Question

Demand for jelly doughnuts on Saturdays at Don’s Doughnut Shoppe is shown in the following table. Demand (dozens) Relative Frequency Demand (dozens) Relative Frequency 19 .01 25 .10 20 .05 26 .11 21 .12 27 .10 22 .18 28 .04 23 .13 29 .02 24 .14 a-1. Determine the optimal number of doughnuts, in dozens, to stock if labor, materials, and overhead are estimated to be $3.80 per dozen, doughnuts are sold for $5.15 per dozen, and leftover doughnuts at the end of each day are sold the next day at half price. (Round your answer to the nearest whole number.) Optimal number of doughnuts a-2. What is the resulting service level? (Round your answer to 2 decimal places.) Service level

Answer 1

The problem will be solved using newsvendor model .

Critical ratio under profit maximization in case of newsvendor model will be defined as :

Critical Ratio =    Cu / ( Cu + Co)

Where ,

Cu = Underage cost i.e. cost of ordering one fewer unit than what one would have ordered had one known the demand (i.e situation under lost sales) = Price/unit – Cost/unit

Co = Overage cost i.e. cost of ordering one more unit than what one would have ordered had one known the demand ( i.e. one overordered) = Cost/unit – Salvage value/unit

Thus,

Critical ratio

= Cu/ ( Cu + Co)

= (Price – Cost ) / ( Price/unit – Cost/unit + Cost/unit – Salvage value/unit )

= (Price/unit – cost/unit )/( Price/unit – Salvage value/unit )

As per relevant data :

Price / unit = $5.15

Cost/ unit = $3.8

Salvage value = $5.15 / 2 = $2.575

Critical ratio

= ( 5.15 – 3.8)/ ( 5.15 – 2.575)

=1.35/2.575

= 0.524

Critical ratio is equivalent to service level . or probability of the optimum stock to be kept

Therefore, Service level = 0.524

It is to be noted that probability that demand to be at least 29 will be 0 and demand to be at lease 19 to be 1

By summing up the frequencies ( from 29 and going backwards), it can also be noted that :

Demand to be at least 24 = 0.51

Demand to be at least 23 = 0.64

Since service level 0.524is nearest to 0.51, this option should be chosen and optimum number of doughnuts 24

Optimum number of doughnuts = 24

Service level = 0.524