Question

Eric Johnson started the Johnson grocery Company after years of not being able to get fresh baked goods in Grand Rapids. Johnson Grocery originally specialized in cakes, tarts, breads, and doughnuts sold from Eric’s home. His success with the affluent, gourmet foodies allowed him to expand to a second store within six months and a half-dozen outlets within the first year. He has expanded into bagels, salads, and gourmet foods such as ice cream and pet food.

            Once Eric got beyond a dozen stores, he moved to a central kitchen/supply location that restocked his stores. The distribution center is causing problems in the form of underutilized employees and long lines for truck loading.

            The loading dock at the distribution center will accommodate only one truck for loading or unloading at a time. Company owned trucks arrive according to a Poisson distribution with a rate of four trucks per day. Presently, the company employs a crew of three to load and unload the trucks, and the unloading/loading rate is Poisson distributed with a mean rate of five trucks per day. The company can employ additional or fewer persons in the loading crew and increase the loading rate by one truck per day for each additional employee up to a maximum of six persons who can be utilized effectively in the process, e.g. a crew of four could unload six trucks per day or a crew of two could unload four trucks per day. The company estimates that the cost of an idle truck and driver is $50 per hour and the company pays $15 per hour (fully fringed) for each employee in the loading crew.

            Eric addressed the problem with Linda Froeb, supervisor of the loading crew. Linda hates to see idle workers and she thinks they’re overstaffed. Barry Bruce supervisors the truck fleet and reported his drivers did not like to wait to be loaded or unloaded

Identify the optimal level of loading/unloading crew .

Is Eric’s business plan the right one to serve his customers?

Answer 1

This is M/M/1 queue model

Arrival rate, a = 4 trucks per day

With a crew (k) of 3 members,

Service rate, (unloading/loading rate), s = 5 trucks per day

Crew cost, Cs = $ 15 per hour

Idle truck and driver cost, Cw = $ 50 per hour  

Average number of trucks in the dock/system (L) = a/(s-a) = 4/(5-4) = 4

Total operating cost per hour (including idle truck and driver cost and employee/crew cost) = Cs*k + Cw*L (k is crew size = 3)

= 15*3 + 50*4

= $ 245

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With a crew (k) of 4 members,

Service rate, (unloading/loading rate), s = 6 trucks per day

Crew cost, Cs = $ 15 per hour

Idle truck and driver cost, Cw = $ 50 per hour  

Average number of trucks in the dock/system (L) = a/(s-a) = 4/(6-4) = 2

Total operating cost per hour (including idle truck and driver cost and employee/crew cost) = Cs*k + Cw*L (k is crew size = 4)

= 15*4 + 50*2

= $ 160

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With a crew (k) of 5 members,

Service rate, (unloading/loading rate), s = 7 trucks per day

Crew cost, Cs = $ 15 per hour

Idle truck and driver cost, Cw = $ 50 per hour  

Average number of trucks in the dock/system (L) = a/(s-a) = 4/(7-4) = 1.33

Total operating cost per hour (including idle truck and driver cost and employee/crew cost) = Cs*k + Cw*L (k is crew size = 4)

= 15*5 + 50*1.33

= $ 141.5

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With a crew (k) of 6 members,

Service rate, (unloading/loading rate), s = 8 trucks per day

Crew cost, Cs = $ 15 per hour

Idle truck and driver cost, Cw = $ 50 per hour  

Average number of trucks in the dock/system (L) = a/(s-a) = 4/(8-4) = 1

Total operating cost per hour (including idle truck and driver cost and employee/crew cost) = Cs*k + Cw*L (k is crew size = 4)

= 15*6 + 50*1

= $ 140

We see that total hourly operating cost is minimum with a crew size of 6. Therefore, optimal number of loading/unloading crew = 6

Eric's current business plan with 3 crew members is not the right one. He needs to hire 3 more crews.