Question

The Mom & Pop Grocery store owners rent space in a local warehouse to which suppliers deliver their goods, and the owners arrange for a truck to pick up the goods for sale at their store. It costs them C $/day/cu.ft. for the storage space, and P$ for every truck pickup. Over the next N days they expect the flow into the warehouse on the morning of day t to be D_t cu.ft. The truck picks up the goods at the end of the day. To save in pickup costs they usually don’t pick up the goods every day, but they never let the shipments remain in the warehouse more than a week. How can Mom & Pop use dynamic programming to determine an optimal schedule for picking up the goods?

Formulate Problem i.e. stages, decision, state and Bellman Equation. Also provide expression for final stage costs.  

data:

Q12	Truck	Capacity (units)	Operating Costs ($)			Type 1	Type 2
(Incorrect - needs revision)	1	15	60		Number of shipments	12	8
	2	18	80		Size (units)	3	5
	3	20	90
	4	30	200
	5	10	40
	6	12	50
	7	23	130
	8	26	170

Answer 1

Brief facts of the case:-

Mom & Pop Grocery Store

Rental Local Warehouse and a store : Supplier deliver goods at the warehouse and the store arrange truck facility to pick up the goods from their warehouse to the store.

Cost Incurred:-

1. Storage Space:- C $/day/cu.ft

2. Truck Pickup:- P$ /truck

Expected flow in warehouse (day t morning):- D_t cu.ft

Operating cost is the cost/expenses incurred in managing/maintaining the day-to-day business activities.

Dynamic Programming:- Optimal Schedule for picking up the goods

Objective:- Maximize profit and Minimize cost

Type 1:-

Number of Shipments:- 12

Units:- 3

Type 2:-

Number of Shipments:- 8

Units:- 5

Truck 1:- Capacity:-15 Units, Operating Cost:- $60 . Per unit Operating cost= $4.00

Truck 2:- Capacity:- 18 Units, Operating Cost:- $80. Per Unit Operating cost= $4.44approx

Truck 3:- Capacity:- 20 Units, Operating Cost:- $90. Per Unit Operating Cost= $4.50

Truck 4:- Capacity:- 30 Units, Operating Cost:- $200. Per Unit Operating Cost= $6.67approx

Truck 5:- Capacity:- 10 Units, Operating Cost:- $40. Per Unit Operating Cost= $4.00

Truck 6:- Capacity:- 12 Units, Operating Cost:- $50. Per Unit Operating Cost= $4.17approx

Truck 7:- Capacity:- 23 Units, Operating Cost:- $130. Per Unit Operating Cost= $5.65approx

Truck 8:- Capacity:- 26 Units, Operating Cost:- $170. Per Unit Operating Cost= $6.54approx

Truck 1 and Truck 5 are optimal.

For Truck 1

Type 1:- Number of shipments:-12, Units:- 3

Truck 1:- Operating Cost=$60, Total cost in Type 1= $60*12= $720

Type 2:- Number of Shipments:-8, Units:-5

Truck 1:- Operating Cost=$60, Total Cost in Type 1= $60*8= $480

Total Cost= $720+ $480= $1200

For Truck 5

Type 1:- Number of shipments:-12, Units:- 3

Truck 5:- Operating Cost=$40, Total Cost in Type 1:- $40*12= $480

Type 2:- Number of Shipments:-8, Units:-5

Truck 5:- Operating Cost:- $40, Total Cost in Type 2:- $40*8= $320

Total Cost= $480+ $320= $800

Truck 5 is Optimal for Mom & Pop Grocery Store.