Bob Brothers, a Maine-based, publicly-held supermarket chain, operates in four s
ID: 458486 • Letter: B
Question
Bob Brothers, a Maine-based, publicly-held supermarket chain, operates in four states in the U.S. Bob’s five retail stores are supplied by three company-owned-and-operated distribution centers. The three full-line grocery distribution centers are located in White Plains, NY; Portsmouth, NH; and Augusta, ME.
The five retail stores are located in Albany, NY; Burlington, VT; Manchester, NH; Portland, ME; and Bangor, ME. The daily demand for groceries in each of these market areas is provided in Table 1. Since the retail stores in these market areas have no space to stockpile extra inventory (and stock-outs are not tolerated), demand must be satisfied each day.
Table 1: Daily Product Demand by Retail Market Area
Retail Market Area
Demand for Grocery Products (truck pallets)
Albany (Al)
260
Burlington (Bu)
200
Manchester (Ma)
300
Portland (Pd)
260
Bangor (Ba)
200
Although each distribution center carries about 14 days’ worth of inventory, their capacity to satisfy store demand is limited by various constraints related to labor and truck availability. So, in practice, the daily shipping capacity of each distribution center does not exceed the figures provided in Table 2.
The unit shipping costs vary depending mostly on the distance between the originating distribution center and the market area destination. On the average, the shipping cost of a pallet is about $0.07/mile. The unit shipping costs from all three distribution centers to all five retail market areas are given in Table 3.
Please use your understanding of linear programming to address the questions which follow.
Please be sure to use sensitivity analysis where possible rather than making additional solver runs.
1.) Formulate the linear programming model to determine how many pallets of groceries Bob should ship daily to satisfy each market area’s demand at the minimum total shipping cost. Use Xij to represent the number of pallets shipped daily from distribution center i to retail market area j. (For example, XWhAl represents the number of pallets shipped daily from the White Plains distribution center to the Albany retail market area.)
a. Explicitly write out the total daily shipping cost objective function?
b. What are the eight structural constraints? Please present them in the order that they are introduced in Table 1 and Table 2.
2.) Run the solver to determine the optimal distribution scheme assuming fractional-pallet answers are acceptable. What is the optimal number of pallets shipped daily from each distribution center to each retail market area? What is the optimal total number of pallets shipped daily from each distribution center? What is the associated minimum total daily shipping cost? Assuming that stores are replenished 365 days/year, what is the total annual shipping cost?
3.) In the optimal distribution scheme determined in question 2, does Bob use all of his daily shipping capacity? Please explain your answer.
4.) Is there another optimal solution? That is, is there a different distribution scheme that results in the same minimum total daily shipping cost? Please explain how you arrived at your answer.
5.) For this questions only, suppose that Bob were considering building a new store in one of the major market areas that would increase the daily demand in that market area by 30 pallets of groceries. In which market area should Bob build the store to minimize the impact on the total daily shipping cost? How would this change the total daily shipping cost from your answer to question 2? Please explain. How would this change the optimal distribution scheme from your answer to question 2? Please explain.
6.) For this question only, suppose that Bob wanted to expand the shipping capacity of one of the distribution centers by 10 pallets/day. Which distribution center should be expanded to minimize the total daily shipping cost? Why? What will the new total daily shipping cost be?
7.) For this question only, suppose that a new east/west highway opened up between Portsmouth and Manchester that decreased the trucking distance (and, thus, the shipping cost) by 20%. How would this impact the optimal distribution scheme found in question 2? Please explain. What would be the impact on the total daily shipping cost?
8.) For this question only, suppose that Bob expanded the shipping capacity of the Portsmouth distribution center by 100 pallets. How would this change the total daily shipping cost from your answer to question 2? Please explain. How does this impact the optimal distribution scheme from question 2? Please interpret your answers.
9.) For this question only, suppose that Bob imposed additional restrictions on the number of White Plains trucks supplying the Manchester market area to take advantage of some lucrative backhaul opportunities (from the Manchester area back to the White Plains area). There have to be at least 6 trucks assigned each day to the Manchester-White Plains route to allow for backhauling. Please reformulate this problem in terms of trucks (not pallets) to accommodate the notion that shipping requires trucks (24-pallet capacity), and incorporate an additional constraint on the minimum number of trucks traveling between White Plains and Manchester. Assuming that stores are replenished 365 days/year, how much does this affect the minimum total annual shipping cost found in question 2? How many trucks are needed in total? In total, how many pallet slots on trucks go unused?
10.) For this question only, assume that Bob wants to eliminate one of the distribution centers. To pick up the slack, he will also increase the total shipping capacity of each distribution center by 300 pallets. Please adjust your formulation from question 1 by introducing three binary variables (YWh, YPh, YAu), adding four new constraints, and increasing all three distribution center capacities by 300 pallets. Run the solver again to solve this reformulated problem after making these adjustments. Which one of the three grocery distribution centers should be eliminated? What are the revised optimal distribution scheme and associated minimum total daily shipping cost?
Retail Market Area
Demand for Grocery Products (truck pallets)
Albany (Al)
260
Burlington (Bu)
200
Manchester (Ma)
300
Portland (Pd)
260
Bangor (Ba)
200
Explanation / Answer
The given question is a transportation LPP. The objective of the problem is to minimise total transporation by satisfying the demand and supply constrains. This is an Unbalanced transporation problem where total supply is (1300) more than total demand (1220).
The transporation table, objective function and constraints are given below.
The solver solution to the above transporation problem is given below.
Albany Burlington Manchester Portland Bangor Supply White Plains 0.7 10.5 10.5 17.5 26.6 610 Portsmouth 21 14 7 0.4 9.1 510 Augusta 22.3 14 10 4.5 6 180 Demand 260 200 300 260 200 Objective Function Min Z= 0.7X11+10.5X12+10.5X13+17.5X14+26.6X15+ 21X21+14X22+7X23+0.424+9.1X25+ 22.3X31+14X32+10X33+4.5X34+6X35 Subject to Constrains 0.7X11+10.5X12+10.5X13+17.5X14+26.6X15<=610 (White Plains Supply) 21X21+14X22+7X23+0.4X24+9.1X25<=510 (Portsmouth Supply) 22.3X31+14X32+10X33+4.5X34+6X35<=180 (Augusta Supply) 0.7X11+21X21+22.3X31=260 Albany 10.5X12+14X22+14X32=200 Burlington 10.5X13+7X23+10X33=300 Manchester 17.5X14+0.4X24+4.5X34=260 Portland 26.6X15+9.1X25+6X35=200 Bangor X11,X12,X13,X14,X15,X21,X22,X23,X24,X25,X31,X32,X33,X34,X35 >=0