United Express Service (UES) uses large quantities of packaging materials at its four distribution hubs. After screening potential suppliers, UES identified six vendors that can provide packaging materials that will satisfy its quality standards. UES asked each of the six vendors to submit bids to satisfy annual demand at each of its four distribution hubs over the next year. The following table lists the bids received (in thousands of dollars). UES wants to ensure that each of the distribution hubs is serviced by a different vendor. Which bids should UES accept, and which vendors should UES select to supply each distribution hub?
| Distribution Hub | ||||
| Bidder | 1 | 2 | 3 | 4 |
| Martin Products | 185 | 180 | 130 | 235 |
| Schmidt Materials | 150 | 240 | 145 | 215 |
| Miller Containers | 220 | 225 | 140 | 240 |
| D&J Burns | 165 | 185 | 185 | 275 |
| Larbes Furnishings | 210 | 180 | 145 | 240 |
| Lawler Depot | 235 | 195 | 140 | 255 |
| Bidder | Decision | Bid |
|---|---|---|
| Martin Products | ||
| Schmidt Materials | ||
| Miller Containers | ||
| D&J Burns | ||
| Larbes Furnishings | ||
| Lawler Depot |
The value of the objective function is _______ thousands of dollars.
This problem will be solved using binary decision variables.
The variables are defined in the table below. It means that if a11 = 1 then Martin Products will service 1st distribution hub. If it is 0 then it will not service the hub.
| 1 | 2 | 3 | 4 | |
| Bidder | ||||
| Martin Products | a11 | a12 | a13 | a14 |
| Schimdt Materials | b11 | b12 | b13 | b14 |
| Miller Containers | c11 | c12 | c13 | c14 |
| D&J Burns | d11 | d12 | d13 | d14 |
| Larbes Furnishings | e11 | e12 | e13 | e14 |
| Lawler Depot | f11 | f12 | f13 | f14 |
Objective function: 185a11+180a12+130a13+235a14+150b11+240b12+145b13+215b14+220c11+225c12+140c13+240c14+165d11+185d12+185d13+275d14+210e11+180e12+145e13+240e14+235f11+195f12+140f13+255f14
This has to be minimized.
Constraints:
1. a11+a12+a13+a14<=1
2. b11+b12+b13+b14<=1
3. c11+c12+c13+c14<=1
4. d11+d12+d13+d14<=1
5. e11+e12+e13+e14<=1
6. f11+f12+f13+f14<=1
7. a11+b11+c11+d11+e11+f11 = 1
8. a12+b12+c12+d12+e12+f12 = 1
9. a13+b13+c13+d13+e13+f13 = 1
10. a14+b14+c14+d14+e14+f14 = 1
11. all variables are binary.
The excel's solver function gives the following solution:
| Variable values (as computed by excel's solver) | ||||
| 1 | 2 | 3 | 4 | |
| Bidder | ||||
| Martin Products | 0 | 0 | 1 | 0 |
| Schimdt Materials | 0 | 0 | 0 | 1 |
| Miller Containers | 0 | 0 | 0 | 0 |
| D&J Burns | 1 | 0 | 0 | 0 |
| Larbes Furnishings | 0 | 1 | 0 | 0 |
| Lawler Depot | 0 | 0 | 0 | 0 |
| 1 | 2 | 3 | 4 | |
| Bidder | ||||
| Martin Products | 185.00 | 180.00 | 130.00 | 235.00 |
| Schimdt Materials | 150.00 | 240.00 | 145.00 | 215.00 |
| Miller Containers | 220.00 | 225.00 | 140.00 | 240.00 |
| D&J Burns | 165.00 | 185.00 | 185.00 | 275.00 |
| Larbes Furnishings | 210.00 | 180.00 | 145.00 | 240.00 |
| Lawler Depot | 235.00 | 195.00 | 140.00 | 255.00 |
| Formula | ||||
| Total cost | 690.00 | 185a11+180a12+130a13+235a14+150b11+240b12+145b13+215b14+220c11+225c12+140c13+240c14+165d11+185d12+185d13+275d14+210e11+180e12+145e13+240e14+235f11+195f12+140f13+255f14 | ||
| Constraints | ||||
| 1 | <= | 1 | a11+a12+a13+a14<=1 | |
| 1 | <= | 1 | b11+b12+b13+b14<=1 | |
| 0 | <= | 1 | c11+c12+c13+c14<=1 | |
| 1 | <= | 1 | d11+d12+d13+d14<=1 | |
| 1 | <= | 1 | e11+e12+e13+e14<=1 | |
| 0 | <= | 1 | f11+f12+f13+f14<=1 | |
| 1 | = | 1 | a11+b11+c11+d11+e11+f11 = 1 | |
| 1 | = | 1 | a12+b12+c12+d12+e12+f12 = 1 | |
| 1 | = | 1 | a13+b13+c13+d13+e13+f13 = 1 | |
| 1 | = | 1 | a14+b14+c14+d14+e14+f14 = 1 | |
Required solution table:
| Bidder | Decision | Bid |
| Martin Products | Select | 130 |
| Schmidt Materials | Select | 215 |
| Miller Containers | Reject | Not applicable as bidder is not selected |
| D&J Burns | Select | 165 |
| Larbes Furnishings | Select | 180 |
| Lawler Depot | Reject | Not applicable as bidder is not selected |
The value of the objective function is 690 thousands of dollars.
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