + 52+48+35 ) * 575000

=297850000

Di = D1+D2+D3 = 500+700+1100 = 2300

Since the annual production is not given, we will assume Ki = twice that of demand

hence Ki = 2*2300 = 4600

dmr = 0.25 * (1-2300/4600) = 0.125

Nmr = 297850000

Qr = sqrt(297850000/0.125)

=48813.9324374

Qr = 48813.93

N = Dr / Qr = 575000/Qr

= 575000/48813.9324374

N = 11.779424 batch intervals per year or 11.78 set ups per year

D$ = Demand per annum in Dollars for the group of items made;

D$1 = Demand per annum in Dollars for the King size beds in group of items made;

D$2 = Demand per annum in Dollars for the Queen size beds in group of items made;

D$3 = Demand per annum in Dollars for the Double size beds in group of items made;

Inventory model = Joint lot sizing = similar to = Manufacturing Model for multi Items Joint Replenishment without shortages;

Shortages are not allowed so as not to lose the customer goodwill;

We can apply the Silver’s Algorithm;

Optimal production cycle = 11.78 times per year;

N = 11.779424 batch intervals per year or 11.78 set ups per year

accuracy 2 decimal places;

Reason for not rounding is that we are finding the number of set ups of intervals between the batches or batch production;

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When they produce q number of Queen Beds they must also produce k number of king size beds and d number of double size beds; Accuracy rounding to the nearest integer as a whole number;

Reason for rounding: As it deals with the number of beds, it needs to be rounded;

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Size

Demand per annum = Di

Custom set up cost = Coi in $

Cost on book

Dr = Di * book cost

King

D1 = 500

Co1 = 52

p1 = 325

500*325 = 162500

Queen

D2 = 700

Co2 = 48

p2 = 275

700*275 = 192500

Double

D3 = 1100

Co3 = 35

200

1100*200 = 220000

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